Existence and Morse Index of least energy nodal solution of the $(p,2)$-laplacian

TL;DR

This paper investigates the existence and Morse index of least energy nodal solutions for a quasilinear $(p,2)$-Laplacian problem, providing new insights especially when the parameter epsilon equals zero.

Contribution

It offers a comprehensive review of the existence of least energy nodal solutions and introduces new Morse index results, notably for the case when epsilon is zero.

Findings

Existence of least energy nodal solutions for the quasilinear problem.

Morse index characterization of these solutions.

Specific Morse index results for the case epsilon=0.

Abstract

In this paper we study the quasilinear equation $- \ep^2 \Delta u-\Delta_p u=f(u)$ in a smooth bounded domain $\Omega$ with Dirichlet boundary condition. For $\ep \geq 0$, we review existence of a least energy nodal solution and then present information about the Morse Index of least nodal energy solutions this BVP. In particular we provide Morse Index information for the case $\ep =0$.

Equations

$$\{\begin{array}{rlrl}-{\epsilon }^2\Delta u-{\Delta }_{p}u=& f(u)& in& \Omega ,\\ u=& 0& on& \partial \Omega ,\end{array}$$

$$p^{*}:=\left\{\begin{array}[]{cc}\frac{Np}{N-p},&\quad\hbox{if}\quad p\in(2,N),\\ +\infty,&\quad\,\,\,\,\hbox{if}\quad p\in[N,\infty)\end{array}\right.$$

$$|f^{\prime }(t)|\le A(1+|t{|}^{q-2}).$$

$$f(t)t\ge mF(t)>0.$$

$$f^{\prime }(t)>(p-1)\frac{f(t)}{t}.$$

$${\int }_{\Omega }\left({\epsilon }^2+|\nabla u{|}^{p-2}\right)\nabla u\cdot \nabla \phi dx={\int }_{\Omega }f(u)\phi dx.$$

$$J_{\epsilon }(v):={\int }_{\Omega }\left(\frac{{\epsilon }^2}{2}|\nabla v{|}^2+\frac{1}{p}|\nabla v{|}^p-F(v)\right)dxforv\in W_0^{1,p}(\Omega ).$$

$$D{J}_{\epsilon }(v)\phi ={\int }_{\Omega }\left({\epsilon }^2+|\nabla v{|}^{p-2}\right)\nabla v\cdot \nabla \phi dx-{\int }_{\Omega }f(v)\phi dx$$

$$\begin{gathered} D^2{J}_{\epsilon }(v)(\phi ,\psi )={\int }_{\Omega }\left({\epsilon }^2+|\nabla v{|}^{p-2}\right)\nabla \phi \cdot \nabla \psi dx \\ +{\int }_{\Omega }(p-2)|\nabla v{|}^{p-4}(\nabla v\cdot \nabla \phi )(\nabla v\cdot \nabla \psi )dx-{\int }_{\Omega }{f}^{\prime }(v)\phi \psi dx \end{gathered}$$

$$D^2{J}_{\epsilon }(u)(\phi ,\phi )<0for\phi \in E.$$

$${\mathcal{N}}_{\epsilon }:=\{v\in W_0^{1,p}(\Omega ):v\ne 0andD{J}_{\epsilon }(v)v=0\}.$$

$${\mathcal{M}}_{\epsilon }:=\{v\in W_0^{1,p}(\Omega ):v^{+}\in {\mathcal{N}}_{\epsilon }and{v}^{-}\in {\mathcal{N}}_{\epsilon }\}.$$

$$u\in {\mathcal{M}}_{\epsilon }and{J}_{\epsilon }(u)=\underset{v\in {\mathcal{M}}_{\epsilon }}{\min }{J}_{\epsilon }(v).$$

$$J_{\epsilon }(u_{\epsilon })=\underset{\begin{array}{c}v\in {\mathcal{M}}_{\epsilon }\end{array}}{\min }{J}_{\epsilon }(v)=\underset{\begin{array}{c}v\in W_0^{1,p}(\Omega ),\\ v^{+},v^{-}\ne 0\end{array}}{\min }\left(\underset{t,s\ge 0}{\max }{J}_{\epsilon }(t{v}^{+}+s{v}^{-})\right).$$

$$\{u\ne 0\}=\{u>0\}\cup \{u<0\}.$$

$$f(t)t\le \mu |t{|}^p+C_{\mu }|t{|}^q.$$

$$F(t)\ge a|t{|}^m-bfort\in \mathbb{R}.$$

$${\gamma }_{\epsilon }(v):=$$

$$=$$

$$D{\gamma }_{\epsilon }(v)\phi ={\int }_{\Omega }\left(\left[2{\epsilon }^2+p|\nabla v{|}^{p-2}\right]\nabla v\cdot \nabla \phi -f^{\prime }(v)v\phi -f(v)\phi \right)dx$$

$${\mathcal{N}}_{\epsilon }:=\{v\in W_0^{1,p}(\Omega )-\{0\}:{\gamma }_{\epsilon }(v)=0\}$$

$${\int }_{\Omega }\left({\epsilon }^2|\nabla v{|}^2+|\nabla v{|}^p\right)dx={\int }_{\Omega }f(v)vdx.$$

$$J_{\epsilon }(v)\ge \left(\frac{1}{p}-\frac{1}{m}\right)\Vert v{\Vert }_{W_0^{1,p}(\Omega )}^p+C.$$

$$\Vert v{\Vert }_{W_0^{1,p}(\Omega )}\ge \rho .$$

$$\Vert v{\Vert }_{L^q(\Omega )}\ge {\rho }_0,$$

$$J_{\epsilon }(v)=$$

$$=$$

$$-\frac{1}{m}{\int }_{\Omega }\left(mF(v)-f(v)v\right)dx$$

$$\ge$$

$$J_{\epsilon }(v)\ge \left(\frac{1}{p}-\frac{1}{m}\right){\int }_{\Omega }|\nabla v{|}^{p}dx+C$$

$${\int }_{\Omega }|\nabla v{|}^{p}dx\le {\int }_{\Omega }\left({\epsilon }^2+|\nabla v{|}^{p-2}\right)|\nabla v{|}^{2}dx={\int }_{\Omega }f(v)vdx$$

$${\int }_{\Omega }|\nabla v{|}^{p}dx\le \mu {\int }_{\Omega }|v{|}^{p}dx+C_{\mu }{\int }_{\Omega }|v{|}^{q}dx.$$

Full text

1. Introduction

Consider the boundary value problem (BVP):

[TABLE]

where $\varepsilon\in\mathbb{R}$ is a parameter, $\Omega\subset\mathbb{R}^{N}$, $N\geq 1$, is a smooth bounded domain when $N\geq 2$ and an open bounded interval when $N=1$. We assume that $p\in(2,+\infty)$ and we denote by $\Delta u:={\rm div}(\nabla u)$ and $\Delta_{p}:={\rm div}(|\nabla u|^{p-2}\nabla u)$ the Laplace and the $p$-Laplace operators of a function $u$, respectively.

In what follows, we set

[TABLE]

and notice that $p\in(2,p^{*})$.

As for the nonlinearity $f:\mathbb{R}\to\mathbb{R}$, define $F(t):=\int_{0}^{t}f(s)ds$ for $t\in\mathbb{R}$ and consider the following set of hypotheses.

1. (f1)

$f\in C^{1}(\mathbb{R})$ and there exist $q\in(p,p^{*})$ and $A>0$ such that for every $t\in\mathbb{R}$,

[TABLE] 2. (f2)

There exist $m\in(p,p^{*})$ and $T>0$ such that for every $t\in\mathbb{R}$ with $|t|\geq T$,

[TABLE] 3. (f3)

$f^{\prime}_{p}(0):=\limsup\limits_{t\to 0}\frac{f(t)}{|t|^{p-2}t}\in(-\infty,\lambda_{1,p})$, where $\lambda_{1,p}>0$ is the first eigenvalue of $-\Delta_{p}$ with homegeneous Dirichlet boundary condition. 4. (f3)’

$f(0)=0$. 5. (f4)

The function $\mathbb{R}-\{0\}\ni t\mapsto\frac{f(t)}{|t|^{p-1}}$ is strictly increasing or equivalently, for every $t\in\mathbb{R}-\{0\}$,

[TABLE]

Notice that (f3) implies (f3)’.

For any function $v:\Omega\to\mathbb{R}$, define $v^{+}(x):=\max\{0,v(x)\}$ and $v^{-}(x):=\min\{0,v(x)\}$ for a.e. $x\in\Omega$ and any connected component of the set $\{v\neq 0\}$ will be called a nodal domain or nodal region of the function $v$.

The concepts of solutions we will be working with throughout this paper are next defined. Let $\varepsilon\in\mathbb{R}$ and assume that $f$ satisfies hypothesis (f1).

Definition 1**.**

A solution of ($\mathcal{P}_{\varepsilon}$) is a function $u\in W_{0}^{1,p}(\Omega)$ such that for every $\varphi\in W_{0}^{1,p}(\Omega)$,

[TABLE]

A nodal solution of ($\mathcal{P}_{\varepsilon}$) is a solution $u$ such that $u^{+},u^{-}\neq 0$ a.e. in $\Omega$.

The energy functional $J_{\varepsilon}:W^{1,p}_{0}(\Omega)\to\mathbb{R}$, associated with (1.3) is defined by

[TABLE]

Since $p>2$ and $f$ satisfies (f1), $J_{\varepsilon}\in C^{2}(W^{1,p}_{0}(\Omega))$ (see for instance [10] and [11]) with first derivative

[TABLE]

and second derivative

[TABLE]

for every $v,\varphi,\psi\in W^{1,p}_{0}(\Omega)$. Solutions to ($\mathcal{P}_{\varepsilon}$) correspond exactly to the critical points of $J_{\varepsilon}$ in $W^{1,p}_{0}(\Omega)$.

In this work we study existence, qualitative properties and Morse index computations for least energy nodal solutions of ($\mathcal{P}_{\varepsilon}$).

If $u\in W^{1,p}_{0}(\Omega)$ is a solution of ($\mathcal{P}_{\varepsilon}$), its Morse index, $m_{\varepsilon}(u)$, is defined as the maximal dimension of a linear subspace $E$ of $W^{1,p}_{0}(\Omega)$ such that

[TABLE]

Similar questions to the ones addressed in this work were studied in [8] in the autonomous case, when $\varepsilon=0$, $p=2$ and under similar hypotheses on the nonlinearity. The non-autonomous case, still for $\varepsilon=0$ and $p=2$, has been treated in [4] under slightly weaker assumptions.

Existence and qualitative properties of nodal solutions to ($\mathcal{P}_{\varepsilon}$) in the case $\varepsilon=0$ and $p>1$ have been studied in [3].

Recently, the authors in [2] studied the existence of a least energy nodal solution for a related BVP were a more general class of diffusions are considered and under similar assumptions as (f1)-(f4), but assuming in (f3) that $f^{\prime}_{p}(0)=0$ and that as $|t|\to\infty$, $f(t)$ has slower growth rate than $|t|^{q-2}t$. We remark that in [2] no information regarding the Morse index of nodal solutions is provided.

The main technique used in the aforementioned works is the Nehari manifold method. Our goal is to adapt this technique to our setting in order to study in more detail nodal solutions of ($\mathcal{P}_{\varepsilon}$) and to extend the results in [8, 4, 2].

In order to state our main results, we introduce some required terminology. For any $\varepsilon\in\mathbb{R}$, the Nehari Manifold associated to the energy $J_{\varepsilon}$ is the set

[TABLE]

Observe that $\mathcal{N}_{\varepsilon}$ contains all the non-zero solutions of ($\mathcal{P}_{\varepsilon}$). Since we are interested in nodal solutions, we consider also the set

[TABLE]

Observe also that $\mathcal{M}_{\varepsilon}\subset\mathcal{N}_{\varepsilon}$ and that $\mathcal{M}_{\varepsilon}$ contains all the nodal solutions of ($\mathcal{P}_{\varepsilon}$). In view of this remark, we say that a solution $u\in W^{1,p}_{0}(\Omega)$ of ($\mathcal{P}_{\varepsilon}$) is a least energy nodal solution if

[TABLE]

Our main results read as follows.

Theorem 1**.**

Let $f$ satisfy hypotheses (f1)-(f4). For any $\varepsilon\in\mathbb{R}$, there exists a least energy nodal solution $u_{\varepsilon}\in W^{1,p}_{0}(\Omega)$ of ($\mathcal{P}_{\varepsilon}$) and

[TABLE]

Although the proof of the existence of the solution $u_{\varepsilon}$, stated in Theorem 1, is essentially contained in Theorem 1.1 in [2], we include the detailed proof since many of the elements in it will be used in the computation of the Morse Index of the solution $u_{\varepsilon}$.

Also, up to our knowledge, the min-max characterization of the energy level $J_{\varepsilon}(u_{\varepsilon})$ in (1.8) is new and it is alternative to Nehari manifold approach.

We point out that Theorem 1 still holds for $\varepsilon\neq 0$ replacing (f3) by (f3)’ and the hypothesis that $f^{\prime}(0)<\varepsilon^{2}\lambda_{1}$, where $\lambda_{1}>0$ is the first eigenvalue of $-\Delta$ with homogeneous Dirichlet boundary condition. We impose (f3) instead of these weaker conditions in order to obtain some uniform estimates which will be used later on.

Our second result is concerned with qualitative properties and the Morse index computation for least energy nodal solutions in the case $\varepsilon\neq 0$.

Theorem 2**.**

Let $f$ satisfy (f1), (f2), (f3)’ and (f4). For any $\varepsilon\neq 0$ and any local minimizer, $u\in\mathcal{M}_{\varepsilon}\cap C^{1}(\overline{\Omega})$, of $J_{\varepsilon}|_{\mathcal{M}_{\varepsilon}}$,

• (i)

$m_{\varepsilon}(u)=2$* and*

• (ii)

$u$* has exactly two nodal domains, i.e. the sets $\{u>0\}$ and $\{u<0\}$ are connected and*

[TABLE]

In particular, any least energy nodal solution $u$ of ($\mathcal{P}_{\varepsilon}$) satisfies (i) and (ii).

For $\varepsilon=0$ and $p=2$ (the semilinear case), Theorem 2 was already hinted in [8] and proved in [4]. In fact our proof of Theorem 2 closely follows the scheme of [4].

Regarding the case $\varepsilon=0$, our next result extends Theorem 1.1 in [3] and Theorem 1.1 in [2] by providing an example of a nodal solution of $(\mathcal{P}_{0})$ having Morse index two.

Theorem 3**.**

Let $f$ satisfy hypotheses (f1)-(f4). Then the BVP $(\mathcal{P}_{0})$ has a least energy nodal solution $u_{0}\in W^{1,p}_{0}(\Omega)\cap C^{1}(\overline{\Omega})$ with $m_{0}(u_{0})=2$ and with two exactly two nodal domains.

If a least energy nodal solutions of $(\mathcal{P}_{0})$ has some nondegeneracy, then it can be approximated by local minimizers associated to ($\mathcal{P}_{\varepsilon}$). This is the content of our next result.

Theorem 4**.**

Let $f$ satisfy hypotheses (f1)-(f4) and let $u\in\mathcal{M}_{0}$ be a strict local minimizer of $J_{0}|_{\mathcal{M}_{0}}$. Then, there exists a decresing sequence $\{\varepsilon_{n}\}_{n\in\bm{\mathbb{N}}}\subset(0,\infty)$ with $\varepsilon_{n}\to 0$ and there exists a sequence of functions $\{u_{n}\}_{n\in\bm{\mathbb{N}}}\subset W^{1,p}_{0}(\Omega)$ such that

• (i)

for every $n\in\bm{\mathbb{N}}$, $u_{n}\in\mathcal{M}_{\varepsilon_{n}}$ and $u_{n}$ is a local minimizer of $J_{\varepsilon_{n}}|_{\mathcal{M}_{\varepsilon_{n}}}$,

• (ii)

$u_{n}\to u$* strongly in $W_{0}^{1,p}(\Omega)$ as $n\to\infty$ and*

• (iii)

$m_{\varepsilon_{n}}(u_{n})=2$* and $u_{n}$ has exactly two nodal domains.*

• (iv)

$m_{0}(u)=2$* and $u$ has exactly two nodal domains.*

We believe the property of the least energy nodal solution stated in Theorem 3 to be generic for such solutions. In this regards, Theorem 4 provides a partial reciprocal to Theorem 3. We remark also that Theorem 4 is true if the solution $u\in\mathcal{M}_{0}$ is a limit point of isolated local minizers of $J_{0}|_{\mathcal{M}_{0}}$ (see Remark at the end of the paper).

For $\varepsilon=1$ and $p>2$, S. Cingolani and G. Vanella in [11], obtained critical groups estimates at any solution of $(\mathcal{P}_{1})$, in the spirit of the generalized Morse lemma and assuming only hypothesis (f1).

Recently the authors in [7], obtained symmetry results and sign changing properties of least energy nodal solutions for the case $\varepsilon=0$ and $p>1$ under similiar assumption on the nonlinearity. In [7] the authors used the domain derivative method to approach a domain optimization problem.

The paper is organized as follows. In section 2 we prove several lemmas that will be useful throught the rest of the paper and we present the proof of the Theorem 1. Section 3 is devoted to some regularity results needed for the computation of the Morse index of the solutions for $\varepsilon\neq 0$. In section 4 we give the detailed proof of the Theorem 2 and in section 5 we handle the limiting case $\varepsilon=0$ and present the proofs of the rest of the Theorems.

2. Qualitative lemmas and existence

In this part, we present the existence of a least energy nodal solution for ($\mathcal{P}_{\varepsilon}$). Our arguments are motivated by those in [8], but we refer the reader to [2], [3] and [4] (and references therein), where existence of least energy nodal solutions was already treated in other settings.

We begin with a some remarks that will be crucial throughout this work and introduce also some further notation.

Assume hypotheses (f1), (f3). Since $2<p<q$, given any $\mu\in(f^{\prime}_{p}(0),\infty)$, there exists $C_{\mu}>0$ such that for any $t\in\mathbb{R}$,

[TABLE]

Assuming hypothesis (f2) and integrating (1.2), we find constants $a,b>0$ such that

[TABLE]

Let $\varepsilon\in\mathbb{R}$ be arbitrary, but fixed and consider the function $\gamma_{\varepsilon}:W^{1,p}_{0}(\Omega)\to\mathbb{R}$ defined by

[TABLE]

for $v\in W^{1,p}_{0}(\Omega)$. Observe that $\gamma_{\varepsilon}\in C^{1}(W^{1,p}_{0}(\Omega))$ with derivative

[TABLE]

for $v,\varphi\in W^{1,p}_{0}(\Omega)$. The set $\mathcal{N}_{\varepsilon}$, defined in (1.5), reads as

[TABLE]

and $v\in\mathcal{N}_{\varepsilon}$ if and only if $v\in W^{1,p}_{0}(\Omega)-\{0\}$ and

[TABLE]

Next lemma states that the energy $J_{\varepsilon}|_{\mathcal{N}_{\varepsilon}}$ is uniformly coercive and the sets $\mathcal{N}_{\varepsilon}$ are uniformly bounded away from zero.

Lemma 5**.**

Let $f$ satisfy (f1)-(f3). Then, the following properties are satisfied.

• (i)

There exist $C\in\mathbb{R}$ such that for any $\varepsilon\in\mathbb{R}$ and any $v\in\mathcal{N}_{\varepsilon}$,

[TABLE]

• (ii)

There exists $\rho>0$ such that for every $\varepsilon\in\mathbb{R}$ and every $v\in\mathcal{N}_{\varepsilon}$,

[TABLE]

Moreover, there exists $\rho_{0}>0$ such that for every $\varepsilon\geq 0$ and every $v\in\mathcal{N}_{\varepsilon}$,

[TABLE]

where $q$ is the exponent introduced in the condition (f1).

• (iii)

For every $\varepsilon\in\mathbb{R}$, $\mathcal{N}_{\varepsilon}$ is a closed subset of $W^{1,p}_{0}(\Omega)$.

Proof.

Let $\varepsilon\in\mathbb{R}$ and $v\in\mathcal{N}_{\varepsilon}$. Using (f2) and the fact that $m>p>2$,

[TABLE]

Since the function $[-T,T]\ni t\mapsto mF(t)-f(t)t$ is bounded, there exists $C\in\mathbb{R}$, depeding only on $f$ and $\Omega$, such that

[TABLE]

and this proves (i).

To prove (ii), we proceed as follows. Let $\mu\in(f^{\prime}_{p}(0),\lambda_{1,p})$ be fixed and choose $C_{\mu}>0$ so that (2.1) holds. Using that $v\in\mathcal{N}_{\varepsilon}$,

[TABLE]

and from (2.1) we find that

[TABLE]

From the Sobolev inequalities, there exists $\tilde{C}_{\mu}$ such that

[TABLE]

so that

[TABLE]

This proves (2.6) with $\rho=\left(1-\frac{\mu}{\lambda_{1,p}}\right)^{\frac{1}{q-p}}$.

To prove (2.7), we argue by cases. If $\|v\|_{L^{q}(\Omega)}\geq 1$, then (2.7) is proven with any $\rho_{0}\in(0,1]$. If $\|v\|_{L^{q}(\Omega)}\in(0,1)$, then $\|v\|_{L^{q}(\Omega)}^{q}<\|v\|_{L^{q}(\Omega)}^{p}$ and using (2.8) and Hölder inequality,

[TABLE]

where ${\rm meas}(\Omega)$ is the Lebesgue measure of $\Omega$.

This implies the existence of $\rho_{0}>0$ such that $\|v\|_{L^{q}(\Omega)}\geq\rho_{0}$ proving (2.7) and consequently concluding the proof of (ii).

Next, we prove (iii). Let $\{v_{n}\}_{n\in\mathbb{N}}\subset\mathcal{N}_{\varepsilon}$ such that $v_{n}\to v$ strongly in $W^{1,p}_{0}(\Omega)$. Since $\gamma_{\varepsilon}\in C^{1}(W^{1,p}_{0}(\Omega))$ and for any $n\in\mathbb{N}$, $\gamma_{\varepsilon}(v_{n})=0$, it follows that $\gamma_{\varepsilon}(v)=0$. Finally, (2.6) yields that $\|v_{n}\|_{W^{1,p}_{0}(\Omega)}\geq\rho$ for every $n\in\bm{\mathbb{N}}$. Therefore, $\|v\|_{W^{1,p}_{0}(\Omega)}\geq\rho>0$ and so $v\in\mathcal{N}_{\varepsilon}$. This completes the proof. ∎

$\blacksquare$

Lemma 6**.**

Let $f$ satisfy (f1)-(f4). For any $\varepsilon\in\mathbb{R}$ and for any $v\in W^{1,p}_{0}(\Omega)$ such that $v\neq 0$ there exists a unique $\tau_{\varepsilon,v}>0$ such that

• (i)

$\tau_{\varepsilon,v}v\in\mathcal{N}_{\varepsilon}$,

• (ii)

The function $[0,\infty)\ni t\mapsto J_{\varepsilon}(tv)$ is strictly increasing in $[0,\tau_{\varepsilon,v})$ and strictly decreasing in $(\tau_{\varepsilon,v},\infty)$ and

• (iii)

if $DJ_{\varepsilon}(v)(v)>0$ then $\tau_{\varepsilon,v}>1$ and if $DJ_{\varepsilon}(v)(v)<0$ then $\tau_{\varepsilon,v}<1$.

Proof.

Consider the function $g_{v}:[0,\infty)\to\mathbb{R}$ defined by

[TABLE]

Since $J\in C^{2}(W^{1,p}_{0}(\Omega))$, it follows that $g_{v}\in C^{2}([0,\infty))$ with

[TABLE]

and

[TABLE]

for any $t\geq 0$.

Let $t^{*}>0$ be such that $\frac{dg_{v}}{dt}(t^{*})=0$. From (2.9) and (2.10),

[TABLE]

Consequently, there exists at most one critical point of $g_{v}$ in $(0,\infty)$, and if such point exists, it must necessarily be the unique global strict maximum of $g_{v}$ in $(0,\infty)$. Next, we show the existence of such a critical point. Using (2.2),

[TABLE]

for any $t\geq 0$.

Since $m>p$, taking $t\to\infty$ yields

[TABLE]

so that there exists $\tau_{\varepsilon,v}\in[0,\infty)$ such that

[TABLE]

Now we show that $\tau_{\varepsilon,v}>0$. Observe that $g_{v}(0)=0$, so it suffices to prove that $\frac{dg_{v}}{dt}(t)>0$ for $t\in(0,\delta)$ with $\delta$ small enough. Indeed from (2.1) and (2.9) we have for $t>0$

[TABLE]

From this it is clear that there exists the required $\delta>0$. Since $v\neq 0$, $g_{v}(\tau_{\varepsilon,v})>0$ and so $\tau_{\varepsilon,v}>0$. Therefore, $\frac{dg_{v}}{dt}(\tau_{\varepsilon,v})=0$, i.e. $DJ_{\varepsilon}(\tau_{\varepsilon,v}v)(\tau_{\varepsilon,v}v)=0$ and hence $\tau_{\varepsilon,v}v\in\mathcal{N}_{\varepsilon}$ as claimed.

Finally, from the previous discussion, we conclude that $g_{v}$ is strictly increasing in $(0,\tau_{\varepsilon,v})$ and strictly decreasing in $(\tau_{\varepsilon,v},\infty)$. On the other hand, $\frac{dg_{v}}{dt}(1)=DJ_{\varepsilon}(v)v$ and $\tau_{\varepsilon,v}>0$ is the only critical point of $g_{v}$. Thus, if $\frac{dg_{v}}{dt}(1)>0$, then $0<1<\tau_{\varepsilon,v})$. If $\frac{dg_{v}}{dt}(1)<0$, then $1<\tau_{\varepsilon,v}<\infty$ and this concludes the proof. $\blacksquare$ ∎

Lemma 7**.**

Let $f$ satisfy hypotheses (f1)-(f4) and let $\varepsilon\in\mathbb{R}$. Then,

• (i)

$\mathcal{N}_{\varepsilon}$* is a $C^{1}$-manifold embedded in $W^{1,p}_{0}(\Omega)$,*

• (ii)

the tangent space $T_{v}\mathcal{N}_{\varepsilon}$ of $\mathcal{N}_{\varepsilon}$ at $v$ is the set $\{\varphi\in W^{1,p}_{0}(\Omega)\,:\,D\gamma_{\varepsilon}(v)\varphi=0\}$ and

• (iii)

for any $v\in\mathcal{N}_{\varepsilon}$, $v\notin T_{v}\mathcal{N}_{\varepsilon}$.

Proof.

First, we prove (i) and (ii) by showing that zero is regular value of $\gamma_{\varepsilon}$. Let $v\in\mathcal{N}_{\varepsilon}$ be arbitrary. Using (2.3) and (2.4),

[TABLE]

and from hypothesis (f4),

[TABLE]

Since $W^{1,p}_{0}(\Omega)=\mathbb{R}v\oplus{\rm Ker}D\gamma_{\varepsilon}(v)$ and $D\gamma_{\varepsilon}(v)|_{\mathbb{R}v}:\mathbb{R}v\to\mathbb{R}$ is bijective, the Implicit Function Theorem and the fact that $v\in\mathcal{N}_{\varepsilon}$ is arbitrary yield part (i).

We also conclude that for any $v\in\mathcal{N}_{\varepsilon}$,

[TABLE]

proving (ii). To prove (iii), observe from (2.12) that $v\notin T_{v}\mathcal{N}_{\varepsilon}$. This completes the proof of the lemma. $\blacksquare$

∎

Denote

[TABLE]

Lemma 8**.**

Let $f$ satisfy (f1)-(f4). For any $\varepsilon\in\mathbb{R}$, the manifold $\mathcal{N}_{\varepsilon}$ is diffeomorphic to $S_{\infty}$. In particular, $\mathcal{N}_{\varepsilon}$ is path-connected.

Proof.

In this proof we use the same notations as in Lemma 6. Let $\varepsilon\in\mathbb{R}$ be fixed and consider the function $\lambda_{\varepsilon}:S_{\infty}\to\mathcal{N}_{\varepsilon}$ defined by

[TABLE]

where $\tau_{\varepsilon,v}>0$ is the unique positive value such that $\tau_{\varepsilon,v}v\in\mathcal{N}_{\varepsilon}$ as described in Lemma 6.

It is direct to verify that $\lambda_{\varepsilon}:S_{\infty}\to\mathcal{N}_{\varepsilon}$ is injective. On the other hand, given ${\rm v}\in\mathcal{N}_{\varepsilon}$, $\tau_{\varepsilon,{\rm v}}=1$. Set

[TABLE]

so that $\tau_{\varepsilon,{v}}=\|{\rm v}\|_{W^{1,p}_{0}(\Omega)}$ and $\lambda_{\varepsilon}\left({v}\right)=\tau_{\varepsilon,{v}}{v}={\rm v}$. Therefore, $\lambda_{\varepsilon}$ is bijective.

Next, we prove that $\lambda_{\varepsilon}\in C^{1}(S_{\infty})$. Consider the function

[TABLE]

Observe that for any $\varepsilon\in\mathbb{R}$, $t>0$ and any $v\in W^{1,p}_{0}(\Omega)$ with $v\neq 0$,

[TABLE]

Also, $\xi$ is a $C^{1}-$function and from (2.3) for any $\varepsilon\in\mathbb{R}$, any $t>0$ and $v\in W^{1,p}_{0}(\Omega)$ with $v\neq 0$ and $\xi(\varepsilon,t,v)=0$,

[TABLE]

From hypothesis (f4),

[TABLE]

Thus, the Implicit Function Theorem yields that the mapping

[TABLE]

belongs to $C^{1}\left(\mathbb{R}\times\left(W^{1,p}_{0}(\Omega)\setminus\{0\}\right)\right)$ and we conclude, in particular, that $\lambda_{\varepsilon}\in C^{1}(S_{\infty})$. Finally, from Lemma 5, ${\rm Img}(\lambda_{\varepsilon})\subset W^{1,p}_{0}(\Omega)-B_{\rho}(0)$ and since $(\lambda_{\varepsilon})^{-1}(\rm v)=\frac{{\rm v}}{\|{\rm v}\|_{W^{1,p}_{0}(\Omega)}}$, we conclude that $\lambda_{\varepsilon}$ is a diffeomorphism. $\blacksquare$ ∎

Remark. An important corollary of the previous proof, is the $C^{1}-$ dependence on $\varepsilon\in\mathbb{R}$ of $\tau_{\varepsilon,v}$.

Now, we study the set $\mathcal{M}_{\varepsilon}$ defined in (1.6) and the minimization problem (1.7). Observe that

[TABLE]

where we recall that for any $v\in W^{1,p}_{0}(\Omega)$,

[TABLE]

Lemma 9**.**

Let $f$ satisfy (f1)-(f4). For any $\varepsilon\in\mathbb{R}$, the numbers

[TABLE]

are well defined and $\alpha_{\varepsilon}=\beta_{\varepsilon}$.

Proof.

Let $\varepsilon\in\mathbb{R}$ be arbitrary, but fixed and let $v\in W^{1,p}_{0}(\Omega)$ be also arbitrary and such that $v^{+},v^{-}\neq 0$. Consider the function $h_{\varepsilon}:[0,\infty)\times[0,\infty)\to\mathbb{R}$ defined by

[TABLE]

Observe that

[TABLE]

so that $h_{\varepsilon}(t,s)=J_{\varepsilon}(tv^{+})+J_{\varepsilon}(sv^{-})$ for any $t,s\geq 0$. Using the the notation of the proof of Lemma 6 we get the decomposition $h_{\varepsilon}(t,s)=g_{v^{+}}(t)+g_{v^{-}}(s)$. Also, since $J_{\varepsilon}\in C^{2}(W^{1,p}_{0}(\Omega))$, $h_{\varepsilon}\in C^{2}\left([0,\infty)\times[0,\infty)\right)$ and

[TABLE]

Hence, since $g_{v^{+}}$ and $g_{v^{-}}$ have only one critical point each, namely $t_{\varepsilon,v}>0$ and $s_{\varepsilon,v}>0$, respectively, we have that this pair provides the unique critical point of $h$. On the other hand, we have that $h_{\varepsilon}(t,s)=g_{v^{+}}(t)+g_{v^{-}}(s)\to-\infty$, as $|(t,s)|\to+\infty$. Also, $h_{\varepsilon}(0,0)=0$ and by (2.11), we find $\delta>0$ small and such that if $s\geq 0$ and $t\in(0,\delta)$ then $\partial_{t}h_{\varepsilon}(t,s)>0$. Similarly, if $t\geq 0$ and $s\in(0,\delta)$ then $\partial_{s}h_{\varepsilon}(t,s)>0$.

We conclude that for any $v\in W^{1,p}_{0}(\Omega)$, with $v^{+},v^{-}\neq 0$, there exists a unique pair $(t_{\varepsilon,v},s_{\varepsilon,v})\in(0,\infty)\times(0,\infty)$ such that

[TABLE]

In particular, $t_{\varepsilon,v}v^{+}+s_{\varepsilon,v}v^{-}\in\mathcal{M}_{\varepsilon}$. From this fact it follows that $\mathcal{M}_{\varepsilon}$ is non-empty and combining it with (2.5) it follows that $\alpha_{\varepsilon}$ and $\beta_{\varepsilon}$ are well defined.

Moreover,

[TABLE]

Since $v$ is arbitrary, we conclude that $\alpha_{\varepsilon}\leq\beta_{\varepsilon}$.

Next, we prove the reverse inequality. Let $v\in\mathcal{M}_{\varepsilon}$ be arbitrary. Since $v^{+},v^{-}\in\mathcal{N}_{\varepsilon}$, the proof of Lemma 6 yields that $J_{\varepsilon}(v^{+})=\max\limits_{t\geq 0}J_{\varepsilon}(tv^{+})$ and $J_{\varepsilon}(v^{-})=\max\limits_{s\geq 0}J_{\varepsilon}(sv^{-})$, i.e. $\tau_{\varepsilon,v^{+}}=\tau_{\varepsilon,v^{-}}=1$.

Let $t_{\varepsilon,v},s_{\varepsilon,v}>0$ be such that $J_{\varepsilon}(t_{\varepsilon,v}v^{+}+s_{\varepsilon,v}v^{-})=\max\limits_{t,s\geq 0}J_{\varepsilon}(tv^{+}+sv^{-})$. Then,

[TABLE]

implying that $(t_{\varepsilon,v},s_{\varepsilon,v})=(1,1)$.

Therefore,

[TABLE]

Since $v\in\mathcal{M}_{\varepsilon}$ is arbitrary, we find that $\beta_{\varepsilon}\leq\alpha_{\varepsilon}$. This concludes the proof. $\blacksquare$ ∎

As pointed out in [4], due to the lack of regularity of the functions

[TABLE]

(even in the case $p=2$), the set $\mathcal{M}_{\varepsilon}$ is not a submanifold of $W^{1,p}_{0}(\Omega)$ and hence it is not automatically clear that $u\in\mathcal{M}_{\varepsilon}$ solving (1.7) is a critical point of $J_{\varepsilon}$. For this reason we use a version of the deformation lemma (see Lemma 2.13 in [22]) to show that any local minimizer of $J_{\varepsilon}|_{\mathcal{M}_{\varepsilon}}$ is a critical point of $J_{\varepsilon}$.

Lemma 10**.**

Let $f$ satisfy (f1)-(f4) and let $\varepsilon\in\mathbb{R}$. Assume $u\in\mathcal{M}_{\varepsilon}\cap V$, where $V\subset W^{1,p}_{0}(\Omega)$ is an open set. If

[TABLE]

then $u$ is a critical point of $J_{\varepsilon}$ in $W^{1,p}_{0}(\Omega)$.

Proof.

Arguing by contradiction, assume that $u$ is not a critical point in $W^{1,p}_{0}(\Omega)$ of $J_{\varepsilon}$ so that

[TABLE]

Choose $r_{\varepsilon}>0$ such that $B_{r_{\varepsilon}}(u)\subset V$ and for every $v\in B_{r_{\varepsilon}}(u)$,

[TABLE]

Fix $a,b\in\mathbb{R}$ with $a<1<b$. Set $D:=(a,b)\times(a,b)\subset\mathbb{R}^{2}$ and assume further that $a,b$ are such that $\{tu^{+}+su^{-}\,:\,(t,s)\in\overline{D}\}\subset V$.

Consider the function $h_{\varepsilon}(t,s):=J_{\varepsilon}(tu^{+}+su^{-})$ for $(t,s)\in[0,\infty)\times[0,\infty)$, introduced in the proof of Lemma 9 and recall that for any $(t,s)\in\mathbb{R}^{2}-\{(1,1)\}$,

[TABLE]

In particular

[TABLE]

Let $\delta\in\left(0,\frac{r_{\varepsilon}}{3}\right)$ be fixed and small enough so that $B_{3\delta}(u)\subset V$ and

[TABLE]

Set $S:=\overline{B}_{\delta}(u)$, $c:=J_{\varepsilon}(u)$ and fix $\theta_{\varepsilon}\in\left(0,\frac{\delta\eta_{\varepsilon}}{8}\right)$. Using Lemma 2.3 in [22] we find a continuous deformation $\Lambda_{\varepsilon}:[0,1]\times W^{1,p}_{0}(\Omega)\to W^{1,p}_{0}(\Omega)$ such that

• (a)

$\Lambda_{\varepsilon}(1,v)=v$ for $v\notin J_{\varepsilon}^{-1}\left([c-2\theta_{\varepsilon},c+2\theta_{\varepsilon}]\right)\cap B_{3\delta}(u)$,

• (b)

$\Lambda_{\varepsilon}(1,J_{\varepsilon}^{c+\theta_{\varepsilon}}\cap\overline{B}_{\delta}(u))\subset J_{\varepsilon}^{c-\theta_{\varepsilon}}$,

• (c)

$J_{\varepsilon}(\Lambda_{\varepsilon}(1,v))\leq J_{\varepsilon}(v)$ for every $v\in W^{1,p}_{0}(\Omega)$,

• (d)

$\Lambda_{\varepsilon}(t,\cdot)$ is an homeomorphism of $W_{0}^{1,p}(\Omega)$ for every $t\in[0,1]$.

From (c) and (2.13) for every $(t,s)\in\overline{D}-\{(1,1)\}$,

[TABLE]

On the other hand, item (c) also implies that $J_{\varepsilon}(\Lambda_{\varepsilon}(1,u))\leq J_{\varepsilon}(u)$ and since $u\in J_{\varepsilon}^{c+\theta_{\varepsilon}}\cap B_{\delta}(u)$ then item (b) yields

[TABLE]

Therefore,

[TABLE]

Consider the mapping {\color[rgb]{1,0,0}\sigma}_{\varepsilon}(t,s):=\Lambda_{\varepsilon}(1,tu^{+}+su^{-}) for $(t,s)\in\overline{D}$. Next, we claim that {\color[rgb]{1,0,0}\sigma}_{\varepsilon}(D)\cap\mathcal{M}_{\varepsilon}\cap V is non-empty. If this is the case, then

[TABLE]

and hence contradicting (2.15).

To prove the claim we proceed as follows. Consider the functions

[TABLE]

for $(t,s)\in\overline{D}$.

Since $u\in\mathcal{M}_{\varepsilon}$, $(1,1)\in D$ is the unique point of maximum of $h_{\varepsilon}$. We conclude that

[TABLE]

From (2.14) and item (a), {\color[rgb]{1,0,0}\sigma}_{\varepsilon}(t,s)=tu^{+}+su^{-} for $(t,s)\in\partial D$, so that $\Psi_{0}=\Psi_{1}$ on $\partial D$ and consequently,

[TABLE]

Therefore, for some $(t_{1},s_{1})\in D$, $\Psi_{1}(t_{1},s_{1})=(0,0)$, i.e. {\color[rgb]{1,0,0}\sigma}_{\varepsilon}(t_{1},s_{1})\in\mathcal{M}_{\varepsilon}. Finally, items (a) and (d) imply that $\Lambda_{\varepsilon}(1,\cdot)$ is a homeomorphism with $\Lambda_{\varepsilon}(1,\cdot)|_{W^{1,p}_{0}(\Omega)-B_{3\delta}(u)}=Id|_{W^{1,p}_{0}(\Omega)-B_{3\delta}(u)}$. Consequently, $\Lambda_{\varepsilon}(1,V)\subset V$ and since $(t_{1},s_{1})\in D$, then $\Lambda_{\varepsilon}(1,t_{1}u^{+}+s_{1}u^{-})\in V$, i.e $\sigma_{\varepsilon}(t_{1},s_{1})\in V$. This concludes the proof of the claim and also the proof of the lemma. $\blacksquare$ ∎

Lemma 11**.**

Let $f$ satisfy (f1)-(f4) and let $\varepsilon\in\mathbb{R}$. Assume that $B\subset W^{1,p}_{0}(\Omega)$ is either an open ball or $B=W^{1,p}_{0}(\Omega)$. Let $\{v_{n}\}_{n\in\bm{\mathbb{N}}}\subset\mathcal{M}_{\varepsilon}\cap B$ and $v\in W^{1,p}_{0}(\Omega)$ be such that $v_{n}\rightharpoonup v$ weakly in $W_{0}^{1,p}(\Omega)$ and let $\tau_{\varepsilon,v^{+}},\tau_{\varepsilon,v^{-}}>0$ be such that $\tau_{\varepsilon,v^{+}}v^{+}+\tau_{\varepsilon,v^{-}}v^{-}\in\mathcal{M}_{\varepsilon}\cap B$. Then the following holds true.

• (i)

$v^{+}\neq 0$* and $v^{-}\neq 0$.*

• (ii)

If $\{v_{n}\}_{n\in\bm{\mathbb{N}}}$ is a minimizing sequence for $J_{\varepsilon}|_{\mathcal{M}_{\varepsilon}\cap B}$, i.e. $J_{\varepsilon}(v_{n})\to\min\limits_{w\in\mathcal{M}_{\varepsilon}\cap B}J_{\varepsilon}(w)$, then $v_{n}\to v$ strongly in $W^{1,p}_{0}(\Omega)$, $v\in\mathcal{M}_{\varepsilon}\cap B$ and $J_{\varepsilon}(v)=\min\limits_{w\in\mathcal{M}_{\varepsilon}\cap B}J_{\varepsilon}(w)$.

In particular, if $J_{\varepsilon}(v_{n})\to\min\limits_{w\in\mathcal{M}_{\varepsilon}}J_{\varepsilon}(w)$, then $v\in\mathcal{M}_{\varepsilon}$ and $J_{\varepsilon}(v)=\min\limits_{w\in\mathcal{M}_{\varepsilon}}J_{\varepsilon}(w)$.

Proof.

We proceed as in section 3 of [8]. First, notice that

[TABLE]

and

[TABLE]

The compactness of the Sobolev embeddings and (2.16) imply that, up to a subsequence,

[TABLE]

for any $r\in[1,p^{*})$. Also, $v^{+}\geq 0$ and $v^{-}\leq 0$ a.e. in $\Omega$. By taking further subsequences, if necessary, we may also assume that the sequences $\{\|v^{+}_{n}\|\}_{n}$ and $\{\|v^{-}_{n}\|\}_{n}$ converge in $\mathbb{R}$.

Vainberg’s Lemma (see [21]) yields

[TABLE]

[TABLE]

as $n\to\infty$. Since $v_{n}\in\mathcal{M}_{\varepsilon}$,

[TABLE]

and using Lemma 5 we find that

[TABLE]

so that $v^{\pm}\neq 0$ in $W^{1,p}_{0}(\Omega)$. This proves (i).

Next, we prove (ii). First we prove that $v_{n}\to v$ strongly in $W^{1,p}_{0}(\Omega)$. We proceed by showing that in (2.17) both equalities hold.

Let us argue by contradiction. Assume either

[TABLE]

Set $a:=\tau_{\varepsilon,v^{+}}$ and $b:=\tau_{\varepsilon,v^{-}}$. Since $av^{+}+bv^{-}\in\mathcal{M}_{\varepsilon}\cap B$,

[TABLE]

which is a contradiction. Thus, the equalities hold in (2.17). Since $W_{0}^{1,p}(\Omega)$ is uniformly convex (see Theorem 2.6 in [1]) the convergence of $\{v_{n}\}_{n\in\bm{\mathbb{N}}}$ to $v$ in $W^{1,p}_{0}(\Omega)$ is strong.

Since $v_{n}\to v$ strongly in $W^{1,p}_{0}(\Omega)$ and $J_{\varepsilon}$ is a $C^{1}-$ function, we conclude that $v^{+},v^{-}\in\mathcal{N}_{\varepsilon}$, i.e. $v\in\mathcal{M}_{\varepsilon}$ or equivalently $t_{\varepsilon,v}=s_{\varepsilon,v}=1$. From our assumptions, this in turn implies that $v\in B$.

Finally, If $\{v_{n}\}_{n\in\bm{\mathbb{N}}}$ is a minimizing sequence for $J_{\varepsilon}|_{\mathcal{M}_{\varepsilon}\cap B}$, then $J_{\varepsilon}(v)=\min\limits_{w\in\mathcal{M}_{\varepsilon}\cap B}J_{\varepsilon}(v)$. This completes the proof of the lemma. $\blacksquare$ ∎

Proof.

Proof of Theorem 1. Let $\{v_{n}\}_{n\in\bm{\mathbb{N}}}\subset\mathcal{M}_{\varepsilon}$ be a minimizing sequence, i.e. $J_{\varepsilon}(v_{n})\to\alpha_{\varepsilon}$. From Lemma 5, $J_{\varepsilon}|_{\mathcal{N}_{\varepsilon}}$ is coercive. Consequently, $\{v_{n}\}_{n\in\bm{\mathbb{N}}}$ is bounded. From the reflexivity of $W^{1,p}_{0}(\Omega)$, there exists $u\in W_{0}^{1,p}(\Omega)$ such that, up to a subsequence $v_{n}\rightharpoonup u$ weakly in $W_{0}^{1,p}(\Omega)$. A direct application of Lemma 11 implies that $u\in\mathcal{M}_{\varepsilon}$ solves (1.7). Finally, Lemma 10 implies that $u$ is a least energy nodal solution of ($\mathcal{P}_{\varepsilon}$) and Lemma 9 yields (1.8). This completes the proof of the theorem. $\blacksquare$ ∎

We conclude this section providing a result that relates the number of nodal regions of a solution with its Morse index.

Lemma 12**.**

Let $f$ satisfy (f1) and (f4) and let $\varepsilon\in\mathbb{R}$. Assume that $u\in W_{0}^{1,p}(\Omega)$ is a weak solution of ($\mathcal{P}_{\varepsilon}$) with $N_{\varepsilon}(u)$ nodal regions. Then $N_{\varepsilon}(u)\leq m_{\varepsilon}(u)$.

Proof.

Let $C$ be a nodal region of $u$ and define $v:=\mathds{1}_{C}u$. Lemma 1 in [18] yields that $v\in W_{0}^{1,p}(\Omega)$ and

[TABLE]

Thus, if $D$ is another nodal region of $u$ and $w:=\mathds{1}_{D}u$, then from (1.4) we have that

[TABLE]

We conclude that given any $t,s\in\mathbb{R}$,

[TABLE]

Therefore, to prove the lemma, it suffices to show that $D^{2}J_{\varepsilon}(u)$ is negative definite along any direction $v$ of the form $\mathds{1}_{C}u$ with $C$ an arbitrary nodal region of $u$.

Notice that since $u$ is critical point, using the definition of $v$

[TABLE]

Using the latter equation and (f4) we obtain (see also the proof of Lemma 6)

[TABLE]

Proceeding inductively on the number of nodal regions the result follows. $\blacksquare$ ∎

3. Regularity

In this part we obtain regularity results for solutions of ($\mathcal{P}_{\varepsilon}$). This results will play a crucial role when computing the Morse index of least energy nodal solutions in Section 4. The first result states the uniform boundedness of $u$.

Lemma 13**.**

Let $\varepsilon\in\mathbb{R}$ and $f$ satisfy hypothesis (f1). Any weak solution $u\in W^{1,p}_{0}(\Omega)$ of ($\mathcal{P}_{\varepsilon}$), in the sense of the Definition 1, belongs to $L^{\infty}(\Omega)$.

Proof.

The reader is referred to Lemma 3.1 in [11], but for the sake of completeness and clarity we present a brief sketch of it.

Let $u\in W^{1,p}_{0}(\Omega)$ satisfy the integral identity (1.3). If $p>N$, the Sobolev embedding $W^{1,p}_{0}(\Omega)\hookrightarrow C^{0,1-\frac{N}{p}}(\overline{\Omega})$ yields the conclusion.

Assume now that $p\in(1,N]$ and that $q\in(p,p^{*})$. Let $j\in\mathbb{N}$ be arbitrary, but fixed and consider the function

[TABLE]

Since $\chi_{j}\in W^{1,\infty}_{loc}(\overline{\mathbb{R}})$, $\chi^{\prime}_{j}\in L^{\infty}(\mathbb{R})$ and $\chi_{j}(0)=0$,

[TABLE]

where $\Omega_{j}:=\{|u|>j\}$ and $\mathds{1}_{\Omega_{j}}$ is the characteristic function of $\Omega_{j}$.

Proceeding in the same fashion as in the proof of the Proposition 9.5 in [6], we conclude that $\chi_{j}(u)\in W^{1,p}_{0}(\Omega)$ and using (1.3) with $\varphi=\chi_{j}(u)$,

[TABLE]

Hypothesis (f1) yields the existence of $C>0$, depending only on $q$, such that

[TABLE]

and since $j\geq 1$ and $2<p<q<p^{*}$, we can select $C>0$ larger if necessary, but yet independent of $j$, so that

[TABLE]

Next, let $r\in(\frac{N}{p},\infty)$ such that $l:=\frac{rq}{r-1}\in(q,p^{*})$. Using Hölder ’s inequality and the fact $u\in L^{l}(\Omega)$, we estimate

[TABLE]

Therefore,

[TABLE]

where $C_{1}:=2^{q-1}C{\rm meas}(\Omega)^{\frac{q}{r-1}}\|u\|^{q-p}_{L^{l}(\Omega)}>0$.

Since (3.1) holds for any $j\in\mathbb{N}$, Theorem 5.1 from Chapter 2 and the footnote in page 71 both from [16] imply that $u\in L^{\infty}(\Omega)$. This concludes the proof of the lemma. $\blacksquare$ ∎

For any $\varepsilon\in\mathbb{R}$ arbitrary, consider the function $A_{\varepsilon}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ defined by

[TABLE]

Observe that $A_{\varepsilon}(0)=0$. Also $A_{0}\in C^{0}(\mathbb{R}^{N})\cap C^{\infty}(\mathbb{R}^{n}-\{0\})$ and $A_{\varepsilon}\in C^{1}(\mathbb{R}^{N})$.

A direct computation yields that

[TABLE]

where $I_{N\times N}$ is the identity matrix and the symbol $\otimes$ represents the tensor product between two vectors of $\mathbb{R}^{N}$.

Lemma 14**.**

For any $\varepsilon\in\mathbb{R}$ and any $z\in\mathbb{R}^{N}$, the symmetric matrix $DA_{\varepsilon}(z)$ has no negative eigenvalues and if $\varepsilon\neq 0$ they are strictly positive.

Proof.

From (3.2), if $z=0$ the result is immediate with eigenvalues $\lambda_{1}=\cdots=\lambda_{N}=\varepsilon^{2}$, so we assume that $z\neq 0$. Let $\lambda\in\mathbb{R}$ be such that

[TABLE]

If $\lambda=\varepsilon^{2}+|z|^{p-2}$, then $\lambda\geq 0$ and it is strictly positive for $\varepsilon>0$. Assume next that $\lambda\neq\varepsilon^{2}+|z|^{p-2}$.

From the matrix determinant formula, see for instance [15], it follows that

[TABLE]

We conclude that $\lambda=\varepsilon^{2}+\left(p-1\right)|z|^{p-2}$ so that $\lambda\geq 0$ and it is strictly positive if $\varepsilon>0$ proving the lemma. $\blacksquare$ ∎

Lemma 15**.**

Let $p>2$. There exist constants $\gamma,\Gamma>0$ such that for any $\varepsilon$ belonging to a bounded interval of $\mathbb{R}$, any $z\in\mathbb{R}^{N}$ and any $\zeta\in\mathbb{R}^{N}$,

[TABLE]

where $\kappa=\varepsilon^{\frac{2}{p-2}}$.

Proof.

Let $\varepsilon$ belong to a fixed bounded interval of $\mathbb{R}$. Lemma 14 yields that $DA_{\varepsilon}(z)$ is a semipositive definite matrix. Let $\lambda_{1}(z)\leq\cdots\leq\lambda_{N}(z)$ denote the eigenvalues of $DA_{\varepsilon}(z)$.

Lemma 14 implies that $\lambda_{1},\lambda_{N}:\mathbb{R}^{N}\to\mathbb{R}$ are continuous in $z$. On the other hand, the characterization

[TABLE]

and Lemma 14 implies that for any $z\in\mathbb{R}^{N}$ and $\zeta\in\mathbb{R}^{N}$,

[TABLE]

Recall that for any $\beta>0$ and for any $a,b\geq 0$

[TABLE]

Using (3.4) and that $p>2$,

[TABLE]

for some constant $C_{p}>0$ depending only on $p>2$. Consequently,

[TABLE]

On the other hand, defining

[TABLE]

we get for $a,b\geq 0$ that

[TABLE]

Therefore, taking $\beta=p-2$ we get

[TABLE]

so that

[TABLE]

Set

[TABLE]

to obtain the inequalities in (3.3). This completes the proof. $\blacksquare$ ∎

Remark: From Lemma 13, any solution $u\in W^{1,p}_{0}(\Omega)$ of ($\mathcal{P}_{\varepsilon}$) is bounded. By choosing $\kappa,\gamma>0$ as in Lemma 15 and choosing

[TABLE]

conditions (1.4)-(1.7) in [20] are satisfied. These hypotheses correspond also to the condition (0.3a)-(0.3d) in [17].

Lemma 16**.**

Assume the hypotheses in Lemma 13 and let $\varepsilon$ belong to any bounded interval of $\mathbb{R}$. Then, there exists $\beta=\beta(N,p,\Omega)\in(0,1)$ independent of $\varepsilon>0$ such that any solution of $u\in W^{1,p}_{0}(\Omega)$ of ($\mathcal{P}_{\varepsilon}$) belongs to $C^{1,\beta}(\overline{\Omega})\cap W^{2,2}_{loc}(\Omega)$.

Proof.

From Lemma 13, $u\in L^{\infty}(\Omega)$. Let $\varepsilon\in\mathbb{R}$ be fixed and set $h:=f(u(\cdot))$. The previous remark states that the operator ${\rm div}A_{\varepsilon}(\nabla u)$ and the right-hand side $h$ satisfy the hypotheses in the [Theorem 1,[20]] and in [Theorem 1, [17]].

Since, any bounded solution $u$ of ($\mathcal{P}_{\varepsilon}$) solves weakly the equation

[TABLE]

there exists $\beta=\beta(N,p,\Omega)>0$ such that $u\in C^{1,\beta}(\overline{\Omega})$ and there exists

[TABLE]

such that

[TABLE]

From Proposition 1 in [20] it follows that $u\in W^{2,2}_{loc}(\Omega)$ and this completes the proof. $\blacksquare$ ∎

Proposition 17**.**

Let $p>2$ and let $\varepsilon\neq 0$ belong to a bounded interval of $\mathbb{R}$. Assume hypotheses from Lemma 13 and let $u\in W^{1,p}_{0}(\Omega)$ a solution of ($\mathcal{P}_{\varepsilon}$). Then, for any $r>1$, $u\in W^{2,r}(\Omega)$.

Proof.

From Lemma 16, $u\in C^{1,\beta}(\overline{\Omega})\cap W^{2,2}_{loc}(\Omega)$. Set $h:=f(u(\cdot))$.

Using hypothesis (f1), for $x,y\in\Omega$,

[TABLE]

so that $h\in C^{0,{\beta}}(\overline{\Omega})$.

Consider the BVP for $w$,

[TABLE]

where summation over repeated indices is understood.

Observe that $u\in W^{2,2}_{loc}(\Omega)\cap C^{1,\beta}(\overline{\Omega})$ is a strong solution of (3.5). Set

[TABLE]

so that equation (3.5) reads as

[TABLE]

Since $p>2$ and $\varepsilon^{2}>0$, $a_{ij,\varepsilon}\in C(\overline{\Omega})$. Also, Lemma 15 implies that $a_{ij,\varepsilon}$ is strictly elliptic. The fact that $h\in L^{\infty}(\Omega)$ and Theorem 9.15 in [14] yield that given any $r\geq 2$, there exists a unique strong solution $w\in W^{2,r}(\Omega)$ of (3.5).

From the Theorem 8.9 in [14] and the remark after it, we conclude that $w$ is the unique strong solution of (3.5) in $W^{2,2}_{loc}(\Omega)\cap W^{1,2}_{0}(\Omega)$. Therefore, $w=u$ a.e. in $\Omega$ and so it is also unique in $W^{2,r}(\Omega)$. Since $\Omega$ is a bounded domain, $u\in W^{2,r}(\Omega)$ for any $r>1$ and this proves the result. $\blacksquare$ ∎

The following lemma provides a useful result for the computations presented in the next section (see the proof of Lemma 19 below).

Lemma 18**.**

If $p\geq 2$, then for any $\phi\in W^{1,p}(\Omega)\cap W^{2,2}(\Omega)$,

[TABLE]

a.e. in $\Omega$. In particular, if $\phi\in W^{1,p}(\Omega)\cap W^{2,p}(\Omega)$ then $\Delta_{p}\phi\in L^{p^{\prime}}(\Omega)$, where $p^{\prime}=\frac{p}{p-1}$.

Proof.

To verify this claim, we first let $\mu>0$ be arbitrary, but fixed. Next, integrating by parts we find that for any $\psi\in C^{\infty}_{c}(\Omega)$,

[TABLE]

We estimate pointwise the integrand in the left-hand side of (3.6), to find that

[TABLE]

a.e. in $\Omega$.

On the other hand, since $\phi\in W^{2,2}(\Omega)$, we compute

[TABLE]

a.e in $\Omega$. Using this computation and Young inequality, we estimate

[TABLE]

a.e. in $\Omega$. Therefore, a direct application of the Dominated Convervenge Theorem taking $\mu\to 0^{+}$ yields that

[TABLE]

Since $\psi\in C^{\infty}_{c}(\Omega)$ is arbitrary, we conclude that

[TABLE]

a.e. in $\Omega$ and this proves the first part of the claim. Now, if $\phi\in W^{1,p}(\Omega)\cap W^{2,p}(\Omega)$ then a direct application of Höder’s inequality shows that $\Delta_{p}\phi\in L^{p^{\prime}}(\Omega)$. $\blacksquare$ ∎

4. Morse index in the case $\varepsilon\neq 0$

Let $\varepsilon\neq 0$ and let $u_{\varepsilon}\in\mathcal{M}_{\varepsilon}$ be a local minimizer for $J_{\varepsilon}|_{\mathcal{M}_{\varepsilon}}$. Since $\mathcal{M}_{\varepsilon}$ is not a smooth manifold of $W_{0}^{1,p}(\Omega)$, we cannot infer directly the local behavior of $J_{\varepsilon}|_{\mathcal{M}_{\varepsilon}}$ around $u_{\varepsilon}$ to estimate its Morse index. We overcome this issue by adapting the approach in [4] to our setting and by working on a suitable dense subspace of $W^{1,p}_{0}(\Omega)$ containing $u_{\varepsilon}$.

Consider the functionals $I_{\varepsilon},H:W_{0}^{1,p}(\Omega)\to\mathbb{R}$ defined by

[TABLE]

for every $v\in W^{1,p}_{0}(\Omega)$.

Observe that $J_{\varepsilon}=I_{\varepsilon}-H$ and $I_{\varepsilon},H\in C^{2}(W_{0}^{1,p}(\Omega))$ with

[TABLE]

for every $v,\varphi\in W^{1,p}_{0}(\Omega)$.

Consider also the Banach space $W:=W_{0}^{1,p}(\Omega)\cap W^{2,p}(\Omega)$ endowed with the norm of $W^{2,p}(\Omega)$.

Our first lemma is a technical result concerned with the regularity of $I_{\varepsilon}$ and $H$ in $W$. This lemma will be used to prove that $\mathcal{M}_{\varepsilon}\cap W$ is a $C^{1}$-manifold embedded in $W$.

Lemma 19**.**

Let $f$ satisfy (f1) and let $\varepsilon\neq 0$. Consider the functions $P^{\pm}_{\varepsilon},Q^{\pm}:W_{0}^{1,p}(\Omega)\to\mathbb{R}$ defined by:

[TABLE]

for every $v\in W^{1,p}_{0}(\Omega)$. Then,

1. (a)

$P_{\varepsilon}^{\pm}|_{W}\in C^{1}(W)$. Moreover, for every $v\in W$, $DP^{\pm}_{\varepsilon}(v)\in(W_{0}^{1,p}(\Omega))^{*}$ with

[TABLE]

for every $\varphi\in W_{0}^{1,p}(\Omega)$. 2. (b)

$Q^{\pm}\in C^{1}(W_{0}^{1,p}(\Omega))$* and*

[TABLE]

Proof.

We proceed as in the proof of the Lemma 3.1 in [4]. First, we prove (a). Hölder’s inequality implies that for each $v\in W$, $P_{\varepsilon}^{\pm}(v)$ given by (4.1) is well defined and continuous in $W_{0}^{1,p}(\Omega)$.

Let $v,\varphi\in W$ and $t\in\mathbb{R}-\{0\}$ be arbitrary. Consider the sets

[TABLE]

with corresponding characteristic functions $\mathds{1}_{C^{t}_{1}},\mathds{1}_{C^{t}_{2}},\mathds{1}_{C^{t}_{3}},\mathds{1}_{C_{4}}:\Omega\to\mathbb{R}$.

Observe that

[TABLE]

[TABLE]

After lenghty, but straightforward computations involving integration by parts, the use of Lemma 18 and rearranging of the terms, we find that

[TABLE]

From Lemma 7.7 of [14] and Lemma 18 in Section 3, $\nabla v$, $\Delta v$ and $\Delta_{p}v$ are zero a.e. on $C_{4}$.

First we estimate $I_{t}$. Observe that

[TABLE]

Notice that either in $C_{2}^{t}$ or in $C_{3}^{t}$ we have that $|v|\leq|t||\varphi|$ implying that

[TABLE]

Also, since for $i=1,2,3$

[TABLE]

and as $t\to 0$, $\mathds{1}_{C_{1}^{t}}\to\mathds{1}_{\{v>0\}}$ and for $i=2,3$ $\mathds{1}_{C_{i}^{t}}\to 0$ a.e. in $\Omega$, the Dominated Convergence Theorem yields

[TABLE]

Next, we estimate $II_{t}$. Since $p>2$,

[TABLE]

Thus,

[TABLE]

Notice that there exists $C_{p}>0$ such that for $i=1,2$

[TABLE]

[TABLE]

Also, since

[TABLE]

and

[TABLE]

a.e. in $\Omega$, as $t\to 0$, we conclude that

[TABLE]

As for $III_{t}$, observe that

[TABLE]

and proceeding in a similar fashion as above,

[TABLE]

Finally, a direct application of the Dominated Convergence Theorem proves that

[TABLE]

From (4.2),(4.3), (4.4) and (4.5), and using that $v,\varphi$ are arbitrary, we conclude that $DP_{\varepsilon}^{+}$ is Gateaux differentiable in $W$ and (4.1) is satisfied in the Gateaux sense.

Next, we prove that $P^{+}_{\varepsilon}|_{W}\in C^{1}(W)$. Let $v_{n},v\in W$ be such that $v_{n}\to v$ strongly in $W$ and let $\varphi\in W$ be such that $\|\varphi\|_{W}=1$. With no loss of generality assume that $v_{n}(x)\to v(x)$ and $\nabla v_{n}(x)\to\nabla v(x)$ as $n\to\infty$ for a.e. $x\in\Omega$. Then,

[TABLE]

Recall that $v_{n},v\in W$. Using Höder’s inequality, we find a constant $C_{p}>0$ independent of $n$ such that

[TABLE]

Proceeding in the same fashion as above, using Lemma 18 and taking $C_{p}>0$ larger if necessary,

[TABLE]

as $n\to\infty$.

Arguing in the same fashion as in (4.7), we conclude also that as $n\to\infty$,

[TABLE]

We estimate the last four integrals as follows. Cauchy-Schwarz, Hölder’s inequality and the fact that $p>2$ imply that

[TABLE]

Since $v_{n}(x)\to v(x)$ for a.e. $x\in\Omega$ and $\nabla v,\Delta v=0$ a.e. in the set $\{v=0\}$, the Dominated Convergence Theorem yields

[TABLE]

Similarly we conclude that

[TABLE]

On the other hand, using Hölder’s inequality and the fact that $\Delta_{p}v\in L^{\frac{p}{p-1}}(\Omega)$ we get

[TABLE]

and arguing as above, as $n\to\infty$

[TABLE]

The same argument yields

[TABLE]

Putting together (4.6)-(4.11),

[TABLE]

as $n\to\infty$ and since $\varphi\in W$ with $\|\varphi\|_{W}=1$ is arbitrary, we conclude that $DP_{\varepsilon}^{+}$ is continuous.

The same argument with obvious changes yields also that $P_{\varepsilon}^{-}$ belongs to $C^{1}(W)$ and (4.1) holds true.

The proof of (b) goes along the same lines as the proof of (a). This concludes the proof. $\blacksquare$ ∎

Remark: We point out that all the computations in the previous proof hold true taking $\varphi\in W_{0}^{1,p}(\Omega)$ with $\|\varphi\|_{W^{1,p}_{0}(\Omega)}=1$.

Lemma 20**.**

Let $f$ satisfy (f1) and (f4) and let $\varepsilon\neq 0$. Assume that $\mathcal{M}_{\varepsilon}\cap W\neq\emptyset$. Then, $\mathcal{M}_{\varepsilon}\cap W$ is a $C^{1}-$manifold in $W$ of codimension two.

Proof.

Following the notations from Lemma 19, recall that

[TABLE]

Since $\mathcal{M}_{\varepsilon}\cap W$ is contained in an open subset of $W$, in view of the Lemma 19 and the Implicit Function Theorem, it suffices to prove that given any $v\in\mathcal{M}_{\varepsilon}\cap W$, the function $(D\left(P^{+}_{\varepsilon}-Q^{+}\right)(v),D\left(P^{-}_{\varepsilon}-Q^{-}\right)(v))$, which belongs to $B(W,\mathbb{R}^{2})$, is a surjective function.

Fix $v\in\mathcal{M}_{\varepsilon}\cap W$ and let $\lambda,\eta\in\mathbb{R}$. Observe that

[TABLE]

can be written in matrix form as

[TABLE]

Using that $v\in W$ and integrating by parts,

[TABLE]

Using that $v\in\mathcal{M}_{\varepsilon}$ and (f4),

[TABLE]

Therefore, the matrix in (4.12) is invertible. From this, we can not directly conclude the desired surjectivity since the functions $v^{+},v^{-}$ do not necessarily belong to $W$. We overcome this obstacle by considering the continuous function $D_{v}:W_{0}^{1,p}(\Omega)\times W_{0}^{1,p}(\Omega)\longrightarrow\mathbb{R}$ defined as

[TABLE]

The previous considerations say $D_{v}(v^{+},v^{-})>0$. Aproximating $v^{+}$ and $v^{-}$ by functions $\varphi$ and $\psi$ in $W$, it follows that $(D(P_{\varepsilon}^{+}-Q^{+})(v),D(P_{\varepsilon}^{-}-Q^{-})(v))|_{span\{\varphi,\psi\}}$ is invertible. Hence $(D\left(P^{+}_{\varepsilon}-Q^{+}\right)(v),D\left(P^{-}_{\varepsilon}-Q^{-}\right)(v))\in B(W,\mathbb{R}^{2})$ is surjective. This concludes the proof. $\blacksquare$ ∎

Before proving the main theorem of this section we state a technical lemma that will help us to get an upper bound for the Morse Index of $u_{\varepsilon}$.

Lemma 21**.**

Let $\left(X,\|\cdot\|\right)$ be a Banach space and let $Y$ and $Z$ subspaces of $X$ such that $Y\leq Z$ and the codimension of $Y$ in $Z$ is $T\in\bm{\mathbb{N}}$. Then $\overline{Y}$, the closure of $Y$ in $X$, has codimension at most $T$ in $\overline{Z}$.

Proof.

Assume first that $Y$ has codimension one in $Z$ so that there exists $z_{0}\in Z$, $z_{0}\notin Y$, for which $Z=Y\oplus\mathbb{R}z_{0}$.

If $z_{0}\in\overline{Y}$, then $\overline{Z}=\overline{Y}$ and the result follows. Assume then that $z_{0}\notin\overline{Y}$. Given $z\in\overline{Z}$, there exist a sequence $\{t_{n}z_{0}+y_{n}\}_{n\in\bm{\mathbb{N}}}$, with $t_{n}\in\mathbb{R}$ and $y_{n}\in Y$ such that $t_{n}z_{0}+y_{n}\to z$ and in particular $\{t_{n}z_{0}+y_{n}\}_{n\in\bm{\mathbb{N}}}$ is bounded.

We claim that the sequence $\{t_{n}\}_{n\in\bm{\mathbb{N}}}$ is bounded. If it were not the case, we could find a subsequence $\{t_{n_{k}}\}_{k\in\bm{\mathbb{N}}}$ such that $\lim\limits_{k\to\infty}t_{n_{k}}=\infty$.

Thus, there would exist $M>0$ such that for every $k\in\bm{\mathbb{N}}$,

[TABLE]

Implying that $\|z_{0}+\frac{1}{t_{n_{k}}}y_{n_{k}}\|_{X}\to 0$ as $k\to\infty$. This is a contradiction with the fact that we have assumed $z_{0}\notin\overline{Y}$ and so $\{t_{n}\}_{n\in\bm{\mathbb{N}}}$ is bounded.

Hence, up to subsequences, we may assume that $t_{n}\to t_{0}\in\mathbb{R}$ and $y_{n}\to y_{0}\in\overline{Y}$ as $n\to\infty$, so that $z=t_{0}z_{0}+y_{0}$.

Since $z\in\overline{Z}$ is arbitrary, we conclude that $\overline{Z}=\mathbb{R}z_{0}\oplus\overline{Y}$. The general case follows from the above discussion and an induction argument. This completes the proof of the lemma. $\blacksquare$ ∎

Proof.

Proof of Theorem 2. First we prove (i). Since $u\in\mathcal{M}_{\varepsilon}$, hypothesis (f4), (specifically the properties of the functions proof $g_{u^{+}}(t)$ and $g_{u^{-}}(s)$ introduced in Lemma 6) imply that for any $(\lambda,\eta)\in\mathbb{R}^{2}-(0,0)$,

[TABLE]

Therefore, $m(u_{\varepsilon})\geq 2$. In virtue of Proposition 17, $u\in\mathcal{M}_{\varepsilon}\cap W$, therefore Lemma 20 implies that $\mathcal{M}_{\varepsilon}\cap W$ is a non-empty $C^{1}-$manifold of $W$. Then, for every $\varphi\in T_{u}\left(\mathcal{M}_{\varepsilon}\cap W\right)$ there exists $\delta>0$ and a $C^{1}$ curve $c:(-\delta,\delta)\to W$ with $c(t)\in\mathcal{M}_{\varepsilon}$ for every $t\in(-\delta,\delta)$ such that $c(0)=u$ and $c^{\prime}(0)=\varphi$. Since $u$ is a local minimizer of $J_{\varepsilon}$ on the $C^{1}$-manifold $\mathcal{M}_{\varepsilon}\cap W$, then $\beta(t):=J_{\varepsilon}(c(t))$ has a local minimum at [math]. By the chain rule $\beta$ is a $C^{1}$ function and notice that since $u$ is a critical point of $J_{\varepsilon}$ then

[TABLE]

This implies that $D^{2}J(u)(\varphi,\varphi)\geq 0$ for every $\varphi\in T_{u}\left(\mathcal{M}_{\varepsilon}\cap W\right)$. From Lemma 20, $T_{u}\left(\mathcal{M}_{\varepsilon}\cap W\right)$ has codimension 2 in $W$. Finally, using that $W$ is dense in $W_{0}^{1,p}(\Omega)$ and applying Lemma 21 to $T_{u}\left(\mathcal{M}_{\varepsilon}\cap W\right)\leq W$ we get that the closure of $T_{u}\left(\mathcal{M}_{\varepsilon}\cap W\right)$ has codimension at most two in $W^{1,p}_{0}(\Omega)$ and so $m_{\varepsilon}(u)\leq 2$.

Next, we prove (ii). Since $u\in\mathcal{M}_{\varepsilon}$ is a local minimizers, we conclude from part (i) in Lemma 11 that $u$ has at least two nodal regions. Using part (i) from this result, $m_{\varepsilon}(u)=2$ and directly from Lemma 12 we conclude that $u$ has exactly two nodal regions. This completes the proof of the theorem. $\blacksquare$ ∎

5. Proof of Theorem 3

Throughout the rest of the developments, we assume that $f$ satisfy (f1)-(f4) and we remind the reader that for $\varepsilon\in\mathbb{R}$,

[TABLE]

For any $\varepsilon\in\mathbb{R}$, let $u_{\varepsilon}\in\mathcal{M}_{\varepsilon}$ be the least energy nodal solution predicted by Theorem 1. From Lemma 9,

[TABLE]

Also, a consequence of the proof of Lemma 9 is that

[TABLE]

Next two lemmas are concerned with the convergence of the sequences $\{\alpha_{\varepsilon}\}_{\varepsilon\geq 0}$ and $\{u_{\varepsilon}\}_{\varepsilon>0}$

Lemma 22**.**

The following assertions hold true:

• (i)

the family $\{\alpha_{\varepsilon}\}_{\varepsilon}$ is strictly increasing in $\varepsilon>0$, $\alpha_{\varepsilon}\to\alpha_{0}$, as $\varepsilon\to 0^{+}$ and

• (ii)

there exist $C>0$ and $\varepsilon_{0}>0$ such that for any $\varepsilon\in(0,\varepsilon_{0})$,

[TABLE]

Proof.

First, we prove (i). Let $\varepsilon_{1},\varepsilon_{2}\in[0,\infty)$ be arbitrary with $\varepsilon_{1}<\varepsilon_{2}$. Let also $\tau_{\varepsilon,v}>0$ denote the projection scalar defined in Lemma 8, associated with the energy $J_{\varepsilon}$ for the function $v$. Clearly $\tau_{\varepsilon_{1},u_{\varepsilon_{2}}^{+}}u_{\varepsilon_{2}}^{+}+\tau_{\varepsilon_{1},u_{\varepsilon_{2}}^{-}}u_{\varepsilon_{2}}^{-}\in\mathcal{M}_{\varepsilon_{1}}$. Then,

[TABLE]

From (5.1),

[TABLE]

and hence $\alpha_{\varepsilon_{1}}<\alpha_{\varepsilon_{2}}$.

Next, we prove that $\alpha_{\varepsilon}\to\alpha_{0}$, as $\varepsilon\to 0^{+}$. The proof of Lemma 8 and the remark after it imply that $\tau_{\varepsilon,u_{0}^{+}}\to\tau_{0,u_{0}^{+}}$ and $\tau_{\varepsilon,u_{0}^{-}}\to\tau_{0,u_{0}^{-}}$ as $\varepsilon\to 0^{+}$. Since $u_{0}\in\mathcal{M}_{0}$, $\tau_{0,u_{0}^{+}}=\tau_{0,u_{0}^{-}}=1$ and hence the definition of $J_{\varepsilon}$ yields that

[TABLE]

as $\varepsilon\to 0^{+}$. This proves (i).

To prove (ii), we notice that for any $\varepsilon_{0}>0$ fixed, from the previous discussion $\{\alpha_{\varepsilon}\,:\,\varepsilon\in(0,\varepsilon_{0})\}$ is bounded and from Lemma 5 the conclusion follows. This completes the proof of the lemma. $\blacksquare$ ∎

Remark: More precise information about the asymptotics of $\alpha_{\varepsilon}$ as $\varepsilon\to 0^{+}$ can be obtain as follows. Following the notations in the previous proof and using Lemma 9,

[TABLE]

Since $u_{0}\in\mathcal{M}_{0}$, from (5.1) for $\varepsilon=0$,

[TABLE]

Therefore,

[TABLE]

The convergence of $\{t_{\varepsilon,u_{0}}\}_{\varepsilon>0}$ and $\{s_{\varepsilon,u_{0}}\}_{\varepsilon>0}$, as $\varepsilon\to 0^{+}$, implies that for some $\varepsilon_{0}>0$ small enough we have that $\{t_{\varepsilon,u_{0}}\}_{\varepsilon\in(0,\varepsilon_{0})}$ and $\{s_{\varepsilon,u_{0}}\}_{\varepsilon\in(0,\varepsilon_{0})}$ are bounded. Thus, fixing $\varepsilon_{0}>0$ as above and using (5.2), we find a constant $C>0$ such that for any $\varepsilon\in(0,\varepsilon_{0})$,

[TABLE]

Lemma 23**.**

There is a sequence $\{u_{\varepsilon_{n}}\}_{n\in\bm{\mathbb{N}}}$ converging strongly to some {\color[rgb]{1,0,0}{\rm u}_{0}}\in\mathcal{M}_{0} which is a least energy nodal solution for ($\mathcal{P}_{\varepsilon}$) with $\varepsilon=0$.

Proof.

This proof follows essentially the arguments of the proof of Lemma 11. From Lemma 22, we find that $\{u_{\varepsilon}\}_{\varepsilon>0}$ is bounded in $W^{1,p}_{0}(\Omega)$. The reflexivity of $W^{1,p}_{0}(\Omega)$ and the compactness of the Sobolev embedding yield the existence of ${\rm u}_{0}\in W^{1,p}_{0}(\Omega)$ such that, up to a subsequence $u_{\varepsilon_{n}}\rightharpoonup{\rm u}_{0}$ weakly in $W^{1,p}_{0}(\Omega)$ and $u_{\varepsilon_{n}}^{+}\to{\rm u}_{0}^{+}$, $u_{\varepsilon_{n}}^{-}\to{\rm u}_{0}^{-}$ strongly in $L^{r}(\Omega)$ for any $r\in[1,p^{*})$.

Vainberg’s lemma (see [21]) and condition (f1) implies that as $n\to\infty$,

[TABLE]

[TABLE]

For $n\in\bm{\mathbb{N}}$, denote $v_{n}:=u_{\varepsilon_{n}}$ and set $v_{0}={\rm u}_{0}$. By taking further subsequences if necessary, we may assume that $\{\|v_{n}^{+}\|\}_{n}$ and $\{\|v_{n}^{-}\|\}_{n}$ converge in $\mathbb{R}$ so that

[TABLE]

Also, observe that $v_{0}^{+}\geq 0$ and $v_{0}^{-}\leq 0$ a.e. in $\Omega$. Since $v_{n}\in\mathcal{M}_{\varepsilon_{n}}$,

[TABLE]

and using Lemma 5 we find that

[TABLE]

and hence $v_{0}^{\pm}\neq 0$ in $W^{1,p}_{0}(\Omega)$.

Next we prove that $v_{n}\to v_{0}$ strongly in $W^{1,p}_{0}(\Omega)$. It suffices to prove both equalities in (5.3) hold. To this end, let us argue by contradiction: assume either

[TABLE]

Set $a:=\tau_{0,v_{0}^{+}}$ and $b:=\tau_{0,v_{0}^{-}}$ the unique positive numbers such that $av_{0}^{+},bv_{0}^{-}\in\mathcal{N}_{0}$. Consequently, $av_{0}^{+}+bv_{0}^{-}\in\mathcal{M}_{0}$ and then

[TABLE]

This contradiction implies that both equalities hold in (5.3). Since $W_{0}^{1,p}(\Omega)$ is uniformly convex (see Theorem 2.6 in [1]) the convergence of $\{v_{n}\}_{n\in\bm{\mathbb{N}}}$ to $v_{0}$ is strong.

Finally, using that $v_{n}\to v_{0}$ strongly in $W^{1,p}_{0}(\Omega)$, we conclude that $DJ_{0}(v_{0}^{\pm})v_{0}^{\pm}=0$ and $J_{0}(v_{0})=\alpha_{0}$. This completes the proof. $\blacksquare$ ∎

Proof.

Proof of Theorem 3. Let $\{u_{\varepsilon_{n}}\}_{n\in\bm{\mathbb{N}}}$ and ${\rm u}_{0}$ be as in Lemma 23 so that $u_{\varepsilon_{n}}\in\mathcal{M}_{\varepsilon_{n}}\cap W^{2,p}(\Omega)$ with $J_{\varepsilon_{n}}=\alpha_{\varepsilon_{n}}$, ${\rm u}_{0}\in\mathcal{M}_{0}$ with $J_{0}({\rm u}_{0})=\alpha_{0}$ and $u_{\varepsilon_{n}}\to{\rm u}_{0}$ strongly in $W^{1,p}_{0}(\Omega)$.

Recall that

[TABLE]

for every $\varphi\in W^{1,p}_{0}(\Omega)$.

Since ${\rm u}_{0}^{+},{\rm u}_{0}^{+}\neq 0$, we test (5.5) against $\varphi={\rm u}_{0}^{\pm}$ and use hypothesis (f4) to conclude that $m_{0}({\rm u}_{0})\geq 2$. Thus, it suffices to show that for some subspace $V$ of $W^{1,p}_{0}(\Omega)$ with codimension two,

[TABLE]

We proceed as follows using the same notations as in the proof of Theorem 2 in Section 4.

For every $n\in\bm{\mathbb{N}}$, consider the functions $\Phi_{\varepsilon_{n}}^{\pm}:W^{1,p}_{0}(\Omega)\to\mathbb{R}$ defined by

[TABLE]

for $\varphi\in W^{1,p}_{0}(\Omega)$.

From Lemma 19, for every $\varphi\in W$,

[TABLE]

and

[TABLE]

We conclude then that the tangent space $T_{\varepsilon_{n}}(\mathcal{M}_{\varepsilon_{n}}\cap W)$ in $W$, consists on functions $\varphi\in W$ such that

[TABLE]

and from the proof of part (i) in Theorem 2, for every $\varphi\in T_{u_{\varepsilon_{n}}}\left(\mathcal{M}_{\varepsilon_{n}}\cap W\right)$, $D^{2}J_{\varepsilon_{n}}(u_{\varepsilon_{n}})(\varphi,\varphi)\geq 0$.

Let $n\in\bm{\mathbb{N}}$ be arbitrary, but fixed and set

[TABLE]

We claim that $V_{n}=E_{n}$, where

[TABLE]

To prove the claim, we notice first that $V_{n}\subseteq E_{n}$. Also, from Lemma 21, ${\rm codim}V_{n}\leq 2$ in $W^{1,p}_{0}(\Omega)$.

In order to prove the reverse inclusion, we prove that ${\rm codim}E_{n}=2$. Arguing in the same fashion as in the proof of (ii) and (iii) in Lemma 7, we find that

[TABLE]

Therefore, given any $\varphi\in W^{1,p}_{0}(\Omega)$, we can set

[TABLE]

so that $w_{n}:=\varphi-au^{+}_{\varepsilon_{n}}-bu^{-}_{\varepsilon_{n}}$ satisfies $\Phi^{+}_{\varepsilon_{n}}(w_{n})=\Phi^{-}_{\varepsilon_{n}}(w_{n})=0$. This proves that given any $\varphi\in W^{1,p}_{0}(\Omega)$, there exist unique $a,b\in\mathbb{R}$ and $w_{\varepsilon_{n}}\in E_{n}$ such that $\varphi=au^{+}_{\varepsilon_{n}}+bu_{\varepsilon_{n}}^{-}+w_{\varepsilon_{n}}$, i.e.

[TABLE]

Even more, from (5.7) it follows that for any $\varphi\in E_{n}$, $a=b=0$.

Since $V_{n},E_{n}$ are closed subspaces of $W^{1,p}_{0}(\Omega)$, $V_{n}\subset E_{n}$ and

[TABLE]

we conclude that $V_{n}=E_{n}$ and this proves the claim.

We finish the proof of Theorem 3 by a limiting process. Define

[TABLE]

for $\varphi\in W^{1,p}_{0}(\Omega)$ and set

[TABLE]

Proceeding as above, it follows that ${\rm codim}V=2$ in $W^{1,p}_{0}(\Omega)$. We prove next that inequality (5.6) holds true for this choice of $V$. Let $\varphi\in V$ be arbitrary, but fixed. Using (5.7), we set

[TABLE]

so that $w_{n}:=\varphi-a_{n}u_{\varepsilon_{n}}^{+}-b_{n}u_{\varepsilon_{n}}^{-}\in V_{n}$.

Since $u_{{\varepsilon_{n}}}\to{\rm u}_{0}$ strongly in $W^{1,p}_{0}(\Omega)$, as $n\to\infty$ and $\Phi^{\pm}_{0}({\rm u}_{0}^{\pm})<0$,

[TABLE]

Consequently, as $n\to\infty$, $\varphi+a_{n}u_{\varepsilon_{n}}^{+}+b_{n}u_{\varepsilon_{n}}^{-}\to\varphi$ strongly in $W_{0}^{1,p}(\Omega)$. On the other hand, notice that

[TABLE]

Since $u_{\varepsilon_{n}}\to{\rm u}_{0}$ strongly in $W^{1,p}_{0}(\Omega)$ and $a_{n},b_{n}\to 0$, as $n\to\infty$, we conclude that $D^{2}J_{\varepsilon_{n}}(u_{\varepsilon_{n}})(w_{n},w_{n})\to D^{2}J_{0}({\rm u}_{0})(\varphi,\varphi)$ as $n\to\infty$. Using that $w_{n}\in V_{n}$ and that $D^{2}J_{\varepsilon_{n}}(u_{\varepsilon_{n}})(\cdot,\cdot)|_{V_{n}}\geq 0$, we conclude that $D^{2}J_{0}({\rm u}_{0})(\varphi,\varphi)\geq 0$. Since $\varphi\in V$ is arbitrary and $V$ has codimension two in $W^{1,p}_{0}(\Omega)$, we find that $m_{0}(u_{0})\geq 2$.

Therefore, $m_{0}({\rm u}_{0})=2$. Finally, Lemma 12 implies that ${\rm u}_{0}$ has exactly two nodal domains. This completes the proof of the theorem. $\blacksquare$ ∎

6. Proof of Theorem 4.

We begin this section with some comments that are crucial for subsequent developments.

• (a)

From Lemma 10, for every $\varepsilon\geq 0$, any local minimizer of $J_{\varepsilon}|_{\mathcal{M}_{\varepsilon}}$ is a critical point of $J_{\varepsilon}$.

• (b)

In particular, for $\varepsilon=0$, if $u\in\mathcal{M}_{0}$ is a local minimizers of $J_{0}|_{\mathcal{M}_{0}}$, which is in addition isolated critical point of $J_{0}$, then it must be a strict local minimizer of $J_{0}|_{\mathcal{M}_{0}}$.

• (c)

Since $p>2$ and $f\in C^{1}(\mathbb{R})$, the energy $J_{0}:W^{1,p}_{0}(\Omega)\to\mathbb{R}$ and it derivative $DJ_{0}:W^{1,p}_{0}(\Omega)\to(W^{1,p}_{0}(\Omega))^{*}$ are uniformly continuous on bounded subsets of $W^{1,p}_{0}(\Omega)$.

Proof.

Proof of Theorem 4. For the sake of clarity, we present this proof splited in five steps. Let $f$ satisfy (f1)-(f4).

Claim 1. Let $R>0$ and $\varepsilon^{\prime}>0$ be arbitrary, but fixed. There exists $K>0$, depending only on $R$ such that for every $\varepsilon\in[0,\varepsilon^{\prime}]$ and every $v\in B_{R}(0)\cap\mathcal{N}_{\varepsilon^{\prime}}$,

[TABLE]

where $\tau_{\varepsilon,v},\tau_{0,v}>0$ are such that $\tau_{\varepsilon,v}v\in\mathcal{N}_{\varepsilon}$ and $\tau_{0,v}v\in\mathcal{N}_{0}$ are the scalars described in Lemma 6.

Proof Claim 1. First we remark that the implicit function theorem applied to the function $\xi$, introduced in the proof of Lemma 8, yields that the function

[TABLE]

is continuously differentiable with

[TABLE]

Using hypothesis (f4), we conclude that both integrals on the denominator of (6.2) are positive and therefore for any fixed $v\in W^{1,p}_{0}(\Omega)$, $v\neq 0$, the function $[0,\infty)\ni\varepsilon\mapsto\tau_{\varepsilon,v}>0$ is a strictly decreasing function in $\varepsilon$.

Let $\varepsilon\in[0,\varepsilon^{\prime}]$ and $v\in\mathcal{N}_{\varepsilon^{\prime}}\cap B_{R}(0)$. Part (ii) in Lemma 6 implies that $DJ_{\varepsilon}(v)(v)\leq 0$ and part (iii) from the same Lemma yields $\tau_{\varepsilon,v}\in(0,1]$.

Using the Mean Value Theorem and (6.1), we find $\eta\in(0,\varepsilon)$ such that

[TABLE]

Observe that

[TABLE]

Also, We rewrite the second term of the denominator in (6.2) as

[TABLE]

From part (iii) in Lemma 5,

[TABLE]

and since $0<\tau_{\eta,v}\leq\tau_{\varepsilon,v}\leq 1$ and $\tau_{\eta,v}v\in\bigcup_{\hat{\varepsilon}\geq 0}\mathcal{N}_{\hat{\varepsilon}}$,

[TABLE]

where

[TABLE]

Next, we prove that $M>0$. Arguing by contradiction, assume that $\{w_{n}\}_{n\in\bm{\mathbb{N}}}\subset B_{R}(0)$ with $\|w_{n}\|_{L^{q}(\Omega)}\geq\rho$ is a minimizing sequence for (6.5) such that $\lim\limits_{n\to\infty}I(w_{n})=0$.

The compactness of the Sobolev embedding $W^{1,p}_{0}(\Omega)\hookrightarrow L^{q}(\Omega)$ implies that there exists $w\in W_{0}^{1,p}(\Omega)$ such that, up to a subsequence, $w_{n}\to w$ strongly in $L^{q}(\Omega)$. In particular, $\|w\|_{L^{q}(\Omega)}\geq\rho>0$

On the other hand, Vainberg’s Lemma implies that

[TABLE]

Using again hypothesis (f4),

[TABLE]

and consequently $w=0$ a.e. in $\Omega$. This is clearly a contradiction and hence $M>0$.

From (6.2),(6.3) and (6.4), we conclude that

[TABLE]

and this proves the claim.

Next, let $u\in\mathcal{M}_{0}$ be a strict local minimizer of $J_{0}|_{\mathcal{M}_{0}}$ so that there exists $r>0$ with

[TABLE]

Claim 2. For every $s_{1},s_{2}\in(0,r)$ with $s_{1}<s_{2}$,

[TABLE]

Proof of Claim 2.

Proceeding by contradiction, assume the existence of a sequence $\{v_{n}\}_{n\in\bm{\mathbb{N}}}\subset\mathcal{M}_{0}$ such that $\|v_{n}-u\|_{W_{0}^{1,p}(\Omega)}\in[s_{1},s_{2}]$ and $J_{0}(v_{n})\to J_{0}(u)$, as $n\to\infty$.

Since $\{v_{n}\}_{n\in\bm{\mathbb{N}}}$ is bounded in $W^{1,p}_{0}(\Omega)$, there exists a subsequence, which we denote the same, and there exists $v\in\overline{B_{s_{2}}(u)}$ such that $v_{n}\rightharpoonup v$ weakly in $W^{1,p}_{0}(\Omega)$.

Using that $u\in\mathcal{M}_{0}$ is a least energy nodal solution for $(\mathcal{P}_{0})$ and part (ii) in Lemma 11, we conclude that $v_{n}\to v$ strongly in $W_{0}^{1,p}(\Omega)$. Therefore, $v\in\mathcal{M}_{0}$, $\|v-u\|_{W_{0}^{1,p}(\Omega)}\in[s_{1},s_{2}]$ and $J_{0}(v)=J_{0}(u)$. This contradicts the fact $u$ is a strict minimizer in $B_{r}(u)$. This proves the claim.

Claim 3. For every $s\in(0,\frac{r}{2})$, there exists $\varepsilon_{0}>0$ such that for every $\varepsilon\in(0,\varepsilon_{0})$,

[TABLE]

is well defined and it is attained.

Proof Claim 3. Let $s\in(0,\frac{r}{2})$ be arbitrary, but fixed. Since the function

[TABLE]

is continuously differentiable, there exists $\hat{\varepsilon}_{0}>0$ and $s^{\prime}\in(0,s)$ such that for every $\varepsilon\in[0,\hat{\varepsilon}_{0})$ and every $v\in B_{s^{\prime}}(u)$,

[TABLE]

The continuity of the mappings

[TABLE]

allows to assume further that $s^{\prime}\in(0,s)$ is such that for every $v\in B_{s^{\prime}}(u)$, $v^{\pm}\neq 0$. In particular, $B_{s^{\prime}}(u)\cap\mathcal{M}_{\varepsilon}$ is non-empty and hence the infimum in (6.6) is well defined.

Next, we prove that the infimum in (6.6) is attained. Fix $s_{1}\in(0,s^{\prime})$ and set $s_{2}=\frac{r}{2}$. Claim 1, guarantees the existence of $C_{s_{1}}>0$ such that

[TABLE]

For any given $\varepsilon>0$, set $w_{\varepsilon}:=\tau_{\varepsilon,u^{+}}u^{+}+\tau_{\varepsilon,u^{-}}u^{-}$. Observe that $w_{\varepsilon}\in\mathcal{M}_{\varepsilon}$. The continuity in $\varepsilon\in\mathbb{R}$ of the function in (6.7), fixing $v=u^{+}$ and $v=u^{-}$, yields the existence of $\hat{\varepsilon}_{1}>0$ such that for $\varepsilon\in(0,\hat{\varepsilon}_{1})$, $w_{\varepsilon}\in B_{s_{1}}(u)$ and since and $J_{\varepsilon}(w_{\varepsilon})\to J_{0}(u)$ as $\varepsilon\to 0^{+}$, we may choose $\hat{\varepsilon}_{1}$ smaller if necessary, so that

[TABLE]

Finally, we claim that there exists $\hat{\varepsilon}_{2}>0$ such that for every $\varepsilon\in(0,\hat{\varepsilon}_{2})$ and for every $v\in\left(B_{s}(u)\setminus\overline{B_{s^{\prime}}(u)}\right)\cap\mathcal{M}_{\varepsilon}$,

[TABLE]

Let us assume for the moment that this last claim holds true. We finish the proof of Claim 3 proceeding as follows.

Set $\varepsilon_{0}:=\min\{\hat{\varepsilon}_{0},\hat{\varepsilon}_{1},\hat{\varepsilon}_{2}\}$. Let $\varepsilon\in(0,\varepsilon_{0})$ and $v\in\left(B_{s}(u)\setminus\overline{B_{s^{\prime}}(u)}\right)\cap\mathcal{M}_{\varepsilon}$ be arbitrary, but fixed.

Observe that $\tau_{0,v^{+}}v^{+}+\tau_{0,v^{-}}v^{-}\in B_{s_{2}}(u)\setminus\overline{B_{s_{1}}(u)}$. Combining (6.10), (6.11) and (6.9), the choice of $C_{s_{1}}$,

[TABLE]

We conclude that for every $\varepsilon\in(0,\varepsilon_{0})$ and every $v\in\left(B_{s}(u)\setminus\overline{B_{s^{\prime}}(u)}\right)\cap\mathcal{M}_{\varepsilon}$,

[TABLE]

Let $\varepsilon\in(0,\varepsilon_{0})$ and let $\{v_{n}\}_{n\in\bm{\mathbb{N}}}\subset\mathcal{M}_{\varepsilon}\cap\overline{B_{s}(u)}$ be a minimizing sequence for (6.6). We may assume with no loss of generality that $v_{n}\rightharpoonup v$ weakly in $W^{1,p}_{0}(\Omega)$ for some $v$ satisfying that $\|v-u\|_{W^{1,p}_{0}(\Omega)}\leq s$.

Therefore, from (6.8) and the part (ii) in Lemma 11, the infimum (6.6) is attained at $v\in\overline{B_{s}(u)}\cap\mathcal{M}_{\varepsilon}$. Finally, from (6.12) we conclude that $v\in B_{s^{\prime}}(u)$ and this completes the proof of Claim 3.

Next, we prove (6.11). Notice first that for any $\varepsilon>0$ and any $v\in\mathcal{M}_{\varepsilon}$, $\tau_{0,v^{+}}\leq\tau_{\varepsilon,v^{+}}=1$ and $\tau_{0,v^{-}}\leq\tau_{\varepsilon,v^{-}}=1$. Thus, the Mean Value Theorem and Claim 1, with $R=r$, imply that for any $\varepsilon>0$ and any $v\in B_{r}(0)\cap\mathcal{M}_{\varepsilon}$,

[TABLE]

and

[TABLE]

Consequently, using part (c) in the last remark, we find $C_{r}>0$ such that for any $\varepsilon>0$ and any $v\in\left(B_{s}(u)\setminus\overline{B_{s^{\prime}}(u)}\right)\cap\mathcal{M}_{\varepsilon}$

[TABLE]

Therefore, there exists $\hat{\varepsilon}_{2}>0$ such that for $\varepsilon\in(0,\hat{\varepsilon}_{2})$ small enough such that, (6.11) is satisfied.

Proof of (i)-(iii). Let $\varepsilon_{0}>0$ small be as in Claim 3. For any $\varepsilon\in(0,\varepsilon_{0})$, Claim 3 and Lemma 10 implies that the infimum in (6.6) is actually attained by a critical point $u_{\varepsilon}\in\mathcal{M}_{\varepsilon}\cap B_{s}(u)$ of $J_{\varepsilon}$.

On the other hand, from Theorem 2, $m_{\varepsilon}(u_{\varepsilon})=2$, $u_{\varepsilon}$ is sign changing and $u_{\varepsilon}$ has exactly two nodal domains. To finish the proof of this step, take $s=s_{n}$ in (6.6), with $s_{n}\to 0^{+}$ as $n\to\infty$. Applying Claim 3 and the previous argument to the sequence $\{s_{n}\}_{n\in\bm{\mathbb{N}}}$, we find the desired sequence $\{u_{\varepsilon_{n}}\}_{n\in\bm{\mathbb{N}}}$. This concludes the proof of this step.

**Proof of **(iv). Using parts (i)-(iii) from this theorem, we can select an approximating sequence $\{u_{\varepsilon_{n}}\}_{n\in\bm{\mathbb{N}}}\subset\mathcal{M}_{\varepsilon_{n}}$ with $u_{\varepsilon_{n}}\to u_{0}$ and $\varepsilon_{n}\to 0^{+}$, as $n\to\infty$. The rest of the proof follows the same lines as in the proof of Theorem 3 with only slight and obvious changes. This completes the proof of (iv) and thus the proof of the theorem. $\blacksquare$ ∎

Remark: The ideas of the proof of Theorem 3 can be easily addapted to show that limit points of least energy nodal solutions in which the energy functional behaves like in the case $\varepsilon\neq 0$ also have Morse index 2. More precisely, if $w\in\mathcal{M}_{0}$ with $J_{0}(w)=\alpha_{0}$ and there is a sequence of $w_{n}\in\mathcal{M}_{\varepsilon_{n}}$ such that:

• (i)

$\varepsilon_{n}\to 0$ (with $\varepsilon_{n}$ possibly 0).

• (ii)

$J_{\varepsilon_{n}}(w_{n})=\alpha_{n}$.

• (iii)

$w_{n}\to w$ strongly in $W_{0}^{1,p}(\Omega)$.

• (iv)

$D^{2}J(w_{n})|_{V_{n}}\geq 0$ with $V_{n}$ defined as in Theorem 3 (including the case $\varepsilon_{n}=0$).

Then we have that $m_{0}(w)=2$. In particular, combining this idea with Theorem 4 we conclude that least energy nodal solutions in $\mathcal{M}_{0}$ which are limit points of isolated least energy nodal solutions in $\mathcal{M}_{0}$ also have Morse index 2.

Acknowledgments.