Least energy sign-changing solution of fractional $p$-Laplacian problems involving singularities

TL;DR

This paper establishes the existence of the least energy sign-changing solutions for a nonlocal fractional p-Laplacian problem with singularities using variational and topological methods.

Contribution

It introduces a novel approach combining Nehari manifold, variational constraints, and Brouwer degree to handle singular nonlocal PDEs.

Findings

Existence of least energy sign-changing solutions proven

Application of Nehari manifold and degree theory to nonlocal PDEs

Handling of singularities in fractional p-Laplacian context

Abstract

In this paper we study the existence of a least energy sign-changing solution to a nonlocal elliptic PDE involving singularity by using the Nehari manifold method, the constraint variational method and Brouwer degree theory.

Equations

$$(P)\{\begin{array}{rl}(-{\Delta }_p{)}^{\alpha }u& =\lambda g(u)+f(x,u)\text{mbox}in\Omega ,\\ u& =0\text{mbox}in{\mathbb{R}}^N\setminus \Omega ,\end{array}$$

$$(-{\Delta }_p{)}^{\alpha }u(x)=C_{N,\alpha }P.V.{\int }_{{\mathbb{R}}^N}\frac{|u(x)-u(y){|}^{p-2}(u(x)-u(y))}{|x-y{|}^{N+p\alpha }}dy,x\in {\mathbb{R}}^N,$$

$$(-{\Delta }_p{)}^{s}u$$

$$u$$

$$\displaystyle\left\{\begin{array}[]{ll}(-\Delta)_{p}^{\alpha}u=f(x,u){}{}&\mbox{in}~{}\Omega,\\ u=0{}{}&\mbox{on}~{}\mathbb{R}^{N}\setminus\partial\Omega,\end{array}\right.$$

$$\Vert u{\Vert }_{W^{\alpha ,p}(\Omega )}=\Vert u{\Vert }_{L^p(\Omega )}+({\int }_{\Omega \times \Omega }\frac{|u(x)-u(y){|}^p}{|x-y{|}^{N+\alpha p}}dxdy{)}^{\frac{1}{p}}.$$

$$X$$

$$\Vert u{\Vert }_X$$

$$X_0$$

$$\Vert u\Vert$$

$$S=\underset{u\in X_0\setminus \{0\}}{\inf }\frac{{\int }_Q\frac{|u(x)-u(y){|}^p}{|x-y{|}^{N+p\alpha }}dxdy}{{\left({\int }_{\Omega }|u{|}^{p_{\alpha }^{\ast }}dx\right)}^{\frac{p}{p_{\alpha }^{\ast }}}}$$

$$|f(x,u)|\le C_0(1+|u{|}^{q-1}),\forall u\in \mathbb{R},\forall x\in \Omega ;$$

$$t^2{f}^{\prime }(t)-(p-1)f(t)t>0,\forall |t|>0.$$

$${\int }_Q\frac{|u(x)-u(y){|}^{p-2}(u(x)-u(y))(\varphi (x)-\varphi (y))}{|x-y{|}^{N+p\alpha }}dxdy-\lambda {\int }_{\Omega }g(u)\varphi -{\int }_{\Omega }f(x,u)\varphi =0$$

$$I_{\lambda }(u)=\frac{1}{p}{\int }_{\Omega }\frac{|u(x)-u(y){|}^p}{|x-y{|}^{N+p\alpha }}dxdy-\lambda {\int }_{\Omega }{G}_{\lambda }(u)dx-{\int }_{\Omega }{F}_{\lambda }(x,u)dx.$$

$$\Lambda =\inf \{\lambda >0:\text{The problem (ref1.1) has no weak solution}\}.$$

$$(-{\Delta }_p{)}^{s}w$$

$$w$$

$$\overline{g}(t)=\{\begin{array}{ll}g(t),& \text{if}t>{\underline{u}}_{\lambda }\\ g({\underline{u}}_{\lambda }),& \text{if}t\le {\underline{u}}_{\lambda }\end{array}$$

$$\overline{f}(x,t)=\{\begin{array}{ll}f(x,t),& \text{if}t>{\underline{u}}_{\lambda }\\ f(x,{\underline{u}}_{\lambda }),& \text{if}t\le {\underline{u}}_{\lambda }\end{array}$$

$$\Phi (u)=\frac{1}{p}\Vert u{\Vert }^p-\lambda {\int }_{\Omega }G(u)dx-{\int }_{\Omega }F(x,u)dx.$$

$$\zeta (u)$$

$$\mathcal{N}$$

$$\mathcal{M}=\{u\in X_0:u^{\pm }\ne 0,⟨{\Phi }^{\prime }(u),u^{+}⟩=⟨{\Phi }^{\prime }(u),u^{-}⟩=0\},$$

$$\begin{array}{rl}& (|\xi {|}^{p-2}\xi -|\eta {|}^{p-2}\eta )(\xi -\eta )\ge d_1|\xi -\eta {|}^p,\\ & ||\xi {|}^{p-2}\xi -|\eta {|}^{p-2}\eta |\le d_2(|\xi |+|\eta |{)}^{p-2}|\xi -\eta |\end{array}$$

$$\begin{array}{rl}& (|\xi {|}^{p-2}\xi -|\eta {|}^{p-2}\eta )(\xi -\eta )\ge d_3\frac{|\xi -\eta {|}^2}{(|\xi |+|\eta |{)}^{2-p}},\\ & ||\xi {|}^{p-2}\xi -|\eta {|}^{p-2}\eta |\le d_4|\xi -\eta {|}^{p-1}\end{array}$$

$$⟨(-{\Delta }_p{)}^{s}v,\varphi ⟩-{\int }_{\Omega }\frac{\lambda \varphi }{v^{\gamma }}dx$$

$$⟨(-{\Delta }_p{)}^{s}v-(-{\Delta }_p{)}^{s}u,(u-v{)}^{+}⟩-{\int }_{\Omega }\lambda (u-v{)}^{+}\left(\frac{1}{v^{\gamma }}-\frac{1}{u^{\gamma }}\right)dx$$

$$|b{|}^{p-2}b-|a{|}^{p-2}a$$

$$|u(x)-u(y){|}^{p-2}(u(x)-u(y))-|v(x)-v(y){|}^{p-2}(u(x)-u(y))$$

$$=(p-1)\{(u(y)-v(y))-(u(x)-v(x))\}Q(x,y)$$

Full text

keywords: Sign-changing solutions; Fractional $p$-Laplacian; Nehari manifold.

AMS classification: 35J60, 35J92, 35R11, 47J30.

1 Introduction and Main results

In this paper we consider the following fractional $p$-Laplacian problem involving singularity and a power nonlinearity.

[TABLE]

where, $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with smooth boundary $\partial\Omega$, $\lambda>0$, $p\in(1,+\infty)$, $\alpha\in(0,1)$, $N>p\alpha$, $0<\delta<1,$ $f:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ is continuous, $g:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is continuous, nonincreasing on $(0,+\infty)$ such that $c_{1}=\liminf_{t\to 0^{+}}g(t)t^{\delta}\leq\limsup_{t\to 0^{+}}g(t)t^{\delta}=c_{2}$, for some $c_{1},c_{2}>0$ and the fractional $p$-Laplacian operator, $(-\Delta)_{p}^{\alpha}$ is defined as,

[TABLE]

where $C_{N,\alpha}$ is a normalizing constant.

One of the classical topic in the analysis of PDEs is the study of existence and multiplicity of nonnegative solutions for both the $p$-Laplacian and the fractional $p$-Laplacian operator involving concave-convex nonlinearity and singularity-power nonlinearity. In the recent past there has been considerable interest in studying the following general fractional $p$-Laplacian problem involving singularity.

[TABLE]

where $N>ps$, $M\geq 0$, $a:\Omega\rightarrow\mathbb{R}$ is a nonnegative bounded function. Ghanmi & Saoudi [15] guaranteed the existence of multiple weak solutions to the problem (1), for $0<\gamma<1$ and $1<p-1<q\leq p_{s}^{*}$ by using the Nehari manifold method. Recently, multiplicity and Hólder regularity of solutions to the problem (1) has been studied by Saoudi et al. [25]. On the other hand, for $p=2$, the problems of the type (1), have been investigated by many researchers. For references see [22, 24, 25] and the references therein.

The existence of a sign-changing solution of nonlinear elliptic PDEs with power nonlinearities has been studied extensively for the $p$-Laplacian operator as well as the fractional $p$-Laplacian operator. We refer the reader to see [2, 4, 5, 10, 14, 20, 29, 31] and the references therein. Consider the nonlocal problem

[TABLE]

For $p=2$, the authors in [9], has studied the problem (1.5), where the fractional Laplacian operator is defined through spectral decomposition to obtain the sign-changing solution. The method of harmonic extension was introduced by Caffarelli and Silvestre [7] to transform the nonlocal problem in $\Omega$ to a local problem in the half cylinder $\Omega\times(0,+\infty)$, by using an equivalent definition of the fraction Laplacian operator [6].

For $p\in(1,\infty)$, the problem studied by Chang et al. [10], where the authors have guaranteed the existence of a sign-changing solutions by using Nehari manifold method. Recently, the study of the nonlocal problems with singularity has drawn interest to many researchers. For recent studies on nonlocal PDEs involving singularities, we refer [11, 12, 16, 17, 18, 19, 25] and the references therein.

The main goal of this article is to obtain a sign-changing solutions to the nonlocal problem (1.1) involving singularity. For $p\neq 2$, the harmonic extension method can non be applied on an equivalent definition of$(-\Delta)_{p}^{\alpha}$. On a similar note, we can not have the decomposition $\Phi(u)=\Phi(u^{+})+\Phi(u^{-})$ for $u=u^{+}+u^{-}$, where $\Phi$ is the corresponding energy functional to the problem (1.1). Therefore, by using the method as in [9], one can not guarantee the existence of a sign-changing solution.

Therefore, we will apply the Nehari manifold method combining with a constrained variational method and Brouwer degree theory to obtain a least energy sign-changing solution.

We first recall some preliminary results on the fractional Sobolev space [1, 13]. Let $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with smooth boundary and $\alpha\in(0,1)$. We denote the fractional Sobolev space by $W^{\alpha,p}(\Omega)$ equipped with the norm

[TABLE]

We set, $Q=\mathbb{R}^{2N}\setminus((\mathbb{R}^{N}\setminus\Omega)\times(\mathbb{R}^{N}\setminus\Omega))$, then the space $(X,\|..\|_{X})$ is defined by

[TABLE]

equipped with the Gagliardo norm

[TABLE]

Here $\|u\|_{p}$ refers to the $L^{p}$-norm of $u$. We then define the space

[TABLE]

equipped with the norm

[TABLE]

The best Sobolev constant is defined as

[TABLE]

For $p>1$, the space $X_{0}$ is a uniformly convex Banach space [26, 27] and the embedding $X_{0}\hookrightarrow L^{q}(\Omega)$ is compact for $q\in[1,p_{\alpha}^{*})$ and is continuous for $q\in[1,p_{\alpha}^{*}]$, where $p_{\alpha}^{*}$ is the Sobolev conjugate of $p$, defined as $p_{\alpha}^{*}=\frac{Np}{N-p\alpha}$.

Henceforth, we have the following assumptions on $f$ and $g$.

($f_{1}$)

$f\in C(\overline{\Omega}\times\mathbb{R})$, $\lim\limits_{|u|\to 0}\frac{f(x,u)}{|u|^{p-2}u}=0$, uniformly in $x$;

($f_{2}$)

there exist constants $C_{0}>0$ and $q\in(p,p_{\alpha}^{*})$ with $p_{\alpha}^{*}=\frac{pN}{N-p\alpha}$ such that

[TABLE]

($f_{3}$)

there exist $\mu>p$ and $M_{0}>0$ such that $f(x,u)u\geq\mu F(x,u)>0$ for $|u|\geq M_{0}$, uniformly in $x$, where $F(x,u)\doteq\int_{0}^{u}f(x,\tau)d\tau$;

($f_{4}$)

$\lim\limits_{|u|\rightarrow+\infty}\frac{f(x,u)}{|u|^{p-2}u}=+\infty$ uniformly in $x$;

($f_{5}$)

$\frac{f(x,u)}{|u|^{p-2}u}$ is strictly increasing on $(0,+\infty)$ and strictly decreasing on $(-\infty,0)$, uniformly in $x$.

($g_{1}$)

$g:\,\mathbb{R}\setminus\{0\}\to\mathbb{R}^{+}$ continuous on $\mathbb{R}\setminus\{0\}$, $g$ is nondecreasing on $(-\infty,0)$ and $g$ is nonincreasing on $(0,+\infty)$,

($g_{2}$)

$c_{1}\leq\liminf_{t\to 0^{+}}g(t)t^{\delta}\leq\limsup_{t\to 0^{+}}g(t)t^{\delta}=c_{2}$ for some $c_{1},c_{2}>0$ and

$d_{1}\leq\liminf_{t\to 0^{-}}g(t)t^{\delta}\leq\limsup_{t\to 0^{-}}g(t)t^{\delta}=d_{2}$ for some $d_{1},d_{2}>0$.

Remark 1.1**.**

By $(f_{5})$ it follows that

[TABLE] 2. 2.

From $(g_{2})$, $g$ is singular at the origin and $\lim_{t\to 0^{\pm}}g(t)=\pm\infty$.

We now define a weak solution to the problem defined in (1.1).

Definition 1.2**.**

A function $u\in X_{0}$ is a weak solution to the problem (1.1), if

[TABLE]

for each $\phi\in X_{0}.$ The corresponding Euler-Lagrange energy functional is

[TABLE]

It is easy to observe that $I_{\lambda}$ is not $C^{1}$ due the presence of the singular term in it but $I_{\lambda}$ is continuous and Gâteaux differentiable (see Corollary 6.3 of [25]). Therefore, we can not apply Nehari manifold method corresponding to the functional $I_{\lambda}$. Hence, we will establish the existence of a sign-changing solution to the problem (1.1) by obtaining a critical point to a $C^{1}$ cutoff functional. We define,

[TABLE]

We now state the existence of a unique solution due to [8] to the following problem.

[TABLE]

Lemma 1.3**.**

Assume $0<\delta<1$ and $\lambda>0$. Then the problem (1) has a unique solution, $\underline{u}_{\lambda}\in W_{0}^{s,p}(\Omega)$, such that for every $K\subset\subset\Omega$, $ess.\underset{K}{\inf}\,\underline{u}_{\lambda}>0$.

Define,

[TABLE]

and

[TABLE]

where, $\underline{u}_{\lambda}$ is the solution to (1). Let $G(s)=\int_{0}^{s}\overline{g}(t)dt$ and $F(x,s)=\int_{0}^{s}\overline{f}(x,t)dt$. We now define the energy functional $\Phi:X_{0}\to\mathbb{R}$ by

[TABLE]

Under the assumptions $(f_{1})-(f_{5})$ and $(g_{1})-(g_{2})$, the functional $\Phi$ is $C^{1}$ on $X_{0}$ (see Lemma 6.4 in [25]) and weakly lower semicontinuous by a standard arguments. Define

[TABLE]

where $X_{0}^{*}$ is the dual space of $X_{0}$. For simplicity, we will denote $\langle\cdot,\cdot\rangle_{X_{0}^{*},X_{0}}$ by $\langle\cdot,\cdot\rangle$. Clearly, every nontrivial solutions of (1.1) belongs to $\mathcal{N}$.

Define the set of sign-changing solutions of (1.1) as

[TABLE]

where $u^{+}(x)\doteq\max\{u(x),0\}$, $u^{-}(x)\doteq\min\{u(x),0\}$. We set $m_{\alpha}\doteq\inf\limits_{u\in\mathcal{M}}\Phi(u)$ and $c_{\alpha}\doteq\inf\limits_{u\in\mathcal{N}}\Phi(u)$. The main result proved in this article is the following.

Theorem 1.4**.**

Suppose that the assumptions $(f_{1})-(f_{5})$ and $(g_{1})-(g_{2})$ holds. Then there exists a $\Lambda>0$, such that for $\lambda\in(0,\Lambda)$, the problem (1.1) admits one sign-changing solution $u^{*}\in X_{0}$ and $\Phi(u^{*})=m_{\alpha}$.

The paper is organized as follows. In Section 2, we present some useful notations and give some preliminary results. In Section 3, we apply the method of Nehari manifold to prove Theorem 1.4. Throughout the paper, we always denote by $C_{1},C_{2},\cdots$ positive constants (possibly different in different places) and let $|\cdot|_{p}$ denote the usual $L^{p}(\Omega)$ norm for all $p\in[1,+\infty]$.

2 Important Lemmas

We begin this section with the following Lemma.

Lemma 2.1**.**

Assume $0<\delta<1<q<p_{s}^{*}-1$. Then $0<\varLambda<\infty$.

Proof.

This result can be proved by working on the similar lines as of [25]. ∎

The following Lemma due to [3], will be useful in the proof of Theorem 1.4.

Lemma 2.2**.**

1. (i)

For $2<p<\infty$, there exists $d_{1},d_{2}>0$ such that, for all $\xi,\eta\in\mathbb{R}^{N}$,

[TABLE] 2. (ii)

For $1<p\leq 2$, there exist $d_{3},d_{4}>0$ such that, for all $\xi,\eta\in\mathbb{R}^{N}$,

[TABLE]

We have the following comparison principle for the fractional $p$-Laplacian operator.

Lemma 2.3** (Weak Comparison Principle).**

Let $u,v\in X_{0}$. Suppose, $(-\Delta_{p})^{s}v-\frac{\lambda}{v^{\gamma}}\geq(-\Delta_{p})^{s}u-\frac{\lambda}{u^{\gamma}}$ weakly with $v=u=0$ in $\mathbb{R}^{N}\setminus\Omega$. Then $v\geq u$ in $\mathbb{R}^{N}.$

Proof.

Since, $(-\Delta_{p})^{s}v-\frac{\lambda}{v^{\gamma}}\geq(-\Delta_{p})^{s}u-\frac{\lambda}{u^{\gamma}}$ weakly with $u=v=0$ in $\mathbb{R}^{N}\setminus\Omega$, we have

[TABLE]

In particular choose $\phi=(u-v)^{+}$. To this choice, (2.3) looks as follows.

[TABLE]

Let $\psi=u-v$. The identity

[TABLE]

with $a=v(x)-v(y)$, $b=u(x)-u(y)$ gives

[TABLE]

where

[TABLE]

We choose the test function $\phi=(u-v)^{+}$. We express,

[TABLE]

to further obtain

[TABLE]

The equation (2.8) implies

[TABLE]

This leads to the conclusion that the Lebesgue measure of $\Omega^{+}$, i.e., $|\Omega^{+}|=0$. In other words $v\geq u$ a.e. in $\Omega$. ∎

3 Proof of Theorem 1.4

In this section we prove the existence of a sign-changing solution for (1.1) by obtaining a minimizer of the energy functional $\Phi$ over

[TABLE]

Further we will verify that the obtained minimizer is a sign-changing solution to (1.1). Since, it is difficult to show that $\mathcal{M}\neq\emptyset$, we will prove that $\mathcal{M}\neq\emptyset$ by using the parametric method. We prove that, if $u\in X_{0}$ with $u^{\pm}\neq 0$, the there exists a unique pair $(s,t)\in\mathbb{R}^{+}\times\mathbb{R}^{+}$, such that $su^{+}+tu^{-}\in\mathcal{M}$. Finally to conclude that the minimizer of the constrained problem is a sign-changing solution, we use the quantitative deformation lemma (see Lemma 2.3 of [32]) and Brouwer degree theory.

Lemma 3.1**.**

Let the assumptions of Theorem 1.4 holds, then there exist $\mu_{1},\mu_{2}>0$ and $\lambda_{1}>0$ such that

(i)

$\|u^{\pm}\|\geq\mu_{1},\forall u\in\mathcal{M}$;

(ii)

$\int_{\Omega}|u^{\pm}|^{q}dx\geq\mu_{2},\forall u\in\mathcal{M}$.

Proof.

We have $\langle\Phi^{\prime}(u),u^{\pm}\rangle=0$ for every $u\in\mathcal{M}$. Therefore,

[TABLE]

By a simple computation one can obtain

[TABLE]

where,

[TABLE]

Therefore, $\langle\Phi^{\prime}(u^{+}),u^{+}\rangle<0$, and hence it follows that

[TABLE]

Similarly, we obtain

[TABLE]

Now by the assumptions $(f_{1})-(f_{2})$, we have for every $\epsilon>0$, there exists $C_{\epsilon}>0$ such that

[TABLE]

Therefore, by the Sobolev inequality and the growth condition of $g$, there exists $C_{1}\,,C_{2}>0$ such that

[TABLE]

We now choose, $\lambda>0$ (say, $\lambda_{1}$) very small such that

[TABLE]

Since, $q\in(p,p_{\alpha}^{*})$, by (3.2) and (3.3) and for $\epsilon=\frac{1}{2C_{1}}$, one can see (i) holds. Again, by (3.1), (3.2) and (3.3), we have

[TABLE]

Therefore, for $\epsilon=\frac{1}{2C_{1}}$, we can obtain that

[TABLE]

This completes the proof. ∎

Lemma 3.2**.**

Let $u\in X_{0}$ be such that $u^{\pm}\neq 0$. Then there exists a unique pair $(t_{u},s_{u})\in\mathbb{R^{+}}\times\mathbb{R^{+}}$ such that $t_{u}u^{+}+s_{u}u^{-}\in\mathcal{M}$.

Proof.

For every $t,s>0$, let us define $g_{1}$ and $g_{2}$ as

[TABLE]

and

[TABLE]

Now by using $(f_{4})$, we have for any $C_{1}>0$, there exists $C_{2}>0$ such that

[TABLE]

Therefore, by using $q\in(p,p_{\alpha}^{*})$, (3.1), (3.4) and Lemma 3.1, there exist $r_{1}>0$, $\lambda>0$ small enough and $R_{1}>0$ large enough such that

[TABLE]

Observe that, for a fixed $t>0$, $g_{1}(t,s)$ is increasing in $s$ on $(0,+\infty)$ and for a fixed $s>0$, $g_{2}(t,s)$ is increasing in $t$ on $(0,+\infty)$. Therefore, by using (3.5) and (3.6) there exist $\lambda>0$, $r>0$ and $R>0$ with $r<R$ such that

[TABLE]

Now, on applying the Miranda’s theorem [21], $g_{1}(t_{u},s_{u})=g_{2}(t_{u},s_{u})=0$, for some $t_{u},s_{u}\in[r,R]$. This implies that $t_{u}u^{+}+s_{u}u^{-}\in\mathcal{M}$.

We now prove the uniqueness. Assume there exists $(t_{1},s_{1})$ and $(t_{2},s_{2})$ such that $t_{i}u^{+}+s_{i}u^{-}\in\mathcal{M}$, $i=1,2$. We prove the uniqueness by dividing into two cases.

Case 1. Let $u\in\mathcal{M}$.

Without loss of generality, we assume $(t_{1},s_{1})=(1,1)$ and $t_{2}\leq s_{2}$. Now, for $u\in X_{0}$, we define

[TABLE]

and

[TABLE]

Since, $u\in\mathcal{M}$, therefore, by using $\langle\Phi^{\prime}(u),u^{+}\rangle=\langle\Phi^{\prime}(u),u^{-}\rangle=0$, we get

[TABLE]

Again by using $<\Phi^{\prime}(t_{2}u^{+}+s_{2}u^{-}),t_{2}u^{+}>=0=<\Phi^{\prime}(t_{2}u^{+}+s_{2}u^{-}),s_{2}u^{-}>$ we have

[TABLE]

where,

[TABLE]

Furthermore, $t_{2}\leq s_{2}$, implies that $B_{1}^{+}(u),B_{2}^{+}(u)\geq 0$. Hence, from (3.9) and (3.11), we have

[TABLE]

where, $I_{1}=\left(\frac{f(x,t_{2}u^{+})}{|t_{2}u^{+}|^{p-2}t_{2}u^{+}}-\frac{f(x,u^{+})}{|u^{+}|^{p-2}u^{+}}\right)|u^{+}|^{p}$ and $I_{2}=\lambda\left(\frac{g(t_{2}u^{+})}{|t_{2}u^{+}|^{p-2}t_{2}u^{+}}-\frac{g(u^{+})}{|u^{+}|^{p-2}u^{+}}\right)|u^{+}|^{p}$. Claim. $t_{2}\geq 1$.

To prove our claim, we consider the following four possibilities.

1. I.

When $I_{1}>0$, $I_{2}>0$: Now, $I_{1}>0$ implies that $t_{2}\geq 1$ by $(f_{5})$. Again, $I_{2}>0$ implies $t_{2}\leq 1$ by using $(g_{1})$-$(g_{2})$. Therefore, on combining both the cases, we get $t_{2}=1$. 2. II.

When $I_{1}>0$, $I_{2}<0$: Since, $I_{2}<0$, we have $\int_{\Omega}I_{1}+\int_{\Omega}I_{2}\leq\int_{\Omega}I_{1}$. Therefore, by $(f_{5})$, $I_{1}>0$ implies $t_{2}\geq 1$ and similarly, $I_{2}<0$ implies $t_{2}\geq 1$ by $(g_{1})$-$(g_{2})$. Thus $t_{2}\geq 1$. 3. III.

When $I_{1}<0$, $I_{2}>0$: Since, $I_{1}<0$, we may choose, $\lambda>0$ small enough such that $\int_{\Omega}I_{1}+\int_{\Omega}I_{2}\leq 0$, which is a contradiction to (3). 4. IV.

When $I_{1}<0$, $I_{2}<0$: In this case, both $I_{1}<0$ and $I_{2}<0$, yield $\int_{\Omega}I_{1}+\int_{\Omega}I_{2}\leq 0$, which is a contradiction to (3).

Therefore, we can conclude that $t_{2}\geq 1$. Again, from $B_{1}^{-}(u),B_{2}^{-}(u)\leq 0$, we have by (3.10) and (3.12) that

[TABLE]

on proceeding as the above proof together with $(f_{5})$, $(g_{1})$-$(g_{2})$, one can prove that $s_{2}\leq 1$. Hence, $t_{2}=s_{2}=1$.

Case 2. Let $u\notin\mathcal{M}$.

Let $v_{1}=t_{1}u^{+}+s_{1}u^{-}$ and $v_{2}=t_{2}u^{+}+s_{2}u^{-}$. Again, by using above arguments, it is easy to prove that $\frac{t_{2}}{t_{1}}=\frac{s_{2}}{s_{1}}=1$. Hence, $(t_{1},s_{1})=(t_{2},s_{2})$. This completes the proof. ∎

Lemma 3.3**.**

Assume $(f_{1})$-$(f_{5})$ and $(g_{1})$-$(g_{2})$ holds. Then there exists $u\in\mathcal{M}$ such that $\Phi(u)=m_{\alpha}$, where, $m_{\alpha}\doteq\inf\limits_{u\in\mathcal{M}}{\Phi(u)}$.

Proof.

Clearly, by the above Lemma 3.2, we have $\mathcal{M}\neq\emptyset$. Consider a minimizing sequence $\{u_{n}\}\subset\mathcal{M}$ such that $\Phi(u_{n})\to m_{\alpha}$ as $n\to+\infty.$

Claim: The sequence $\{u_{n}\}$ is uniformly bounded in $X_{0}$.

Proof. We will prove by contradiction. Let us assume that $\|u_{n}\|\rightarrow\infty$. We set $w_{n}=\frac{u_{n}}{\|u_{n}\|}$. Clearly, $\|z_{n}\|=1$, and upto a subsequence, there exists $w_{0}\in X_{0}$ such that

1. (i)

$w_{n}\rightarrow w_{0}$ in $X_{0}$, 2. (ii)

$w_{n}\rightarrow w_{0}$ in $L^{r}(\Omega)$ for all $r\in[1,p_{\alpha}^{*})$ and 3. (iii)

$w_{n}(x)\rightarrow w_{0}(x)$ almost everywhere in $\Omega$.

We further claim that $w_{0}=0$. Suppose not, define $\Omega_{1}=\{x\in\Omega:w_{0}(x)\neq 0\}$, then by $(f_{4})$ and Fatou’s lemma, we get,

[TABLE]

which is a contradiction. Thus, $w_{0}\equiv 0$. Therefore, $\{u_{n}\}$ is uniformly bounded in $X_{0}$. Then there exists $u^{*}\in X_{0}$ such that

[TABLE]

From Lemma 3.1, we have $(u^{*})^{\pm}\neq 0$. In addition, under the assumptions $(f_{1})$-$(f_{2})$ and $(g_{1})$-$(g_{2})$, by using the compact embedding of $X_{0}\hookrightarrow L^{r}(\Omega)$ for $r\in[1,p_{\alpha}^{*})$ and by applying some standard arguments (see [32]), we get that

[TABLE]

From Lemma 3.2, we have the existence of $t^{*},s^{*}>0$ such that $t^{*}(u^{*})^{+}+s^{*}(u^{*})^{-}\in\mathcal{M}$. This implies

[TABLE]

We now prove that $t^{*},s^{*}\leq 1$. Since, the minimizing sequence $\{u_{n}\}\subset\mathcal{M}$, we get $\langle\Phi^{\prime}(u_{n}),u_{n}^{\pm}\rangle=0$, which implies that

[TABLE]

Therefore, by using the above inequalities (3.14)-(3.21) and Fatou’s lemma, we obtain

[TABLE]

Furthermore, without loss of generality, assume $t^{*}\leq s^{*}$. Again from (3.22) and (3.24) and the fact $B_{1}^{-}(u^{*}),B_{2}^{-}(u^{*})\leq 0$, we get

[TABLE]

Now proceeding on similar arguments as in the proof of Lemma 3.2 and using (3), one can easily obtain $s^{*}\leq 1$. Therefore, we have $0<t^{*}\leq s^{*}\leq 1$.

Let us define, $\mathcal{H}(x,\tau)=f(x,\tau)\tau-pF(x,\tau)$ and $\mathcal{H}_{1}(\tau)=g(\tau)\tau-pG(\tau)$. Then, from $(f_{5})$, we have $\mathcal{H}(x,\tau)$ is increasing with respect to $\tau$ on $(0,+\infty)$, decreasing on $(-\infty,0)$ and $\mathcal{H}(x,\tau)\geq 0$. Again by $(g_{1})$-$(g_{2})$, we have $\mathcal{H}_{1}(\tau)\leq 0$ for $\tau\in\mathbb{R}\setminus\{0\}$. Therefore, by the definition of $\Phi$ and Fatou’s lemma, we get

[TABLE]

[TABLE]

Thus, we conclude $t^{*}=s^{*}=1$ and hence $\Phi(u^{*})=m_{\alpha}$. This completes the proof. ∎

Lemma 3.4**.**

Let $u\in\mathcal{M}$. Then for every $t,s\geq 0$ with $(t,s)\neq(1,1)$, we have

[TABLE]

Proof.

For each $u\in X_{0}$ such that $u^{\pm}\neq 0$, let us define $I_{u}:[0,+\infty)\times[0,+\infty)\to\mathbb{R}$ as

[TABLE]

Observe that from $(f_{4})$, we get

[TABLE]

Therefore, $I_{u}$ admits a global maximum at some $(t_{0},s_{0})\in[0,+\infty)\times[0,+\infty)$. We now prove that $t_{0}>0,s_{0}>0$ by showing that the other three possibilities can not hold, which are as follow.

(i)

$t_{0}=s_{0}=0$;

(ii)

$t_{0}>0,s_{0}=0$;

(iii)

$t_{0}=0,s_{0}>0$.

Let $s_{0}=0$. Since, $I_{u}$ has a global maximum at $(t_{0},s_{0})$, then $\Phi(t_{0}u^{+})\geq\Phi(tu^{+})$ for every $t>0$,. Therefore, we have $\langle\Phi^{\prime}(t_{0}u^{+}),t_{0}u^{+}\rangle=0$, which implies,

[TABLE]

Again, since $u\in\mathcal{M}$, we get $\langle\Phi^{\prime}(u^{+}),u^{+}\rangle<0$, i.e.

[TABLE]

Now, using this inequality and (3.26), we get

[TABLE]

Again, on repeating similar arguments as in Lemma 3.2, together with $(g_{1})$-$(g_{2})$ and $(f_{5})$, we get $t_{0}\leq 1$. Furthermore, $\mathcal{H}(x,\tau)\geq 0,\forall(x,\tau)\in\overline{\Omega}\times\mathbb{R}$ and $\mathcal{H}_{1}(\tau)\leq 0,\forall\,\tau\in\mathbb{R}\setminus\{0\}$. In addition, $\mathcal{H}(x,\tau)$ is increasing on $(0,+\infty)$ and decreasing on $(-\infty,0)$ with respect to $\tau$. Therefore, we have

[TABLE]

This contradicts that $I_{u}$ has a global maximum at $(t_{0},s_{0})$. Hence, $s_{0}>0$. Similarly, we can prove that $t_{0}>0$. Finally, Lemma 3.2, guarantees that $(1,1)$ is the unique critical point of $I_{u}$ in $(0,+\infty)\times(0,+\infty)$. This readily implies that, if $t_{0},s_{0}\in(0,1]$ such that $(t_{0},s_{0})\neq(1,1)$, then we have

[TABLE]

This completes the proof. ∎

The following lemma concludes the existence of a critical point of $\Phi$, which is a least energy solution to our problem.

Lemma 3.5**.**

Let there exists $u^{*}\in\mathcal{M}$ such that $\Phi(u^{*})=m_{\alpha}$. Then $u^{*}$ is a critical point of $\Phi$, i.e., $\Phi^{\prime}(u^{*})=0$.

Proof.

We will prove by method of contradiction. Let $\Phi^{\prime}(u^{*})\neq 0$, then there exist $\rho_{1},\mu_{1}>0$ such that

[TABLE]

where $B_{3\mu_{1}}(u^{*})=\{u\in X_{0}:\|u-u^{*}\|\leq 3\mu_{1}\}$ a closed ball of radius $3\mu$ in $X_{0}$ centered at $u^{*}$. Now, $u^{*}\in\mathcal{M}$ implies that $(u^{*})^{\pm}\neq 0$, then we can choose a sufficiently small $\mu_{1}>0$ such that $u^{\pm}\neq 0$ for all $u\in B_{3\mu_{1}}(u^{*})$. For sufficiently small $\delta_{1}\in(0,\frac{1}{2})$, let us define, $D=(1-\delta_{1},1+\delta_{1})\times(1-\delta_{1},1+\delta_{1})$ such that $t(u^{*})^{+}+s(u^{*})^{-}\in B_{3\mu_{1}}(u^{*})$ for all $(t,s)\in\overline{D}$. From Lemma 3.4, one can say

[TABLE]

Choose, $\epsilon_{1}\doteq\min\{\frac{m_{\alpha}-\tilde{m}_{\alpha}}{2},\frac{\rho_{1}\mu_{1}}{8}\}$. Therefore, from the quantitative deformation lemma (see Lemma 2.3 of [32]) it follows that there exists a continuous map $\eta:\mathbb{R}\times X_{0}\to X_{0}$ such that

(i)

$\eta(1,u)=u$ if $u\not\in\Phi^{-1}[m_{\alpha}-2\epsilon_{1},m_{\alpha}+2\epsilon_{1}]\cap B_{2\mu_{1}}(u^{*})$;

(ii)

$\eta(1,\Phi^{m_{\alpha}+\epsilon_{1}}\cap B_{\mu_{1}}(u^{*}))\subset\Phi^{m_{\alpha}-\epsilon_{1}}$;

(iii)

$\Phi(\eta(1,u))\leq\Phi(u),\forall u\in X_{0}$.

We define, $\sigma(t,s)\doteq\eta(1,t(u^{*})^{+}+s(u^{*})^{-}),\forall\,(t,s)\in\overline{D}$. Thus from the Lemma 3.4 together with (ii)-(iii) of the deformation lemma, we get

[TABLE]

which implies that $\{\sigma(t,s)\}_{(t,s)\in\overline{D}}\cap\mathcal{M}=\emptyset$. Again, we will prove by the following argument that $\{\sigma(t,s)\}_{(t,s)\in\overline{D}}\cap\mathcal{M}\neq\emptyset$ to arrive at a contradiction. Now, for $(t,s)\in\overline{D}$, we define

[TABLE]

Since $f,g\in C^{1}$, the functional $J_{1}$ is $C^{1}$. Therefore, from $\langle\Phi^{\prime}(u^{*}),(u^{*})^{\pm}\rangle=0$, we get

[TABLE]

[TABLE]

From $(g_{1})$-$(g_{2})$ and $(f_{5})$, we have $\mathcal{H}^{\prime}(x,\tau)\tau=f^{\prime}(x,\tau)\tau^{2}-(p-1)f(x,\tau)\tau>0$ for all $\tau\in\mathbb{R}\setminus\{0\}$. We denote

[TABLE]

It is easy to observe that

[TABLE]

Hence, we get

[TABLE]

Therefore, by using the Brouwer degree theory, we get $\deg(J_{1},D,0)=1$. Again, from (3.28), we have $\sigma(t,s)=t(u^{*})^{+}+s(u^{*})^{-},\forall\,(t,s)\in\partial D$. Hence,

[TABLE]

Therefore, there exists $(t_{0},s_{0})\in D$ such that $J_{2}(t_{0},s_{0})=0$. On using the conditions (i)-(ii) in the deformation lemma, one can obtain that

[TABLE]

Therefore, we can say $\langle\Phi^{\prime}(u_{0}),u_{0}^{+}\rangle=\langle\Phi^{\prime}(u_{0}),u_{0}^{-}\rangle=0$ such that $u_{0}^{\pm}\neq 0$, that is, $u_{0}\in\{\eta(t,s)\}_{(t,s)\in\overline{D}}\cap\mathcal{M}$. Hence, we have a contradiction. Thus we conclude that $u^{*}$ is a critical point of $\Phi$ and a least energy sign-changing solution of problem corresponding to $\Phi$. Finally, since the critical points of $\Phi$ are also critical points of $I_{\lambda}$, we have $u^{*}$ is a critical point of $I_{\lambda}$. Hence $u^{*}$ is a sign-changing solution to the problem (1.1). ∎

Acknowledgement

The author S. Ghosh, thanks the Council of Scientific and Industrial Research (C.S.I.R), India, for the financial assistantship received to carry out this research work.