Nontrivial solutions of quasilinear elliptic equations with natural growth term

TL;DR

This paper establishes the existence of multiple solutions for a class of quasilinear elliptic equations with natural growth terms, using an adapted degree theory approach under invariant assumptions.

Contribution

It introduces a novel adaptation of degree theory to prove multiple solutions for quasilinear elliptic equations with natural growth, invariant under diffeomorphisms.

Findings

Proved existence of multiple solutions.

Developed an adapted degree theory method.

Applicable under invariant assumptions.

Abstract

We prove the existence of multiple solutions for a quasilinear elliptic equation containing a term with natural growth, under assumptions that are invariant by diffeomorphism. To this purpose we develop an adaptation of degree theory.

Equations

$$\left\{\begin{array}[]{ll}-\mathrm{div}\left[A(x,u)\nabla u\right]+B(x,u)|\nabla u|^{2}=g(x,u)&\qquad\text{in $\Omega$}\,,\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr u=0&\qquad\text{on $\partial\Omega$}\,,\end{array}\right.$$

$$A,B,g:\Omega \times \mathbb{R}\to \mathbb{R}$$

$${\nu }_R\le A(x,s)\le {\beta }_R,|B(x,s)|\le {\beta }_R,|g(x,s)|\le {\beta }_R|s|,$$

$$\begin{gathered} {\int }_{\Omega }\left[A(x,u)\nabla u\cdot \nabla v+B(x,u)|\nabla u{|}^{2}v\right]dx={\int }_{\Omega }g(x,u)vdx \\ \text{for any v∈W01,2(Ω)∩L∞(Ω)}, \end{gathered}$$

$$g(x,\underline{M})\ge 0\ge g(x,\overline{M})\text{for a.e. x∈Ω};$$

$$\left\{\begin{array}[]{ll}-\mathrm{div}\left[A(x,0)\nabla u\right]-D_{s}g(x,0)u=\lambda u&\qquad\text{in $\Omega$}\,,\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr u=0&\qquad\text{on $\partial\Omega$}\,,\end{array}\right.$$

$$\text{u1<0 in Ω, u2>0 in Ω,u3 sign-changing}.$$

$$A^{\phi }(x,s)$$

$$B^{\phi }(x,s)$$

$$g^{\phi }(x,s)$$

$$A^{\phi }(x,0)=({\phi }^{\prime }(0){)}^{2}A(x,0),D_s{g}^{\phi }(x,0)=({\phi }^{\prime }(0){)}^2{D}_{s}g(x,0);$$

$$\underset{k}{\mathrm{limsup}}⟨F(u_k),u_k-u⟩\le 0,$$

$$\underset{k}{\mathrm{limsup}}⟨H_{t_k}(u_k),u_k-u⟩\le 0,$$

$$\{\begin{array}{ll}u\in K,& \\ \text{vskip}6.0ptplus2.0ptminus2.0pt& \\ ⟨F(u),v-u⟩\ge ⟨\phi ,v-u⟩& \forall v\in K.\end{array}$$

$$\left\{u\in \overline{U}:\text{u is a solution ofleavevmodenobreak eqrefeq:vi}\right\}$$

$$\mathrm{deg}((F,K),U,\phi )\in \mathbb{Z}.$$

$$\mathrm{deg}((F,K),U,\phi )=\mathrm{deg}((F-\phi ,K),U,0).$$

$$\hat{U}$$

$$\hat{K}$$

$$\hat{F}(u)$$

$$\mathrm{deg}((\hat{F},\hat{K}),\hat{U},\phi )=\mathrm{deg}((F,K),U,\phi ).$$

$$⟨F(v),v-u_0⟩\ge ⟨\phi ,v-u_0⟩\text{for any v∈K∩∂U},$$

$$\mathrm{deg}((F,K),U,\phi )=\mathrm{deg}((F,K),U_0,\phi )+\mathrm{deg}((F,K),U_1,\phi ).$$

$$\{\begin{array}{l}u\in K_t,\\ \text{vskip}6.0ptplus2.0ptminus2.0pt\\ ⟨H_t(u),v-u⟩\ge ⟨\phi ,v-u⟩\forall v\in K_t,\end{array}$$

$$W_t=\left\{u\in X:(u,t)\in W\right\},$$

$$\begin{array}{rl}⟨H_{t_k}(u_k),u_k-u⟩& =⟨H_{t_k}(u_k),{\hat{u}}_k-u⟩+⟨H_{t_k}(u_k),u_k-{\hat{u}}_k⟩\\ & \le ⟨H_{t_k}(u_k),{\hat{u}}_k-u⟩+⟨\phi ,u_k-{\hat{u}}_k⟩,\end{array}$$

$$\underset{k}{\mathrm{limsup}}⟨H_{t_k}(u_k),u_k-u⟩\le 0.$$

$$⟨H_{t_k}(u_k),v_k-u_k⟩\ge ⟨\phi ,v_k-u_k⟩$$

$$⟨H_t(u),v-u⟩\ge ⟨\phi ,v-u⟩$$

$$\hat{W}$$

$${\hat{K}}_t$$

$${\hat{H}}_t(u)$$

$$\mathrm{deg}(({\hat{H}}_t,{\hat{K}}_t),{\hat{W}}_t,\phi )=\mathrm{deg}((H_t,K_t),W_t,\phi )$$

Full text

The authors are member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

1. Introduction

Consider the quasilinear elliptic problem

[TABLE]

where $\Omega$ is a bounded, connected and open subset of $\mathbb{R}^{n}$ and

[TABLE]

are Carathéodory functions such that: **

1. $(h_{1})$

for every $R>0$ there exist $\beta_{R}>0$ and $\nu_{R}>0$ satisfying

[TABLE]

for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$ with $|s|\leq R$.

The existence of a weak solution $u\in W^{1,2}_{0}(\Omega)\cap L^{\infty}(\Omega)$ of (1.1), namely a solution of

[TABLE]

follows from the results of [6, 7], provided that a suitable a priori $L^{\infty}$-estimate holds, possibly related to the existence of a pair of sub-/super-solutions (see e.g. [6, Théorème 2.1] and [7, Theorems 1 and 2]). Moreover, each weak solution $u\in W^{1,2}_{0}(\Omega)\cap L^{\infty}(\Omega)$ is locally Hölder continuous in $\Omega$ (see e.g. [10, Theorem VII.1.1]).

The existence of multiple solutions, in the semilinear case and under suitable regularity assumptions, has also been proved for instance in [2]. Here we are interested in the existence of multiple nontrivial solutions, as $(h_{1})$ implies that $g(x,0)=0$, under assumptions that do not imply an a priori $W^{1,\infty}$-estimate. Let us state our main result.

Theorem 1.1**.**

Assume $(h_{1})$ and also that:

1. $(h_{2})$

there exist $\underline{M}<0<\overline{M}$ such that

[TABLE] 2. $(h_{3})$

the function $g(x,\cdot)$ is differentiable at $s=0$ for a.e. $x\in\Omega$.

Consider the eigenvalue problem

[TABLE]

and denote by $(\lambda_{k})$, $k\geq 1$, the sequence of the eigenvalues repeated according to multiplicity.

If there exists $k\geq 2$ with $\lambda_{k}<0<\lambda_{k+1}$ and $k$ even, then problem (1.2) admits at least three nontrivial solutions $u_{1},u_{2},u_{3}$ in $W^{1,2}_{0}(\Omega)\cap L^{\infty}(\Omega)\cap C(\Omega)$ with

[TABLE]

If $A$ is constant and $B=0$, the result is essentially contained in [3, 15], which in turn developed previous results of [4]. Actually, in those papers it is enough to assume that $\lambda_{2}<0$, because in that case (1.2) is the Euler-Lagrange equation of a suitable functional and variational methods, e.g. Morse theory, can be applied.

In our case there is no functional and also degree theory arguments cannot be applied in a standard way, because of the presence of the term $B(x,u)|\nabla u|^{2}$. Let us point out that our assumptions do not imply that the solutions $u$ of (1.2) belong to $W^{1,\infty}(\Omega)$, so that the natural growth term $B(x,u)|\nabla u|^{2}$ plays a true role.

Let us also mention that our statement has an invariance property.

Remark 1.2**.**

Let $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ be an increasing $C^{2}$-diffeomorphism with $\varphi(0)=0$. Then the following facts hold:

1. $(a)$

the functions $A$, $B$ and $g$ satisfy the assumptions of Theorem 1.1 if and only $A^{\varphi}$, $B^{\varphi}$ and $g^{\varphi}$, defined as

[TABLE]

do the same; in particular, we have

[TABLE] 2. $(b)$

a function $u\in W^{1,2}_{0}(\Omega)\cap L^{\infty}(\Omega)$ is a solution of (1.2) if and only if $\varphi^{-1}(u)$ is a solution of (1.2) with $A$, $B$ and $g$ replaced by $A^{\varphi}$, $B^{\varphi}$ and $g^{\varphi}$, respectively.

From the definition of $B^{\varphi}$ we see that the term $B(x,u)|\nabla u|^{2}$ cannot be omitted in the equation, if we want to ensure this kind of invariance.

When (1.2) is the Euler-Lagrange equation associated to a functional, the question of invariance under suitable classes of diffeomorphisms has been already treated in [14], where it is shown that problems with degenerate coercivity can be reduced, in some cases, to coercive problems.

In the next sections we develop an adaptation of degree theory suited for our setting and then we prove Theorem 1.1 by a degree argument. Under assumptions that are not diffeomorphism-invariant, a degree theory for quasilinear elliptic equations with natural growth conditions has been already developed in [1]. Here we find it more convenient to reduce the equation (1.2) to a variational inequality possessing as obstacles a pair of sub-/super-solutions, according to an approach already considered for instance in [12].

2. Topological degree in reflexive Banach spaces

Let $X$ be a reflexive real Banach space.

Definition 2.1**.**

A map $F:D\rightarrow X^{\prime}$, with $D\subseteq X$, is said to be of class $(S)_{+}$ if, for every sequence $(u_{k})$ in $D$ weakly convergent to some $u$ in $X$ with

[TABLE]

it holds $\|u_{k}-u\|\to 0$.

More generally, if $T$ is a metrizable topological space, a map $H:D\rightarrow X^{\prime}$, with $D\subseteq X\times T$, is said to be of class $(S)_{+}$ if, for every sequence $(u_{k},t_{k})$ in $D$ with $(u_{k})$ weakly convergent to $u$ in $X$, $(t_{k})$ convergent to $t$ in $T$ and

[TABLE]

it holds $\|u_{k}-u\|\to 0$ (we write $H_{t}(u)$ instead of $H(u,t)$).

Assume now that $U$ is a bounded and open subset of $X$, $F:\overline{U}\rightarrow X^{\prime}$ a bounded and continuous map of class $(S)_{+}$, $K$ a closed and convex subset of $X$ and $\varphi\in X^{\prime}$. We aim to consider the variational inequality

[TABLE]

Remark 2.2**.**

It is easily seen that the set

[TABLE]

is compact (possibly empty).

According to [8, 11, 13], if the variational inequality (2.1) has no solution $u\in\partial{U}$, one can define the topological degree

[TABLE]

Let us recall some basic properties.

Proposition 2.3**.**

If (2.1) has no solution $u\in\partial{U}$, then

[TABLE]

Proposition 2.4**.**

If (2.1) has no solution $u\in\partial{U}$, $u_{0}\in X$ and we set

[TABLE]

then

[TABLE]

Theorem 2.5**.**

If (2.1) has no solution $u\in\overline{U}$, then $\mathrm{deg}((F,K),U,\varphi)=0$.

Theorem 2.6**.**

If (2.1) has no solution $u\in\partial{U}$ and there exists $u_{0}\in K\cap U$ such that

[TABLE]

then $\mathrm{deg}((F,K),U,\varphi)=1$.

Theorem 2.7**.**

If $U_{0}$ and $U_{1}$ are two disjoint open subsets of $U$ and (2.1) has no solution $u\in\overline{U}\setminus(U_{0}\cup U_{1})$, then

[TABLE]

Definition 2.8**.**

Let $K_{k}$, $K$ be closed and convex subsets of $X$. The sequence $(K_{k})$ is said to be Mosco-convergent to $K$ if the following facts hold:

1. $(a)$

if $k_{j}\to\infty$, $u_{k_{j}}\in K_{k_{j}}$ for any $j\in\mathbb{N}$ and $(u_{k_{j}})$ is weakly convergent to $u$ in $X$, then $u\in K$; 2. $(b)$

for every $u\in K$ there exist $\overline{k}\in\mathbb{N}$ and a sequence $(u_{k})$ strongly convergent to $u$ in $X$ with $u_{k}\in K_{k}$ for any $k\geq\overline{k}$.

Theorem 2.9**.**

Let $W$ be a bounded and open subset of $X\times[0,1]$, $H:\overline{W}\rightarrow X^{\prime}$ be a bounded and continuous map of class $(S)_{+}$ and $(K_{t})$, $0\leq t\leq 1$, be a family of closed and convex subsets of $X$ such that, for every sequence $(t_{k})$ convergent to $t$ in $[0,1]$, the sequence $(K_{t_{k}})$ is Mosco-convergent to $K_{t}$.

Then the following facts hold:

1. $(a)$

the set of pairs $(u,t)\in\overline{W}$, satisfying

[TABLE]

is compact (possibly empty); 2. $(b)$

if the problem (2.2) has no solution $(u,t)\in\partial_{X\times[0,1]}\,W$ and we set

[TABLE]

then $\mathrm{deg}((H_{t},K_{t}),W_{t},\varphi)$ is independent of $t\in[0,1]$.

Proof.

If $(u_{k},t_{k})$ is a sequence in $\overline{W}$ constituted by solutions of (2.2), then up to a subsequence $(u_{k})$ is weakly convergent to some $u$ in $X$ and $(t_{k})$ is convergent to some $t$ in $[0,1]$. Then $u\in K_{t}$ and there exists a sequence $(\hat{u}_{k})$ strongly convergent to $u$ in $X$ with $\hat{u}_{k}\in K_{t_{k}}$. It follows

[TABLE]

whence

[TABLE]

Then $\|u_{k}-u\|\to 0$ and $(u,t)\in\overline{W}$. For every $v\in K_{t}$ there exists a sequence $(v_{k})$ strongly convergent to $v$ in $X$ with $v_{k}\in K_{t_{k}}$. From

[TABLE]

it follows

[TABLE]

so that the set introduced in assertion $(a)$ is compact.

Assume now that the problem (2.2) has no solution $(u,t)\in\partial_{X\times[0,1]}\,W$. It is enough to prove that $\left\{t\mapsto\mathrm{deg}((H_{t},K_{t}),W_{t},\varphi)\right\}$ is locally constant.

Suppose first that $K_{t}\neq\emptyset$ for any $t\in[0,1]$. By Michael selection theorem (see e.g. [5, Theorem 1.11.1]) there exists a continuous map $\gamma:[0,1]\rightarrow X$ such that $\gamma(t)\in K_{t}$ for any $t\in[0,1]$.

If we set

[TABLE]

then $\widehat{W}$, $\widehat{K}_{t}$ and $\widehat{H}$ satisfy the same assumptions and

[TABLE]

by Proposition 2.4. Moreover $0\in\widehat{K}_{t}$ for any $t\in[0,1]$. Therefore we may assume, without loss of generality, that $0\in K_{t}$ for any $t\in[0,1]$.

Given $t\in[0,1]$, there exist a bounded and open subset $U$ of $X$ and $\delta>0$ such that

[TABLE]

and such that (2.2) has no solution $(u,\tau)$ in

[TABLE]

with $t-\delta\leq\tau\leq t+\delta$. From Theorem 2.7 we infer that

[TABLE]

From [11, Theorem 4.53 and Proposition 4.61] we deduce that $\left\{\tau\mapsto\mathrm{deg}((H_{\tau},K_{\tau}),U,\varphi)\right\}$ is constant on $[t-\delta,t+\delta]$.

In general, given $t\in[0,1]$, let us distinguish the cases $K_{t}\neq\emptyset$ and $K_{t}=\emptyset$.

If $K_{t}\neq\emptyset$, by the Mosco-convergence there exists $\delta>0$ such that $K_{\tau}\neq\emptyset$ for any $\tau\in[t-\delta,t+\delta]$. By the previous step we infer that $\left\{\tau\mapsto\mathrm{deg}((H_{\tau},K_{\tau}),W_{\tau},\varphi)\right\}$ is constant on $[t-\delta,t+\delta]$.

If $K_{t}=\emptyset$, from Theorem 2.5 we infer that $\mathrm{deg}((H_{t},K_{t}),W_{t},\varphi)=0$. Assume, for a contradiction, that there exists a sequence $(t_{k})$ convergent to $t$ with $\mathrm{deg}((H_{t_{k}},K_{t_{k}}),W_{t_{k}},\varphi)\neq 0$ for any $k\in\mathbb{N}$. Again from Theorem 2.5 we infer that the problem (2.2) has a solution $(u_{k},t_{k})\in\overline{W}$, in particular $u_{k}\in K_{t_{k}}$, for any $k\in\mathbb{N}$. Up to a subsequence, $(u_{k})$ is weakly convegent to some $u$, whence $u\in K_{t}$ by the Mosco-convergence, and a contradiction follows. ∎

Now let $\Omega$ be a bounded and open subset of $\mathbb{R}^{n}$, let $T$ be a metrizable topological space and let

[TABLE]

be two Carathéodory functions. We will denote by $\|\leavevmode\nobreak\ \|_{p}$ the usual norm in $L^{p}$ and write $a_{t}(x,s,\xi)$, $b_{t}(x,s,\xi)$ instead of $a(x,(s,\xi,t))$, $b(x,(s,\xi,t))$.

In this section, we assume that $a_{t}$ and $b_{t}$ satisfy the controllable growth conditions in the sense of [10], uniformly with respect to $t$. In a simplified form enough for our purposes, this means that: **

1. $(UC)$

there exist $p\in]1,\infty[$, $\alpha^{(0)}\in L^{1}(\Omega)$, $\alpha^{(1)}\in L^{p^{\prime}}(\Omega)$, $\beta>0$ and $\nu>0$ such that

[TABLE]

for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$, $\xi\in\mathbb{R}^{n}$, $t\in T$; such a $p$ is clearly unique.

It follows

[TABLE]

and the map $H:W^{1,p}_{0}(\Omega)\times T\rightarrow W^{-1,p^{\prime}}(\Omega)$ defined by

[TABLE]

is continuous and bounded on $B\times T$, whenever $B$ is bounded in $W^{1,p}_{0}(\Omega)$.

Theorem 2.10**.**

Assume $(UC)$ and also that:

1. $(UM)$

we have

[TABLE]

for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$, $\xi,\hat{\xi}\in\mathbb{R}^{n}$, $t\in T$, with $\xi\neq\hat{\xi}$.

Then $H:W^{1,p}_{0}(\Omega)\times T\rightarrow W^{-1,p^{\prime}}(\Omega)$ is of class $(S)_{+}$.

Proof.

See e.g. [13, Theorem 1.2.1]. ∎

3. Quasilinear elliptic variational inequalities with

natural growth conditions

Again, let $\Omega$ be a bounded and open subset of $\mathbb{R}^{n}$ and let now

[TABLE]

be two Carathéodory functions. In this paper we are interested in the case in which $a$ and $b$ satisfy the natural growth conditions in the sense of [10]. More precisely, we assume that: **

1. $(N)$

there exist $p\in]1,\infty[$ and, for every $R>0$, $\alpha^{(0)}_{R}\in L^{1}(\Omega)$, $\alpha^{(1)}_{R}\in L^{p^{\prime}}(\Omega)$, $\beta_{R}>0$ and $\nu_{R}>0$ such that

[TABLE]

for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$, $\xi\in\mathbb{R}^{n}$ with $|s|\leq R$; such a $p$ is clearly unique; 2. $(M)$

we have

[TABLE]

for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$, $\xi,\hat{\xi}\in\mathbb{R}^{n}$ with $\xi\neq\hat{\xi}$.

Then we can define a map

[TABLE]

by

[TABLE]

Remark 3.1**.**

Assume that $\hat{a}$ and $\hat{b}$ also satisfy $(N)$ and $(M)$. An easy density argument shows that, if

[TABLE]

then

[TABLE]

Consider also a $p$-quasi upper semicontinuous function $\underline{u}:\Omega\rightarrow\overline{\mathbb{R}}$ and a $p$-quasi lower semicontinuous function $\overline{u}:\Omega\rightarrow\overline{\mathbb{R}}$, and set

[TABLE]

where $\tilde{u}$ is any $p$-quasi continuous representative of $u$ (see e.g. [9]).

We aim to consider the solutions of the variational inequality

[TABLE]

We denote by $Z^{tot}(F,K)$ the set of solutions $u$ of ($VI$). We will simply write $Z^{tot}$, if no confusion can arise.

For every $u\in K$, we also set

[TABLE]

Proposition 3.2**.**

A function $u\in K$ satisfies ($VI$) if and only if

[TABLE]

Proof.

Assume that $u$ is a solution of ($VI$) and let $v\in T_{u}K$ with $v\leq 0$ a.e. in $\Omega$. Since $\max\{k(\underline{u}-\tilde{u}),\tilde{v}\}$ is a nonincreasing sequence of nonpositive $p$-quasi upper semicontinuous functions converging to $\tilde{v}$ $p$-q.e. in $\Omega$, by [9, Lemma 1.6] there exists a sequence $(v_{k})$ in $W^{1,p}_{0}(\Omega)$ converging to $v$ in $W^{1,p}_{0}(\Omega)$ with $\tilde{v}_{k}\geq\max\{k(\underline{u}-\tilde{u}),\tilde{v}\}$ $p$-q.e. in $\Omega$. Without loss of generality, we may assume that $\tilde{v}_{k}\leq 0$ $p$-q.e. in $\Omega$.

Then it follows that $u+\frac{1}{k}\,v_{k}\in K$, whence

[TABLE]

Going to the limit as $k\to\infty$, we get

[TABLE]

If $v\in T_{u}K$ with $v\geq 0$ a.e. in $\Omega$, the argument is similar. Since every $v\in T_{u}K$ can be written as $v=v^{+}-v^{-}$ with $v^{+},-v^{-}\in T_{u}K$, it follows

[TABLE]

Since $K\subseteq u+T_{u}K$, the converse is obvious. ∎

We are also interested in the invariance of the problem with respect to suitable transformations.

Let us denote by $\Phi$ the set of increasing $C^{2}$-diffeomorphisms $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ such that $\varphi(0)=0$ and by $\Theta$ the set of $C^{1}$-functions $\vartheta:\mathbb{R}\rightarrow]0,+\infty[$.

For any $\varphi\in\Phi$ and $\vartheta\in\Theta$, we define

[TABLE]

by

[TABLE]

If we define the Carathéodory functions

[TABLE]

by

[TABLE]

it easily follows that

[TABLE]

We also set $u^{\varphi}=\varphi^{-1}(u)$ and, given a set $E$ of real valued functions, $E^{\varphi}=\left\{u^{\varphi}:\,\,u\in E\right\}$.

It is easily seen that

[TABLE]

for every $\varphi,\psi\in\Phi$ and $\vartheta,\varrho\in\Theta$.

We also say that $(a,b)$ is of Euler-Lagrange type, if there exists a function

[TABLE]

such that

1. $(a)$

$\left\{x\mapsto L(x,s,\xi)\right\}$ is measurable for every $(s,\xi)\in\mathbb{R}\times\mathbb{R}^{n}$; 2. $(b)$

$\left\{(s,\xi)\mapsto L(x,s,\xi)\right\}$ is of class $C^{1}$ for a.e. $x\in\Omega$; 3. $(c)$

we have

[TABLE]

for a.e. $x\in\Omega$ and every $(s,\xi)\in\mathbb{R}\times\mathbb{R}^{n}$.

Taking into account Proposition 3.2, the next two results are easy to prove.

Proposition 3.3**.**

For every $\varphi\in\Phi$, the following facts hold:

1. $(a)$

the functions $a^{\varphi},b^{\varphi}$ satisfy $(N)$ and $(M)$ with the same $p$; 2. $(b)$

we have

[TABLE]

and $\underline{u}^{\varphi}:\Omega\rightarrow\overline{\mathbb{R}}$ is $p$-quasi upper semicontinuous, while $\overline{u}^{\varphi}:\Omega\rightarrow\overline{\mathbb{R}}$ is $p$-quasi lower semicontinuous (here we agree that $\varphi(-\infty)=-\infty$ and $\varphi(+\infty)=+\infty$); 3. $(c)$

$u$* is a solution of ($VI$) if and only if $u^{\varphi}$ is a solution of the corresponding variational inequality with $a$, $b$ and $K$ replaced by $a^{\varphi}$, $b^{\varphi}$ and $K^{\varphi}$, respectively;* 4. $(d)$

the pair $(a,b)$ is of Euler-Lagrange type if and only if the pair $(a^{\varphi},b^{\varphi})$ does the same; moreover, if $L$ is associated with $(a,b)$, then

[TABLE]

is associated with $(a^{\varphi},b^{\varphi})$.

Proposition 3.4**.**

For every $\vartheta\in\Theta$, the following facts hold:

1. $(a)$

the functions $a_{\vartheta},b_{\vartheta}$ satisfy $(N)$ and $(M)$ with the same $p$; 2. $(b)$

$u$* is a solution of ($VI$) if and only if the same $u$ is a solution of the corresponding variational inequality with $a$ and $b$ replaced by $a_{\vartheta}$ and $b_{\vartheta}$, respectively.*

Remark 3.5**.**

Let $\varphi\in\Phi$ with $\varphi^{\prime}$ nonconstant, let $w\in L^{2}(\Omega)$ and let $a(x,s,\xi)=\varphi^{\prime}(s)\xi$, $b(x,s,\xi)=-w(x)$.

Then $(a,b)$ is not of Euler-Lagrange type. However, if we take $\vartheta(s)=\varphi^{\prime}(s)$, then $(a_{\vartheta},b_{\vartheta})$, which is given by

[TABLE]

is of Euler-Lagrange type with

[TABLE]

Therefore the property of being of Euler-Lagrange type is not invariant under the transformation induced by $\vartheta$, which plays in fact the role of “integrating factor”. By the way, if then we take $\psi=\varphi^{-1}$, we get

[TABLE]

which are simply related to

[TABLE]

Proposition 3.6**.**

For every $R>0$ there exist $\vartheta_{1},\vartheta_{2}\in\Theta$ and $c>0$, depending only on $R$, $\beta_{R}$ and $\nu_{R}$, such that

[TABLE]

for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$, $\xi\in\mathbb{R}^{n}$ with $|s|\leq R$.

Proof.

If we set $\vartheta(s)=\exp(\gamma s)$ with $\gamma\,\nu_{R}\geq\beta_{R}$, we have

[TABLE]

whence the existence of $\vartheta_{2}$. The existence of $\vartheta_{1}$ can be proved in a similar way. ∎

4. Quasilinear elliptic variational inequalities with

natural growth conditions depending on a parameter

Again, let $\Omega$ be a bounded and open subset of $\mathbb{R}^{n}$ and let now $T$ be a metrizable topological space and

[TABLE]

be two Carathéodory functions satisfying $(N)$ and $(M)$ uniformly with respect to $t\in T$.

More precisely, we assume that $a_{t}$ and $b_{t}$ satisfy $(UM)$ and: **

1. $(UN)$

there exist $p\in]1,\infty[$ and, for every $R>0$, $\alpha^{(0)}_{R}\in L^{1}(\Omega)$, $\alpha^{(1)}_{R}\in L^{p^{\prime}}(\Omega)$, $\beta_{R}>0$ and $\nu_{R}>0$ such that

[TABLE]

for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$, $\xi\in\mathbb{R}^{n}$ and $t\in T$ with $|s|\leq R$; again, such a $p$ is clearly unique.

Then we can define a map

[TABLE]

by

[TABLE]

Consider also, for each $t\in T$, a $p$-quasi upper semicontinuous function $\underline{u}_{t}:\Omega\rightarrow\overline{\mathbb{R}}$ and a $p$-quasi lower semicontinuous function $\overline{u}_{t}:\Omega\rightarrow\overline{\mathbb{R}}$, set

[TABLE]

and assume the following form of continuity related to the Mosco-convergence: **

1. $(MC)$

for every sequence $(t_{k})$ convergent to $t$ in $T$, the following facts hold:

• •

if $(u_{k})$ is a sequence weakly convergent to $u$ in $W^{1,p}_{0}(\Omega)$, with $u_{k}\in K_{t_{k}}$ for any $k\in\mathbb{N}$ and $(u_{k})$ bounded in $L^{\infty}(\Omega)$, then $u\in K_{t}$;

• •

for every $u\in K_{t}$ there exist $\overline{k}\in\mathbb{N}$ and a sequence $(u_{k})$ in $W^{1,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$ which is bounded in $L^{\infty}(\Omega)$ and strongly convergent to $u$ in $W^{1,p}_{0}(\Omega)$, with $u_{k}\in K_{t_{k}}$ for any $k\geq\overline{k}$.

Then consider the parametric variational inequality

[TABLE]

Theorem 4.1**.**

Let $(u_{k},t_{k})$ be a sequence of solutions of ($PVI$) with $(u_{k})$ bounded in $L^{\infty}(\Omega)$ and $(t_{k})$ convergent to some $t$ in $T$ with $K_{t}\neq\emptyset$.

Then $(u_{k})$ admits a subsequence strongly convergent in $W^{1,p}_{0}(\Omega)$ to some $u$ and $(u,t)$ is a solution of ($PVI$).

Proof.

Let $w\in K_{t}$ and let $(w_{k})$ be a sequence strongly convergent to $w$ in $W^{1,p}_{0}(\Omega)$, with $w_{k}\in K_{t_{k}}$ for any $k\in\mathbb{N}$ and $(w_{k})$ bounded in $L^{\infty}(\Omega)$. If we set

[TABLE]

and define $\widehat{K}_{k}$, $\widehat{K}_{\infty}$ accordingly, it is easily seen that all the assumptions are still satisfied and now $0\in\widehat{K}_{k}$. Therefore, we may assume without loss of generality that $0\in K_{t_{k}}$ for any $k\in\mathbb{N}$.

Let $R>0$ be such that $\|u_{k}\|_{\infty}\leq R$ for any $k\in\mathbb{N}$. We claim that $(u_{k}^{+})$ is bounded in $W^{1,p}_{0}(\Omega)$. Actually, by Propositions 3.4 and 3.6, we may assume, without loss of generality, that

[TABLE]

Then the choice $v=-u_{k}^{-}$ in ($PVI$) yields

[TABLE]

which implies that $(u_{k}^{+})$ is bounded in $W^{1,p}_{0}(\Omega)$.

In a similar way one finds that $(u_{k}^{-})$ is bounded in $W^{1,p}_{0}(\Omega)$, so that $(u_{k})$ is weakly convergent, up to a subsequence, to some $u$ in $W^{1,p}_{0}(\Omega)$ and $u\in K_{t}$. Let $(z_{k})$ be a sequence strongly convergent to $u$ in $W^{1,p}_{0}(\Omega)$, with $z_{k}\in K_{t_{k}}$ for any $k\in\mathbb{N}$ and $(z_{k})$ bounded in $L^{\infty}(\Omega)$.

Let $\psi:\mathbb{R}\rightarrow[0,1]$ be a continuous function such that $\psi(s)=1$ for $|s|\leq R$ and $\psi(s)=0$ for $|s|\geq R+1$. Then, let

[TABLE]

It is easily seen that each $(u_{k},t_{k})$ is also a solution of ($PVI$) with $a_{t}$ and $b_{t}$ replaced by $\check{a}_{t}$ and $\check{b}_{t}$. Moreover, $\check{a}_{t}$ and $\check{b}_{t}$ satisfy both the assumptions $(UN)$, $(UM)$ and the assumptions of [1, Theorem 4.2]. In particular, there exist $\alpha^{(0)}\in L^{1}(\Omega)$, $\beta>0$ and $\nu>0$ such that

[TABLE]

for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$, $\xi\in\mathbb{R}^{n}$ and $t\in T$.

Now, if $p<n$, the proof of [1, Theorem 4.2] can be repeated in a simplified form, as $(u_{k})$ is bounded in $L^{\infty}(\Omega)$. We have only to observe that, if $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ is the solution of

[TABLE]

then there exists $\tau>0$ such that

[TABLE]

so that

[TABLE]

It follows

[TABLE]

and now the proof of [1, Theorem 4.2] can be repeated with minor modifications, showing that $(u_{k})$ is strongly convergent to $u$ in $W^{1,p}_{0}(\Omega)$. If $p\geq n$, the argument is similar and simpler.

It is easily seen that $(u,t)$ is a solution of ($PVI$). ∎

Remark 4.2**.**

In the previous theorem the assumption $K_{t}\neq\emptyset$ is crucial to ensure that $(u_{k})$ is bounded in $W^{1,p}_{0}(\Omega)$.

Consider $\Omega=]0,1[$, $a_{t}(x,s,\xi)=\xi$, $b_{t}(x,s,\xi)=0$, $\underline{u}_{t}=z-t$ and $\overline{u}_{t}=z+t$, where $z\in C_{c}(]0,1[)\setminus W^{1,2}_{0}(]0,1[)$.

If $t_{k}\to 0$ with $t_{k}>0$ and $(u_{k},t_{k})$ are the solutions of ($PVI$), then the sequence $(u_{k})$ is unbounded in $W^{1,2}_{0}(\Omega)$.

5. Topological degree for quasilinear elliptic variational

inequalities with natural growth conditions

Consider again the setting of Section 3. Throughout this section, we also assume that: **

1. $(B)$

the functions $\underline{u}$ and $\overline{u}$ are bounded.

It follows that $Z^{tot}$ is automatically bounded in $L^{\infty}(\Omega)$.

Theorem 5.1**.**

The set $Z^{tot}$ is (strongly) compact in $W^{1,p}_{0}(\Omega)$ (possibly empty).

Proof.

If $K=\emptyset$, we have $Z^{tot}=\emptyset$. Otherwise the assertion follows from Theorem 4.1. ∎

Definition 5.2**.**

We denote by $\mathcal{Z}(F,K)$ the family of the subsets $Z$ of $Z^{tot}$ which are both open and closed in $Z^{tot}$ with respect to the $W^{1,p}_{0}(\Omega)$-topology. We will simply write $\mathcal{Z}$, if no confusion can arise.

Fix a continuous function $\psi:\mathbb{R}\rightarrow[0,1]$ such that

[TABLE]

then set, for any $t\in[0,1]$ and $s\in\mathbb{R}$,

[TABLE]

If we consider $T=[0,1]$ and define

[TABLE]

it is easily seen that $a_{t}^{\psi},b_{t}^{\psi}$ satisfy $(UN)$ and $(UM)$. Moreover, for every $\underline{t}\in]0,1[$, they satisfy $(UC)$, if $t$ is restricted to $[\underline{t},1]$. In particular, we can define a continuous map $H^{\psi}:W^{1,p}_{0}(\Omega)\times]0,1]\rightarrow W^{-1,p^{\prime}}(\Omega)$ by

[TABLE]

and, by Theorem 2.10, this map is of class $(S)_{+}$. We will simply write $H$, if no confusion can arise.

Proposition 5.3**.**

For every $Z\in\mathcal{Z}$, the following facts hold:

1. $(a)$

there exist a bounded and open subset $U$ of $W^{1,p}_{0}(\Omega)$ and $\overline{t}\in]0,1]$ such that

[TABLE]

and such that the variational inequality ($PVI$) has no solution $(u,t)\in\partial U\times[0,\overline{t}]$; in particular, the degree $\mathrm{deg}((H_{t}^{\psi},K),U,0)$ is defined whenever $t\in]0,\overline{t}]$; 2. $(b)$

if $\psi_{0},\psi_{1}:\mathbb{R}\rightarrow[0,1]$ have the same properties of $\psi$ and $U_{0}$, $\overline{t}_{0}$ and $U_{1}$, $\overline{t}_{1}$ are as in $(a)$, then

[TABLE]

for every $t\in]0,\overline{t}_{0}]$ and $\tau\in]0,\overline{t}_{1}]$; 3. $(c)$

if $\hat{a}$ and $\hat{b}$ also satisfy $(N)$, $(M)$ and

[TABLE]

then we have

[TABLE]

for every $t\in]0,\overline{t}]$ and $\tau\in]0,\hat{t}]$, provided that $U$, $\overline{t}$, $\widehat{U}$, $\hat{t}$ are as in $(a)$ with respect to $a,b$ and $\hat{a},\hat{b}$, respectively.

Proof.

By definition of $\mathcal{Z}$, there exists a bounded and open subset $U$ of $W^{1,p}_{0}(\Omega)$ such that

[TABLE]

If $(u_{k},t_{k})$ is a sequence of solutions of ($PVI$) with $u_{k}\in\partial U$ and $t_{k}\to 0$, then $(u_{k})$ is bounded in $L^{\infty}(\Omega)$ by $(B)$. By Theorem 4.1, up to a subsequence $(u_{k})$ is convergent to some $u$ in $W^{1,p}_{0}(\Omega)$ and $u$ is a solution of ($VI$). Then $u\in\partial U$ and a contradiction follows. Therefore, there exists $\overline{t}\in]0,1]$ such that ($PVI$) has no solution $(u,t)\in\partial U\times[0,\overline{t}]$ and assertion $(a)$ is proved.

To prove $(b)$, define

[TABLE]

and consider $H_{t}^{\psi_{\mu}}$. Arguing as before, we find $\underline{t}>0$ with $\underline{t}\leq\overline{t}_{j}$, $j=0,1$, such that

[TABLE]

has no solution with $u\in(\overline{U_{0}}\setminus U_{1})\cup(\overline{U_{1}}\setminus U_{0})$, $t\in[0,\underline{t}]$ and $\mu\in[0,1]$.

From Theorem 2.7 we infer that

[TABLE]

On the other hand, we have

[TABLE]

by Theorem 2.9. Then assertion $(b)$ also follows.

The proof of $(c)$ is quite similar. ∎

Definition 5.4**.**

For every $Z\in\mathcal{Z}(F,K)$, we set

[TABLE]

where $\psi$, $U$, $\overline{t}$ are as in $(a)$ and $0<t\leq\overline{t}$. We will simply write $\mathrm{ind}(Z)$, if no confusion can arise.

Proposition 5.5**.**

Assume that $a$ and $b$ satisfy, instead of $(N)$, the more specific controllable growth condition:

1. $(C)$

there exist $p\in]1,\infty[$, $\alpha^{(0)}\in L^{1}(\Omega)$, $\alpha^{(1)}\in L^{p^{\prime}}(\Omega)$, $\beta>0$ and $\nu>0$ such that

[TABLE]

for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$, $\xi\in\mathbb{R}^{n}$.

Then the map $F:W^{1,p}_{0}(\Omega)\rightarrow W^{-1,p^{\prime}}(\Omega)$ is continuous, bounded on bounded subsets and of class $(S)_{+}$. Moreover, if $U$ is a bounded and open subset of $W^{1,p}_{0}(\Omega)$ such that ($VI$) has no solution $u\in\partial U$ and

[TABLE]

then $Z\in\mathcal{Z}$ and

[TABLE]

Proof.

It is easily seen that this time $a_{t}$ and $b_{t}$ satisfy $(UC)$ and $(UM)$, for $t$ belonging to all $[0,1]$.

Then the assertions follow from Theorems 2.10 and 2.9. ∎

Theorem 5.6**.**

Let $Z\in\mathcal{Z}$ with $\mathrm{ind}(Z)\neq 0$. Then $Z\neq\emptyset$.

Proof.

Let $U$ and $\overline{t}$ be as in $(a)$ of Proposition 5.3. If $Z=\emptyset$, from Theorem 4.1 we infer that there exists $t\in]0,\overline{t}]$ such that ($PVI$) has no solution $(u,t)$ with $u\in\overline{U}$. From Theorem 2.5 we deduce that

[TABLE]

and a contradiction follows. ∎

Along the same line, the additivity property can be proved taking advantage of Theorems 2.7 and 4.1.

Theorem 5.7**.**

Let $Z_{0},Z_{1}\in\mathcal{Z}$ with $Z_{0}\cap Z_{1}=\emptyset$. Then $Z_{0}\cup Z_{1}\in\mathcal{Z}$ and

[TABLE]

Theorem 5.8**.**

Let

[TABLE]

be two Carathéodory functions satisfying $(UN)$ and $(UM)$ with respect to $T=[0,1]$ and set

[TABLE]

Let also, for each $t\in[0,1]$, $\underline{u}_{t}:\Omega\rightarrow\overline{\mathbb{R}}$ be a $p$-quasi upper semicontinuous function and $\overline{u}_{t}:\Omega\rightarrow\overline{\mathbb{R}}$ a $p$-quasi lower semicontinuous function, define $K_{t}$ as in section 4 and assume that:

• •

the functions $\underline{u}_{t},\overline{u}_{t}$ are bounded uniformly with respect to $t\in[0,1]$, we have $K_{t}\neq\emptyset$ for any $t\in[0,1]$ and assumption $(MC)$ is satisfied.

Then the following facts hold:

1. $(a)$

the set

[TABLE]

is (strongly) compact in $W^{1,p}_{0}(\Omega)\times[0,1]$ (possibly empty); 2. $(b)$

if $\widehat{Z}$ is open and closed in $\widehat{Z}^{tot}$ with respect to the topology of $W^{1,p}_{0}(\Omega)\times[0,1]$ and

[TABLE]

then $\widehat{Z}_{t}\in\mathcal{Z}(H_{t},K_{t})$ for any $t\in[0,1]$ and $\mathrm{ind}((H_{t},K_{t}),\widehat{Z}_{t})$ is independent of $t\in[0,1]$.

Proof.

First of all, the set $\widehat{Z}^{tot}$ is compact by Theorem 4.1. To prove assertion $(b)$, let $W$ be a bounded and open subset of $W^{1,p}_{0}(\Omega)\times[0,1]$ such that

[TABLE]

In particular, we have $\widehat{Z}_{t}\in\mathcal{Z}(H_{t},K_{t})$ for any $t\in[0,1]$.

Let $\widehat{T}=[0,1]\times[0,1]$, let

[TABLE]

for $(t,\tau)\in\widehat{T}$ and define

[TABLE]

It is easily seen that $a_{t,\tau}$ and $b_{t,\tau}$ satisfy $(UN)$ and $(UM)$ with respect to $\widehat{T}$, so that we can consider the problem

[TABLE]

Since $[0,1]$ is compact, by Theorem 4.1 there exists $\overline{\tau}\in]0,1]$ such that (5.1) has no solution $(u,(t,\tau))$ with $(u,t)\in\partial W$ and $0\leq\tau\leq\overline{\tau}$.

By Definition 5.4 we infer that

[TABLE]

and the assertion follows from Theorem 2.9. ∎

Remark 5.9**.**

By Theorem 5.8 and Proposition 5.5, $\mathrm{ind}((F,K),Z)$ can be calculated also by other approximation techniques, with respect to the one used in Definition 5.4.

Theorem 5.10**.**

If $K\neq\emptyset$, then $\mathrm{ind}(Z^{tot})=1$.

Proof.

Define, for $0\leq t\leq 1$,

[TABLE]

It is easily seen that $a_{t}$ and $b_{t}$ satisfy assumptions $(UN)$ and $(UM)$, so that Theorem 5.8 can be applied. If we take $\widehat{Z}=\widehat{Z}^{tot}$, we get

[TABLE]

Let $u_{0}\in K$ and let

[TABLE]

with $r$ large enough to guarantee that $u_{0}\in U$, $\widehat{Z}_{0}\subseteq U$ and

[TABLE]

From Proposition 5.5 and Theorem 2.6 we infer that

[TABLE]

and the assertion follows. ∎

Proposition 5.11**.**

Let $Z\in\mathcal{Z}(F,K)$ and let $\varphi\in\Phi$ and $\vartheta\in\Theta$. Then $Z^{\varphi}\in\mathcal{Z}(F^{\varphi},K^{\varphi})$, $Z\in\mathcal{Z}(F_{\vartheta},K)$ and

[TABLE]

Proof.

If we set

[TABLE]

the assertion follows from Theorem 5.8. ∎

6. Proof of Theorem 1.1

We aim to apply the results of the previous sections to

[TABLE]

By hypothesis $(h_{1})$, assumptions $(N)$ and $(M)$ are satisfied with $p=2$. Moreover, if $\underline{M}$ and $\overline{M}$ are as in hypothesis $(h_{2})$, then $\underline{u}=\underline{M}$ and $\overline{u}=\overline{M}$ satisfy assumption $(B)$.

Denote by $(\lambda_{k})$, $k\geq 1$, the sequence of the eigenvalues of (1.3), repeated according to multiplicity, and set, for a matter of convenience, $\lambda_{0}=-\infty$.

Finally, define $F$, $K$, $Z^{tot}$, $\mathcal{Z}$ and $\mathrm{ind}(Z)$ as before, observe that $K\neq\emptyset$ and set

[TABLE]

It is easily seen that

[TABLE]

Let us also set

[TABLE]

so that

[TABLE]

Proposition 6.1**.**

For every $R>0$ there exist $\vartheta_{1},\vartheta_{2}\in\Theta$, depending only on $\beta_{R}$ and $\nu_{R}$, such that

[TABLE]

Proof.

It is a simple variant of Proposition 3.6. ∎

Proposition 6.2**.**

Let

[TABLE]

be a Carathéodory function such that for every $R>0$ there exists $\beta_{R}>0$ satisfying

[TABLE]

and such that

[TABLE]

If $u$ is a solution of the variational inequality ($VI$) with

[TABLE]

then $u$ satisfies the equation

[TABLE]

Proof.

Let $v\in W^{1,2}_{0}(\Omega)\cap L^{\infty}(\Omega)$ with $v\geq 0$ a.e. in $\Omega$, let $t>0$ and let

[TABLE]

Since $u_{t}\in K$, it follows

[TABLE]

On the other hand, we have

[TABLE]

whence

[TABLE]

Since $0\leq u_{t}-u\leq tv$, we can go to the limit as $t\to 0^{+}$, obtaining

[TABLE]

Arguing on $u_{t}=\max\{u-tv,\underline{M}\}$, one can prove in a similar way that

[TABLE]

whence

[TABLE]

and the assertion follows. ∎

Proposition 6.3**.**

Let $\Omega$ be connected and assume that $u\in W^{1,2}_{0}(\Omega)\cap L^{\infty}(\Omega)$ satisfies $u\geq 0$ a.e. in $\Omega$, $u>0$ on a set of positive measure and

[TABLE]

Then we have

[TABLE]

Proof.

By Propositions 3.4 and 6.1, we may assume without loss of generality that

[TABLE]

Then for every $v\in W^{1,2}_{0}(\Omega)\cap L^{\infty}(\Omega)$ with $v\geq 0$ a.e. in $\Omega$, we have

[TABLE]

with $\gamma(x,u)\in L^{\infty}(\Omega)$.

From [11, Theorem 8.15 and Remark 8.16] the assertion follows. ∎

Lemma 6.4**.**

Assume that $\Omega$ is connected and that $\lambda_{1}<0$. Moreover, according to $(h_{1})$, let $\beta>0$ be such that

[TABLE]

Let also $\psi:\mathbb{R}\rightarrow[0,1]$ be a continuous function, with $\psi(0)>0$ and $\psi(s)=0$ outside $]\underline{M},\overline{M}[$, and consider the problem

[TABLE]

where

[TABLE]

Denote by $\widehat{Z}^{tot}$ the set of solutions $(u,t)$ of (6.1) and let

[TABLE]

Then there exist $0<r_{1}<r_{2}$ such that

[TABLE]

Proof.

If we set

[TABLE]

it is easily seen that assumptions $(UN)$ and $(UM)$ are satisfied. From Theorem 5.8 we infer that $\widehat{Z}^{tot}$ is compact in $W^{1,p}_{0}(\Omega)\times[0,1]$.

First of all, we claim that there exists $r_{2}>0$ such that

[TABLE]

By Propositions 3.4 and 6.1, we may assume without loss of generality that

[TABLE]

Assume, for a contradiction, that $(u_{k},t_{k})$ is a sequence in $\widehat{Z}$ with $\|u_{k}\|_{2}\to 0$. Then we may suppose, without loss of generality, that $(u_{k})$ is convergent to [math] in $W^{1,p}_{0}(\Omega)$ and a.e. in $\Omega$ and that $(t_{k})$ is convergent to some $t\in[0,1]$. Let $u_{k}=\tau_{k}z_{k}$ with $\tau_{k}=\|\nabla u_{k}\|_{2}$ and, up to a subsequence, $(z_{k})$ weakly convergent to some $z$ in $W^{1,2}_{0}(\Omega)$.

If $v\in K\setminus\{0\}$ with $v\geq 0$ a.e. in $\Omega$, we have

[TABLE]

which implies that $(t_{k}/\tau_{k})$ is bounded hence convergent, up to a subsequence, to some $\sigma\geq 0$.

Then we also get

[TABLE]

whence

[TABLE]

If we choose $v=\varphi_{1}$, where $\varphi_{1}$ is a positive eigenfunction of (1.3) associated with $\lambda_{1}<0$, we get

[TABLE]

whence $z=0$.

Finally, the choice $v=0$ in (6.1) yields

[TABLE]

We infer that $\|\nabla z_{k}\|_{2}\to 0$ and a contradiction follows.

With this choice of $r_{2}$, we also have

[TABLE]

for every $r_{1}\in]0,r_{2}[$. Now we claim that there exists $r_{1}\in]0,r_{2}[$ such that

[TABLE]

By Propositions 3.4 and 6.1, now we may assume without loss of generality that

[TABLE]

Assume, for a contradiction, that $(u_{k},t_{k})$ is a sequence in the set at the left hand side with $(u_{k},t_{k})\not\in\widehat{Z}$ and $\|u_{k}^{-}\|_{2}\to 0$. Then, up to a subsequence, $(u_{k})$ is convergent to some $u\in\widehat{Z}$ in $W^{1,p}_{0}(\Omega)$ and a.e. in $\Omega$, while $(t_{k})$ is convergent to some $t\in[0,1]$. By Propositions 6.2 and 6.3 we have $u>0$ a.e. in $\Omega$. If we write $u_{k}^{-}=\tau_{k}z_{k}$ with $\tau_{k}=\|\nabla u_{k}^{-}\|_{2}$, we have that $(z_{k})$ is weakly convergent, up to a subsequence, to some $z$ in $W^{1,2}_{0}(\Omega)$ and, on the other hand, $(z_{k})$ is convergent to [math] a.e. in $\Omega$, as $u>0$. The choice $v=u_{k}^{+}$ in (6.1) yields

[TABLE]

whence

[TABLE]

We infer that $\|\nabla z_{k}\|_{2}\to 0$ and a contradiction follows. ∎

Proposition 6.5**.**

If $\Omega$ is connected and $\lambda_{1}<0$, then we have $Z_{+},Z_{-}\in\mathcal{Z}$ and

[TABLE]

Proof.

Let $g_{t}$, $\widehat{Z}^{tot}$, $\widehat{Z}$, $r_{1}$ and $r_{2}$ be as in Lemma 6.4. We aim to apply again Theorem 5.8 with

[TABLE]

By Lemma 6.4 the set $\widehat{Z}$ is open and closed in $\widehat{Z}^{tot}$. From Theorem 5.8 we infer that

[TABLE]

and that

[TABLE]

Now we claim that $\widehat{Z}_{1}=Z^{tot}(H_{1},K)$. By Propositions 3.4 and 6.1 we may assume without loss of generality that

[TABLE]

If we take $v\in K\setminus\{0\}$ with $v\geq 0$ a.e. in $\Omega$ in (6.1), we see that $0\not\in Z^{tot}(H_{1},K)$. Moreover, if $u\in Z^{tot}(H_{1},K)$, the choice $v=u^{+}$ yields $(v-u)=u^{-}$, hence

[TABLE]

Therefore $u^{-}=0$, whence $u\in\widehat{Z}_{1}$ and the claim is proved.

From Theorem 5.10 we infer that

[TABLE]

and the assertion concerning $Z_{+}$ follows.

The assertion concerning $Z_{-}$ can be proved in a similar way. ∎

Proposition 6.6**.**

If there exists $k\geq 0$ with $\lambda_{k}<0<\lambda_{k+1}$, then $\{0\}\in\mathcal{Z}$ and

[TABLE]

Proof.

If we set

[TABLE]

[TABLE]

[TABLE]

it is easily seen that assumptions $(UN)$ and $(UM)$ are satisfied. We aim to apply Theorem 5.8.

We claim that there exists $r>0$ such that, if $(u,t)\in\widehat{Z}^{tot}$ and $\|\nabla u\|_{2}\leq r$, then $u=0$. Assume, for a contradiction, that $(u_{k},t_{k})$ is a sequence in $\widehat{Z}^{tot}$ with $u_{k}\neq 0$ and $\|\nabla u_{k}\|_{2}\to 0$. Let $u_{k}=\tau_{k}z_{k}$ with $\tau_{k}=\|\nabla u_{k}\|_{2}$ and, up to a subsequence, $(z_{k})$ weakly convergent to some $z$ in $W^{1,2}_{0}(\Omega)$ and $(t_{k})$ convergent to some $t$ in $[0,1]$.

Given $v\in K$, if $t_{k}>0$ we have

[TABLE]

whence

[TABLE]

Going to the limit as $k\to\infty$, we get

[TABLE]

If $t_{k}=0$, we have

[TABLE]

and the same conclusion easily follows.

Then we infer that

[TABLE]

whence $z=0$, as [math] is not in the sequence $(\lambda_{k})$.

By Propositions 3.4 and 6.1, we may assume without loss of generality that

[TABLE]

If $t_{k}>0$, the choice $v=-u_{k}^{-}$ in (6.2) yields

[TABLE]

which implies that $\|\nabla z_{k}^{+}\|_{2}\to 0$. If $t_{k}=0$ the argument is analogous and simpler. In a similar way one can show that $\|\nabla z_{k}^{-}\|_{2}\to 0$ and a contradiction follows. Therefore, there exists $r>0$ with the required property.

In particular, we can apply Theorem 5.8 with

[TABLE]

obtaining

[TABLE]

and

[TABLE]

On the other hand, if we set

[TABLE]

from Proposition 5.5 we infer that

[TABLE]

Now, if we set

[TABLE]

from [11, Theorem 4.53 and Proposition 4.61] we deduce that

[TABLE]

Finally, from [13, Theorem 2.5.2] it follows that

[TABLE]

∎

Proof of Theorem 1.1.

From Proposition 6.5 we know that

[TABLE]

By Theorem 5.6 and Propositions 6.2 and 6.3 we infer that there exist at least two solutions $u_{1}\in Z_{-}$ and $u_{2}\in Z_{+}$ of (1.2) with

[TABLE]

Assume, for a contradiction, that

[TABLE]

with $\mathrm{ind}\left(\{0\}\right)=1$ by Proposition 6.6. From Theorems 5.10 and 5.7 we infer that

[TABLE]

and a contradiction follows. Therefore there exists

[TABLE]

By Proposition 6.2 $u_{3}$ is a sign-changing solution of (1.2). According to [10, Theorem VII.1.1], each $u_{j}$ is locally Hölder continuous in $\Omega$. ∎