On branches of positive solutions for p-Laplacian problems at the extreme value of Nehari manifold method

TL;DR

This paper investigates the existence of positive solution branches for p-Laplacian boundary value problems at the critical parameter value where the Nehari manifold method's applicability threshold is reached.

Contribution

It establishes the existence of two solution branches for parameters above the Nehari manifold threshold, extending understanding of solution structure at this critical point.

Findings

Existence of two positive solution branches above the threshold parameter

Identification of the threshold parameter $\lambda^*$ for the Nehari manifold method

Analysis of solution continuation at the extreme value of the Nehari manifold

Abstract

This paper is concerned with variational continuation of branches of solutions for nonlinear boundary value problems, which involve the p-Laplacian, the indefinite nonlinearity, and depend on the real parameter $\lambda$. A special focus is made on the extreme value of Nehari manifold $\lambda^*$, which determines the threshold of applicability of Nehari manifold method. In the main result the existence of two branches of positive solutions for the cases where parameter $\lambda$ lies above the threshold $\lambda^*$ is obtained.

Equations

$$\{\begin{array}{rlrl}-{\Delta }_{p}u& =\lambda |u{|}^{p-2}u+f|u{|}^{\gamma -2}u& & \text{mbox}in\Omega \\ u& =0& & \text{mbox}on\partial \Omega \end{array}$$

$${\Phi }_{\lambda }(u)=\frac{1}{p}\int |\nabla u{|}^{p}dx-\frac{\lambda }{p}\int |u{|}^{p}dx-\frac{1}{\gamma }\int f|u{|}^{\gamma }dx,$$

$$\min \{{\Phi }_{\lambda }(u):\text{leavevmode}u\in {\mathcal{N}}_{\lambda }\}$$

$${\lambda }^{\ast }=\inf \left\{\frac{\int |\nabla u{|}^{p}dx}{\int |u{|}^{p}dx}:\text{leavevmode}\text{leavevmode}\int f|u{|}^{\gamma }dx\ge 0,\text{leavevmode}u\in W\setminus 0\right\},$$

$${\Omega }^{+}=\{x\in \Omega :f^{+}(x)\ne 0\},\text{leavevmode}\text{leavevmode}{\Omega }^{-}=\{x\in \Omega :f^{-}(x)\ne 0\}$$

$$H_{\lambda }(u)=\int |\nabla u{|}^{p}dx-\lambda \int |u{|}^{p}dx,\text{leavevmode}F(u)=\int f(x)|u{|}^{\gamma },\text{leavevmode}u\in W.$$

$${\Phi }_{\lambda }(u)=\frac{1}{p}{H}_{\lambda }(u)-\frac{1}{\gamma }F(u),\text{leavevmode}\text{leavevmode}\text{leavevmode}{\mathcal{N}}_{\lambda }=\{u\in W\setminus 0:H_{\lambda }(u)-F(u)=0\}.$$

$${\mathcal{N}}_{\lambda }^{+}:=\{u\in {\mathcal{N}}_{\lambda }:D_{uu}{\Phi }_{\lambda }(u)(u,u)>0\}$$

$${\mathcal{N}}_{\lambda }^{+}=\{u=sv:\text{leavevmode}s=s_{\lambda }^{+}(v),\text{leavevmode}v\in {\Theta }_{\lambda }^{+}\},$$

$$s_{\lambda }^{+}(v)={\left(\frac{H_{\lambda }(v)}{F(v)}\right)}^{1/(\gamma -p)}.$$

$$J_{\lambda }^{+}(v)=:{\Phi }_{\lambda }(s_{\lambda }^{+}(v)v)=-c_{p,\gamma }\frac{|H_{\lambda }(v){|}^{\gamma /(\gamma -p)}}{|F(v){|}^{p/(\gamma -p)}},\text{leavevmode}v\in {\Theta }_{\lambda }^{+},$$

$${\Theta }_{\lambda }^{+}=\{v\in W\setminus 0:\text{leavevmode}{H}_{\lambda }(v)<0\}$$

$${\hat{J}}_{\lambda }^{+}:=\min \{J_{\lambda }^{+}(v)|\text{leavevmode}v\in {\Theta }_{\lambda }^{+}\}.$$

$${\mu }_0{D}_v{H}_{{\lambda }^{\ast }}({\varphi }_1^{\ast })={\mu }_1{D}_{v}F({\varphi }_1^{\ast }).$$

$${\hat{J}}_{\lambda }^{+}\to {\bar{J}}^{+}({\lambda }^{\ast })\ge -\infty \text{leavevmode}\text{leavevmode}\text{mbox}as\text{leavevmode}\text{leavevmode}\lambda \uparrow {\lambda }^{\ast }.$$

$${\bar{J}}^{+}({\lambda }^{\ast })-\epsilon <{\hat{J}}_{\lambda }^{+}\le J_{\lambda }^{+}(v_K)<J_{{\lambda }^{\ast }}^{+}(v_K)+\epsilon <{\bar{J}}^{+}({\lambda }^{\ast })-K+\epsilon .$$

$$J_{{\lambda }^{\ast }}^{+}(w_n)=-c_{p,\gamma }\frac{|H_{{\lambda }^{\ast }}(w_n){|}^{\gamma /(\gamma -p)}}{|F(w_n){|}^{p/(\gamma -p)}}=-c_{p,\gamma }{s}_{{\lambda }^{\ast }}^{+}(w_n{)}^p|H_{{\lambda }^{\ast }}(w_n)|.$$

$$-{\Delta }_p{w}_n-{\lambda }^{\ast }|w_n{|}^{p-2}{w}_n-s_{{\lambda }^{\ast }}^{+}(w_n{)}^{\gamma -p}f|w_n{|}^{\gamma -2}{w}_n=0,\text{leavevmode}n=1,2,\dots$$

$$H_{{\lambda }^{\ast }}(v)\le \underset{n\to \infty }{\mathrm{liminf}}{H}_{{\lambda }^{\ast }}(v_n)=\underset{n\to \infty }{\mathrm{liminf}}(H_{\mu }(v_n)+(\mu -{\lambda }^{\ast })\int |v_n{|}^p)<0.$$

$${\hat{J}}_{\lambda }^{+}(\mu )=\inf \{J_{\lambda }^{+}(v):v\in {\Theta }_{\mu }^{+}\}$$

$$J_{\lambda }^{+}(v)\le \underset{n\to \infty }{\mathrm{liminf}}{J}_{\lambda }^{+}(v_n)={\hat{J}}_{\lambda }^{+}(\mu ).$$

$$J_{{\lambda }_0}^{+}(v)\le \underset{n\to \infty }{\mathrm{liminf}}{J}_{{\lambda }_n}^{+}(v_n)=:\tilde{J}<+\infty .$$

$$|(-J_{\lambda }^{+}(w){)}^{\frac{\gamma -p}{\gamma }}-(-J_{{\lambda }_0}^{+}(w){)}^{\frac{\gamma -p}{\gamma }}|=\frac{|\lambda -{\lambda }_0|G(w)}{|F(w){|}^{p/\gamma }}\le \frac{|\lambda -{\lambda }_0|}{{\lambda }_1}\frac{1}{|c_{\mu }{|}^{p/\gamma }}.$$

$${\mu }^{\lambda }=\sup \{\mu \in ({\mu }_0,{\lambda }^{\ast }):{\hat{J}}_{\lambda }^{+}(\mu )={\hat{J}}_{\lambda }^{+}({\mu }_0)\},$$

$$c_{\lambda }=\underset{\eta \in {\Gamma }_{\lambda }}{\inf }\underset{t\in [0,1]}{\max }{\Phi }_{\lambda }(\eta (t)),$$

$${\Gamma }_{\lambda }=\{\eta \in C([0,1],W):\eta (0)=u_{\lambda },\eta (1)=w_{\lambda }\}.$$

$${\Phi }_{\lambda }(u)\ge j_{\lambda }>{\hat{J}}_{\lambda }^{+}({\mu }_0),\text{leavevmode}\text{leavevmode}\text{leavevmode}\forall u\in \partial {\Theta }_{{\mu }_0}^{+}.$$

$$j_{\lambda }:=\inf \{J_{\lambda }^{+}(v):v\in \partial {\Theta }_{\mu }^{+}\}>{\hat{J}}_{\lambda }^{+}({\mu }_0),\text{leavevmode}\text{leavevmode}\text{leavevmode}\forall \lambda \in ({\lambda }^{\ast },\Lambda ).$$

$${\Phi }_{\lambda }(u)\ge {\Phi }_{\lambda }(s_{\lambda }^{+}(u)u)=J_{\lambda }^{+}(u)\ge j_{\lambda }>{\hat{J}}_{\lambda }^{+}({\mu }_0).$$

$$H_{{\lambda }^{\ast }}(\overline{\eta }(t))$$

Full text

Key words: $p$-Laplacian, extreme value of Nehari manifold method, indefinite nonlinearity, branches of solutions

††

Yavdat Il’yasov, Institute of Mathematics, Ufa Scientific Senter, Russian Academy of Sciences, 112, Chernyshevsky str., Ufa, RussiaE-mail address: ilyasov02@gmail.com Kaye Silva, Institute of Mathematics and Statistics, Federal University of Goias, Campus II, CEP 74690-900 E-mail address: kayeoliveira@hotmail.com

1 Introduction

We study the following p-Laplacian problem with indefinite nonlinearity

[TABLE]

Here $\Omega$ denotes a bounded domain in $\mathbb{R}^{N}$ with $C^{1}$-boundary $\partial\Omega$, $\lambda$ is a real parameter, $1<p<\gamma<p^{*}$, where $p^{*}$ is the critical Sobolev exponent, $f\in L^{d}(\Omega)$ where $d\geq p^{*}/(p^{*}-\gamma)$ if $p<N$, and $d>1$ if $p\geq N$. We suppose that (1.1) has an indefinite nonlinearity, i.e., $f$ change sign in $\Omega$. By a solution of (1.1) we mean a critical point $u\in W:=W_{0}^{1,p}(\Omega)$ of the energy functional

[TABLE]

where $W_{0}^{1,p}(\Omega)$ is the standard Sobolev space.

The problems with the indefinite nonliearity of type (1.1) have been intesively studied, see e.g., Alama & Tarantello [1], Berestycki, Capuzzo–Dolcetta & Nirenberg [2], Ouyang [13]. One of the fruitful approaches in the study of such problems is the Nehari manifold method [12] where solutions are obtained through the constrained minimization problem

[TABLE]

with the Nehari manifold $\mathcal{N}_{\lambda}:=\{u\in W\setminus 0:\leavevmode\nobreak\ D_{u}\Phi_{\lambda}(u)(u)=0\}$ (see e.g. Drabek & Pohozhaev [6], Il’yasov [9, 8], Ouyang [13]).

The applicability of NM-method to (1.1) depends on the parameter $\lambda$. Indeed, (1.1) possess the so-called extreme value of the Nehari manifold method [8]

[TABLE]

which was known to be the first found by Ouyang [13]. A feature of $\lambda^{*}$ is that it defines a threshold for the applicability of the Nehari manifold method so that for any $\lambda<\lambda^{*}$ the set $\mathcal{N}_{\lambda}$ is a $C^{1}$-manifold of codimension $1$ in $W$ wherein for any $\lambda\geq\lambda^{*}$ there is $u\in\mathcal{N}_{\lambda}$ such that $\Phi_{\lambda}^{\prime\prime}(u):=D^{2}_{uu}\Phi_{\lambda}(u)(u,u)=0$. Moreover, $\Phi_{\lambda}$ is unbounded from below over $\mathcal{N}_{\lambda}$ if $\lambda\geq\lambda^{*}$ (see e.g [8]). It is remarkable that once the extreme value (1.3) is detected, one is able to directly find solutions for (1.1) as $\lambda<\lambda^{*}$, by means of the Nehari minimization problems (1.2) (see e.g. [13] for $p=2$ and [9] for $1<p<+\infty$).

A natural question which arises from this is whether there are any positive solutions of (1.1) for $\lambda>\lambda^{*}$. An answer for this question, in the case $p=2$, follows from the works of Alama & Tarantello [1], Ouyang [13], where the authors proved that (1.1) possess a branch of minimal positive solution for $\lambda$ belonging to the whole interval $(-\infty,\Lambda)$ and does not admit any positive solutions for $\lambda>\Lambda$. However, one approach used in [1, 13] is based on the application of the local continuation method [3], which essentially involves an analysis of the corresponding linearized problems.

The main aim of the present paper is to give a contribution in the investigation of the branches of solutions for the problems where the application of local continuation methods can cause difficulty. Our approach is based on the development of the Nehari manifold method where we focus also on obtaining a new knowledge on extreme value of the Nehari manifold method.

Let us state our main results. Denote

[TABLE]

and $\Omega^{0}=\Omega\setminus(\overline{\Omega^{+}}\cup\overline{\Omega^{-}})$. We write $U\neq\emptyset$ if the interior int($U$) of a set $U\subset\mathbb{R}^{n}$ is non-empty. We denote ($\lambda_{1}({\rm int}(U))$, $\phi_{1}({\rm int}(U)))$ the first eigenpair of $-\Delta_{p}$ on ${\rm int}(U)$ with zero Dirichlet boundary conditions. It is known that $\lambda_{1}({\rm int}(U))$ is positive, simple and isolated, and $\phi_{1}({\rm int}(U))$ is positive [11]. To simplify notations, we write $\lambda_{1}:=\lambda_{1}(\Omega)$, $\phi_{1}:=\phi_{1}(\Omega)$.

Throughout the paper, we assume that $\Omega^{+}\neq\emptyset$. Furthermore, we shall need the following assumption

($f_{1}$):  If $\Omega^{0}\neq\emptyset$, then $\lambda_{1}({\rm int}(\Omega^{0}\cup\Omega^{+}))<\lambda_{1}({\rm int}(\Omega^{0}))$.

Notice that if $\Omega^{0}\cup\Omega^{+}\neq\emptyset$, then $\lambda^{*}<+\infty$ and there exists $\overline{\lambda}>\lambda_{1}$ such that (1.1) has no positive solutions for any $\lambda>\overline{\lambda}$ (see e.g. [7]).

Our main result is the following

Theorem 1.1**.**

Let $1<p<\gamma<p^{*}$ and suppose that $\Omega^{+}\neq\emptyset$, $F(\phi_{1})<0$ and ($f_{1}$) is satisfied. Then there exists $\Lambda>\lambda^{*}$ such that for all $\lambda\in(\lambda^{*},\Lambda)$ problem (1.1) admits two positive weak solutions $u_{\lambda},\overline{u}_{\lambda}$. Moreover,

(i)

$\Phi_{\lambda}^{\prime\prime}(u_{\lambda})>0,\Phi_{\lambda}^{\prime\prime}(\overline{u}_{\lambda})>0$* and $\Phi_{\lambda}(u_{\lambda})<\Phi_{\lambda}(\overline{u}_{\lambda})<0$ for any $\lambda\in(\lambda^{*},\Lambda)$;*

(ii)

$\Phi_{\lambda}(u_{\lambda})\uparrow\Phi_{\lambda^{*}}(u_{\lambda^{*}})$* as $\lambda\downarrow\lambda^{*}$. *

This paper is organized as follows. Section 2 contains preliminaries results. In Section 3, we show the existence of solutions $u_{\lambda}$. In Section 4 we show the existence of solutions $\overline{u}_{\lambda}$ and conclude the proof of Theorem 1.1. In Appendix we provide some technical and auxiliary results.

2 Preliminaries

Denote

[TABLE]

Then

[TABLE]

To our aims, it is sufficient to use the following Nehari submanifold

[TABLE]

which we shall use in the fibering representation [9, 14]:

[TABLE]

where $\Theta^{+}_{\lambda}=\{v\in W\setminus 0:\leavevmode\nobreak\ H_{\lambda}(v)<0,\leavevmode\nobreak\ F(v)<0\}$ and

[TABLE]

Thus we are able to introduce

[TABLE]

where $c_{p,\gamma}=(\gamma-p)/p\gamma$.

Observe, $J^{+}_{\lambda}$ is the 0-homogeneous functional on $\mathcal{N}_{\lambda}^{+}$, i.e., $J^{+}_{\lambda}(su)=J^{+}_{\lambda}(u)$ for any $s>0$, $u\in\mathcal{N}_{\lambda}^{+}$. It is worth pointing out that (1.3) implies

[TABLE]

for any $\lambda\in(\lambda_{1},\lambda^{*})$. In what follows, we denote $\partial\Theta_{\lambda}^{+}=\{v\in W\setminus 0:\ H_{\lambda}(v)=0\}$.

It is not hard to prove (see e.g. [9])

Proposition 2.1**.**

If $D_{v}J^{+}_{\lambda}(v)(\eta)=0$ for any $\eta\in W\setminus 0$, then $s^{+}_{\lambda}(v)v$ weakly satisfies (1.1).

In what follows, we shall use

Proposition 2.2**.**

Assume $(w_{n})\subset\Theta^{+}_{\lambda}$ for $\lambda>\lambda_{1}$ and $||w_{n}||=1$, $n=1,2,...$. Then there exist $w\in W\setminus 0$ and a subsequence, which we will still denote by $(w_{n})$, such that $w_{n}\rightharpoonup w$ weakly in $W$ and $w_{n}\to w$ strongly in $L^{q}(\Omega)$ for $1<q<p^{*}$.

Proof.

Once $(w_{n})$ is bounded in $W$, the proof of the existence of the limit point $w\in W$ follows from the Eberlein-Šmulian and Sobolev theorems. Since $H_{\lambda}(w_{n})<0$, $n=1,2,...$, it follows that $w\neq 0$. ∎

Consider the following Nehari minimization problem

[TABLE]

Lemma 2.1**.**

Suppose the assumptions of Theorem 1.1 are satisfied. Then

(a)

$\lambda_{1}<\lambda^{*}<\infty$;

(b)

there exists a minimizer $\phi_{1}^{*}$ of the problem (1.3) such that $\phi_{1}^{*}>0$. Moreover, any minimizer $\phi_{1}^{*}$ of (1.3) weakly satisfies, up to scalar multiplier, to (1.1) for $\lambda=\lambda^{*}$ and $H_{\lambda^{*}}(\phi^{*}_{1})=F(\phi^{*}_{1})=0$;

(c)

$\hat{J}_{\lambda^{*}}^{+}>-\infty$* and there exists a minimizer $v_{\lambda^{*}}\in\Theta^{+}_{\lambda^{*}}$ of $J_{\lambda^{*}}^{+}(v_{\lambda^{*}})$ so that $u_{\lambda^{*}}:=s_{2}(v_{\lambda^{*}})v_{\lambda^{*}}$ satisfies (1.1) and $u_{\lambda^{*}}>0$.*

Proof.

The proof of (a) can be found in [9, 7]. Furthermore, by [9], there exists a nonzero minimizer $\phi_{1}^{*}$ of (1.3) such that $\phi_{1}^{*}\geq 0$. Hence by Lagrange multiplier rule there exist $\mu_{0},\mu_{1}\geq 0$, $|\mu_{0}|+|\mu_{1}|\neq 0$ such that

[TABLE]

Since $\phi_{1}^{*}$ is a minimizer of (1.3) then $H_{\lambda^{*}}(\phi^{*}_{1})=0$ and threfore $\mu_{1}F(\phi^{*}_{1})=0$.

Suppose $\mu_{0}=0$, then $f|\phi^{*}_{1}|^{\gamma-2}\phi^{*}_{1}=0$ a.e. in $\Omega$. This is possible only if supp $\phi^{*}_{1}$ $\subset\Omega^{0}$. Thus if $\Omega^{0}=\emptyset$ then we get a contradiciton. Assume that $\Omega^{0}\neq\emptyset$. Then there exist eigenpairs $(\lambda_{1}({\rm int}(\Omega^{0})),\phi_{1}({\rm int}(\Omega^{0})))$ and $(\lambda_{1}({\rm int}(\Omega^{0}\cup\Omega^{+})),\phi_{1}({\rm int}(\Omega^{0}\cup\Omega^{+})))$. Since $\phi_{1}(\Omega^{0})\in W^{1,p}_{0}(\Omega^{0})$ and $H_{\lambda^{*}}(\phi^{*}_{1})=0$, $\lambda_{1}({\rm int}(\Omega^{0}))\leq\lambda^{*}$. On the other hand, the assumption ($f_{1}$) entails the strong inequality $\lambda_{1}({\rm int}(\Omega^{0}\cup\Omega^{+}))<\lambda_{1}({\rm int}(\Omega^{0}))$. Hence we get a contradiction because $\lambda^{*}\leq\bar{\lambda}\leq\lambda_{1}({\rm int}(\Omega^{0}\cup\Omega^{+}))$ (see [7]). Thus $\mu_{0}\neq 0$.

Suppose $\mu_{1}=0$, then $D_{u}H_{\lambda^{*}}(\phi^{*}_{1})=0$. By the Harnack inequality (see [15]) we have $\phi^{*}_{1}>0$ in $\Omega$. But this is possible only if $\lambda^{*}=\lambda_{1}$, $\phi^{*}_{1}=\phi_{1}$. However, by (1.3), $F(\phi^{*}_{1})\geq 0$ which contradicts the assumption $F(\phi_{1})<0$. Hence $\mu_{1}>0$ and therefore $F(\phi^{*}_{1})=0$ and there exists $t(\mu)>0$ such that $t(\mu)\phi_{1}^{*}$ satisfies (1.1). The maximum principle and regularity of solutions for the $p$-Laplacian equation yields that $\phi^{*}_{1}>0$ and $\phi^{*}_{1}\in C^{1,\alpha}(\overline{\Omega})$. Thus we have proved (b).

Let us prove (c). By [9] there is a finite limit

[TABLE]

and there exists a weak positive solution $u_{\lambda^{*}}$ of (1.1) such that $\bar{J}^{+}(\lambda^{*})=J_{\lambda^{*}}^{+}(u_{\lambda^{*}})$. It is clear that $\hat{J}_{\lambda^{*}}^{+}\leq\bar{J}^{+}(\lambda^{*})$. Thus, we will obtain the proof if we show that $\hat{J}_{\lambda^{*}}^{+}=\bar{J}^{+}(\lambda^{*})$. Suppose, contrary to our claim, that $\hat{J}_{\lambda^{*}}^{+}<\bar{J}^{+}(\lambda^{*})$. We prove that this is impossible if $\hat{J}_{\lambda^{*}}^{+}=-\infty$. The proof in the other case is similar.

Since $\hat{J}_{\lambda^{*}}^{+}=-\infty$, for every $K>0$, one can find $v_{K}\in\Theta^{+}_{\lambda^{*}}$ such that $J_{\lambda^{*}}^{+}(v_{K})<\bar{J}^{+}(\lambda^{*})-K$. Since $J_{\lambda}^{+}(v_{K})\to J_{\lambda^{*}}^{+}(v_{K})$, for every $\varepsilon>0$, there exists $\delta>0$ such that $|J_{\lambda}^{+}(v_{K})-J_{\lambda^{*}}^{+}(v_{K})|<\varepsilon$ as $|\lambda-\lambda^{*}|<\delta$. In view of (2.5), we may assume that there holds also $|\hat{J}_{\lambda}^{+}-\bar{J}^{+}(\lambda^{*})|<\varepsilon$ if $|\lambda-\lambda^{*}|<\delta$. Then

[TABLE]

Since $K>0$, $\varepsilon>0$ may be chosen arbitrarily, we get a contradiction.

∎

We need also

Corollary 2.1**.**

There exists $\mu_{0}\in(\lambda_{1},\lambda^{*})$ such that any minimizer $w_{\lambda^{*}}$ of (2.3) for $\lambda=\lambda^{*}$ satisfies $H_{\mu_{0}}(w_{\lambda^{*}})<0$.

Proof.

Suppose the assertion of the corollary is false. Then there exists a sequence $w_{n}\in\Theta^{+}_{\lambda^{*}}$ such that $\hat{J}_{\lambda^{*}}^{+}=J_{\lambda^{*}}^{+}(w_{n})$ and $H_{\lambda^{*}}(w_{n})\to 0$ as $n\to\infty$. By homogeneity of $J^{+}_{\lambda^{*}}(v)$ we may assume $||w_{n}||=1$. Hence by Proposition 2.2, there is $w\in W\setminus 0$ such that $w_{n}\rightharpoonup w$ in $W$ and $w_{n}\to w$ in $L^{q}(\Omega)$ for $1<q<p^{*}$. Observe

[TABLE]

From this and since $J_{\lambda^{*}}^{+}(w_{n})=\hat{J}_{\lambda^{*}}^{+}<0$, it follows that $s^{+}_{\lambda^{*}}(w_{n})\to\infty$ and $F(w_{n})\to 0$. Hence $F(w)=0$ and therefore $H_{\lambda^{*}}(w)=0$ which implies by $(2^{o})$, Lemma 2.1 that $w=\phi_{1}^{*}>0$. Note that

[TABLE]

Thus $s^{+}_{\lambda^{*}}(w_{n})\to\infty$ implies $f|\phi_{1}^{*}|^{\gamma-1}=0$ a.a. in $\Omega$, which is an absurd.

∎

Corollary 2.2**.**

For each $\mu\in(\lambda_{1},\lambda^{*})$, there is $c_{\mu}<0$ such that $F(v)\leq c_{\mu}$ $\forall v\in\overline{\Theta}_{\mu}^{+}$.

Proof.

Let $\mu<\lambda^{*}$ and assume contrary to our claim that there exists a sequence $v_{n}\in\overline{\Theta}_{\mu}^{+}$ such that $F(v_{n})\to 0$ as $n\to\infty$. By homogeneity of $J^{+}_{\lambda^{*}}(v)$ and Proposition 2.2 there is $v\in W\setminus 0$ such that $v_{n}\rightharpoonup v$ weakly in $W$ and $v_{n}\to v$ strongly in $L^{q}(\Omega)$ in $L^{q}(\Omega)$ for $1<q<p^{*}$. Hence, by the weakly lower-semicontinuity of $\int|\nabla v|^{p}dx$ we conclude that

[TABLE]

But this contradicts to the definition of $\lambda^{*}$ since $F(v)=0$.

∎

3 Local Minima Solutions

In this section, we show the existence of local minima type solutions $u_{\lambda}$ for (1.1). Let us consider the following family of constrained minimization problems

[TABLE]

parametrized by $\lambda\geq\lambda^{*}$ and $\mu\in(\lambda_{1},\lambda^{*})$.

Proposition 3.1**.**

For each $\lambda\geq\lambda^{*}$ and $\mu\in(\lambda_{1},\lambda^{*})$ there holds

(a)

$\hat{J}_{\lambda}^{+}(\mu)>-\infty$;

(b)

there exists a minimizer $v_{\lambda}(\mu)$ of (3.1).

Proof.

(a) follows immediately from Lemma 2.2. Let us prove (b). Take a minimizing sequence $v_{n}\in\Theta_{\mu}^{+}$ of (3.1), that is $J^{+}_{\lambda}(v_{n})\to\hat{J}_{\lambda}^{+}(\mu)>-\infty$ as $n\to\infty$. By homogeneity of $J^{+}_{\lambda^{*}}(v)$ and Proposition 2.2 there is $v\in W\setminus 0$ such that $v_{n}\rightharpoonup v$ in $W$ and $v_{n}\to v$ in $L^{q}(\Omega)$ for $1<q<p^{*}$. Hence, by the weak lower-semicontinuity we infer that $H_{\mu}(v)\leq\liminf_{n\to\infty}H_{\mu}(v_{n})\leq 0,\ F(v)=\lim_{n\to\infty}F(v_{n})<0$ and therefore $v\in\overline{\Theta_{\mu}^{+}}$. By the weak lower-semicontinuity of $J_{\lambda}^{+}(v)$,

[TABLE]

In view of (3.1), this is possible only if $J_{\lambda}^{+}(v)=\hat{J}_{\lambda}^{+}(\mu)$, that is $v$ is a minimizer of (3.1).

∎

We denote the set of minimizers for (3.1) by $\mathcal{S}_{\lambda}(\mu)=\{v\in\overline{\Theta}_{\mu}^{+}:\ J_{\lambda}^{+}(v)=\hat{J}_{\lambda}^{+}(\mu)\}$ and let $\mathcal{S}_{\lambda}^{\partial}(\mu)=\{v\in\mathcal{S}_{\lambda}(\mu):\leavevmode\nobreak\ H_{\mu}(v)=0\}.$

Lemma 3.1**.**

Let $\lambda_{0}\geq\lambda^{*}$ and $\mu\in(\lambda_{1},\lambda^{*})$ such that $\mathcal{S}^{\partial}_{\lambda_{0}}(\mu)=\emptyset$. Then there exists $\varepsilon>0$ such that $\mathcal{S}^{\partial}_{\lambda}(\mu)=\emptyset$ for each $\lambda\in[\lambda_{0},\lambda_{0}+\varepsilon)$.

Proof.

Suppose the lemma were false. Then we could find sequences $\lambda_{n}\to\lambda_{0}$ such that $v_{n}:=v_{\lambda_{n}}^{+}(\mu)\in\partial\Theta_{\mu}^{+}$, $n=1,2,...$. By By homogeneity of $J^{+}_{\lambda^{*}}(v)$ we may assume that $||v_{n}||=1$, $n=1,2,...$, and therefore by Proposition 2.2, $v_{n}\rightharpoonup v$ weakly in $W$ and $v_{n}\to v$ strongly in $L^{q}(\Omega)$ for $1<q<p^{*}$ and some $v\in W\setminus 0$. Hence $H_{\mu}(v)\leq\liminf_{n\to\infty}H_{\mu}(v_{n})=0,\ F(v)=\lim_{n\to\infty}F(v_{n})<0$ and therefore $v\in\overline{\Theta_{\mu}^{+}}$. Furthermore, by the weak lower semi-continuity

[TABLE]

Observe, from the Poincare’s inequality and Corollary 2.2, we have that for all $w\in\overline{\Theta_{\mu}^{2}}$ and $\lambda\geq\lambda_{1}$

[TABLE]

Thus, $J_{\lambda_{n}}^{+}(w)\to J_{\lambda_{0}}^{+}(w)$ uniformly on $w\in\overline{\Theta_{\mu}^{+}}$ as $n\to\infty$ and therefore $\tilde{J}=\hat{J}_{\lambda_{0}}^{+}(\mu)$. Hence if $J_{\lambda_{0}}^{+}(v)<\hat{J}_{\lambda_{0}}^{+}(\mu)$, we obtain a contradiction since $v\in\overline{\Theta_{\mu}^{+}}$. The case $J_{\lambda_{0}}^{+}(v)=\hat{J}_{\lambda_{0}}^{+}(\mu)$ entails that $H_{\mu_{0}}(v)=0$ and $v=v_{\lambda_{0}}(\mu)$. Consequently $v_{\lambda_{0}}(\mu)\in\mathcal{S}_{\lambda_{0}}^{\partial}(\mu)$ which contradicts to the assumption $\mathcal{S}_{\lambda_{0}}^{\partial}(\mu)=\emptyset$.

∎

Let us prove the existence of the first solution $u_{\lambda}$ in Theorem 1.1.

Lemma 3.2**.**

Let $1<p<\gamma<p^{*}$ and suppose that $\Omega^{+}\neq\emptyset$, $F(\phi_{1})<0$ and ($f_{1}$) is satisfied. Then there exists $\Lambda>\lambda^{*}$ such that for all $\lambda\in(\lambda^{*},\Lambda)$ problem (1.1) admits positive weak solution $u_{\lambda}$ such that

(li)

$\Phi_{\lambda}^{\prime\prime}(u_{\lambda})>0$* and $\Phi_{\lambda}(u_{\lambda})<0$ for any $\lambda\in(\lambda^{*},\Lambda)$;*

(lii)

$\Phi_{\lambda}(u_{\lambda})\uparrow\Phi_{\lambda^{*}}(u_{\lambda^{*}})$* as $\lambda\downarrow\lambda^{*}$;*

Proof.

From Lemma 2.1 it follows that there exists $\mu_{0}\in(\lambda_{1},\lambda^{*})$ such that $\mathcal{S}_{\lambda^{*}}^{\partial}(\mu_{0})=\emptyset$. Thus Lemma 3.1 implies that there exist $\Lambda>\lambda^{*}$ such that $\mathcal{S}_{\lambda}^{\partial}(\mu_{0})=\emptyset$ for all $\lambda\in(\lambda^{*},\Lambda)$. Since by Proposition 3.1, $\mathcal{S}_{\lambda}(\mu)\neq\emptyset$ for $\lambda\geq\lambda^{*}$, we conclude that for every $\lambda\in(\lambda^{*},\Lambda)$ there exists a minimizer $v_{\lambda}(\mu_{0})$ of (3.1) such that $v_{\lambda}(\mu_{0})\in\Theta^{+}_{\mu_{0}}$. This and Proposition 2.1 yield that $u_{\lambda}=s^{+}_{\lambda}(v_{\lambda}(\mu_{0}))v_{\lambda}(\mu_{0})$ is a weak solution of (1.1) for $\lambda\in(\lambda^{*},\Lambda)$.

In virtue that $J_{\lambda}^{+}(v)=J_{\lambda}^{+}(|v|)$ and $|v|\in\Theta_{\mu_{0}}^{+}$ for any $v\in\Theta_{\mu_{0}}^{+}$, we may assume that $u_{\lambda}\geq 0$ in $\Omega$. Now by the Harnack inequality (see [15]) we conclude that $u_{\lambda}^{+}>0$ in $\Omega$.

Since $v_{\lambda}^{+}\in\Theta_{\lambda}^{+}$, we get (li). Let us prove assertion (lii). Notice that from (3.1) it follows that $\Phi_{\lambda^{*}}(u_{\lambda^{*}})=\hat{J}_{\lambda^{*}}^{+}(\mu_{0})\geq\hat{J}_{\lambda}^{+}(\mu_{0})$ for any $\lambda>\lambda^{*}$. Thus if we suppose that assertion (lii) were false then we could find a sequence $\lambda_{n}\downarrow\lambda^{*}$ such that $J_{\lambda_{n}}^{+}(v_{\lambda_{n}}(\mu_{0}))\to J_{\lambda^{*}}^{+}<\hat{J}_{\lambda^{*}}^{+}(\mu_{0})$. Arguing as above, we may assume that $v_{\lambda_{n}}(\mu_{0})\rightharpoonup v$ weakly in $W$ and $v_{n}\to v$ strongly in $L^{p}(\Omega)$, $L^{\gamma}(\Omega)$ as $n\to\infty$ with $v\neq 0$. This implies that $v\in\overline{\Theta_{\mu_{0}}^{+}}$ and $J_{\lambda^{*}}^{+}(v)<\hat{J}_{\lambda^{*}}^{+}(\mu_{0})$. Thus we get a contradiction. ∎

From the proof of Lemma 3.2 we see that the solution $u_{\lambda}$ may depend on the parameter $\mu\in(\lambda_{1},\lambda^{*})$. However, one can prove that, at least locally by $\mu$, there is no such dependence.

Corollary 3.1**.**

Let $\lambda\geq\lambda^{*}$ and $\mu_{0}\in(\lambda_{1},\lambda^{*})$. Suppose that $\mathcal{S}^{\partial}_{\lambda}(\mu_{0})=\emptyset$. Then there exists $\varepsilon>0$ such that $\mathcal{S}_{\lambda}(\mu_{0})=\mathcal{S}_{\lambda}(\mu)$ for all $\mu\in(\mu_{0}-\varepsilon,\mu_{0}+\varepsilon)$.

Proof.

Conversely, suppose that there is $(\mu_{n})$ such that $\mu_{n}\to\mu_{0}$ and $\exists v_{n}\in\mathcal{S}_{\lambda}(\mu_{n})\setminus\mathcal{S}_{\lambda}(\mu_{0})$. Then $J_{\lambda}^{+}(v_{n})<\hat{J}_{\lambda}^{+}(\mu_{0})$ and $v_{n}\in\Theta^{+}_{\mu_{n}}\setminus\Theta^{+}_{\mu_{0}}$. Arguing as in the proof of Proposition 3.1 it can be shown that there exists a subsequence (which we denote again $(v_{n})$) such that $v_{n}\to v$ strongly in $W^{1,2}$. Hence, $J_{\lambda}^{+}(v)=\hat{J}_{\lambda}^{+}(\mu_{0})$ and $v\in\partial\Theta^{+}_{\mu_{0}}$ that is $v\in\mathcal{S}^{\partial}_{\lambda}(\mu_{0})$ which is a contradiction. ∎

4 Mountain Pass Solutions

In this section, for $\lambda\in(\lambda^{*},\Lambda)$, we will find the second branch of positive solution $\bar{u}_{\lambda}$ of a mountain pass type.

Fix $\mu_{0}\in(\lambda_{1},\lambda^{*})$ such that any minimizer $w_{\lambda^{*}}$ of (2.3) for $\lambda=\lambda^{*}$ satisfies $H_{\mu_{0}}(w_{\lambda^{*}})<0$. The existence of $\mu_{0}$ follows from Corollary 2.1.

Let $\lambda\in(\lambda^{*},\Lambda)$. Define

[TABLE]

Proposition 4.1**.**

For each $\lambda\in(\lambda^{*},\Lambda)$ there holds

(a)

$\mu_{0}<\mu^{\lambda}<\lambda^{*}$;

(b)

$\hat{J}^{+}_{\lambda}(\mu^{\lambda})=\hat{J}^{+}_{\lambda}(\mu_{0})$* and $\mathcal{S}_{\lambda}^{\partial}(\mu^{\lambda})\neq\emptyset$.*

Proof.

(a) By Corollary 2.1, $\mathcal{S}_{\lambda^{*}}^{\partial}(\mu_{0})=\emptyset$ and by Corollary 3.1, $\mathcal{S}_{\lambda}(\mu_{0})=\mathcal{S}_{\lambda}(\mu)$ for $\mu\in(\mu_{0},\mu_{0}+\varepsilon)$ and some $\varepsilon>0$. Hence, $\mu_{0}<\mu^{\lambda}$. Notice that by Proposition 5.1 from Appendix, the function $\hat{J}_{\lambda}^{+}(\mu)$ is continuous with respect to $\mu\in(\mu_{0},\lambda^{*})$. Hence and since $\hat{J}_{\lambda}^{+}(\mu)\to-\infty$ as $\mu\to\lambda^{*}$, there is $\mu^{\prime}\in(\mu_{0},\lambda^{*})$ such that $\hat{J}^{+}_{\lambda}(\mu_{0})>\hat{J}^{+}_{\lambda}(\mu)$ for each $\mu\in(\mu^{\prime},\lambda^{*})$. Thus $\mu^{\lambda}\leq\mu^{\prime}<+\infty$.

(b) Continuity of $\hat{J}_{\lambda}^{+}(\cdot)$ and (4.1) yield $\hat{J}^{+}_{\lambda}(\mu^{\lambda})=\hat{J}^{+}_{\lambda}(\mu_{0})$. Suppose, contrary to our claim, that $\mathcal{S}_{\lambda}^{\partial}(\mu^{\lambda})=\emptyset$. Then by Corollary 3.1, there is $\varepsilon^{\prime}>0$ such that for $\mu\in(\mu^{\lambda},\mu^{\lambda}+\varepsilon^{\prime})$, $\mathcal{S}_{\lambda}(\mu^{\lambda})=\mathcal{S}_{\lambda}(\mu)$ and consequently $\hat{J}^{+}_{\lambda}(\mu)=\hat{J}^{+}_{\lambda}(\mu^{\lambda})=\hat{J}^{+}_{\lambda}(\mu_{0})$ which is a contradiction.

∎

Observe, for any $\lambda\in(\lambda^{*},\Lambda)$ and $\mu\in(\lambda_{1},\lambda^{*})$, if $w\in\mathcal{S}_{\lambda}^{\partial}(\mu^{\lambda})$, then $|w|\in\mathcal{S}_{\lambda}^{\partial}(\mu^{\lambda})$

For each $\lambda\in(\lambda^{*},\Lambda)$, fix $0\leq w_{\lambda}\in\mathcal{S}_{\lambda}^{\partial}(\mu^{\lambda})$ and let $0<u_{\lambda}\in\Theta_{\mu_{0}}^{+}$ be the local minimum found in Lemma 3.2. Define

[TABLE]

where

[TABLE]

Proposition 4.2**.**

For each $\lambda\in(\lambda^{*},\Lambda)$ there exists $j_{\lambda}$ such that

[TABLE]

Proof.

Evidently, $\mathcal{S}^{\partial}_{\lambda}(\mu_{0})=\emptyset$ implies

[TABLE]

Thus for any $u\in\partial{\Theta}_{\mu_{0}}^{+}$, one has

[TABLE]

∎

Let us shows that every path from $\Gamma_{\lambda}$ intersects $\partial{\Theta}_{\mu_{0}}^{+}$.

Proposition 4.3**.**

Let $\lambda\in(\lambda^{*},\Lambda)$. Then for any $\eta\in\Gamma_{\lambda}$ there exists $t_{0}\in(0,1)$ such that $\eta(t_{0})\in\partial{\Theta}_{\mu_{0}}^{+}$.

Proof.

Notice $H_{\mu_{0}}(\eta(0))=H_{\mu_{0}}(v_{\lambda})<0$ while $H_{\mu_{0}}(\eta(1))=H_{\mu_{0}}(w_{\lambda})>0$ because $w_{\lambda}\in\Theta_{\lambda^{*}}^{+}\setminus\overline{\Theta}_{\mu_{0}}^{+}$. Thus by the continuity of $H_{\mu_{0}}(\eta(\cdot))$, there is $t_{0}\in(0,1)$ such that $H_{\mu_{0}}(\eta(t_{0}))=0$.

∎

Using [4] we are able to prove

Proposition 4.4**.**

For each $\lambda\in(\lambda^{*},\Lambda)$, there is $\overline{\eta}\in\Gamma_{\lambda}$ such that $H_{\lambda^{*}}(\overline{\eta}(t))<c<0$ for all $t\in[0,1]$

Proof.

Consider the path $\overline{\eta}(t)=[(1-t)v_{\lambda}^{p}+tw_{\lambda}^{p}]^{1/p}$, $t\in[0,1]$. Once $v_{\lambda}>0$, $\{x\in\Omega:\ u_{\lambda}(x)=w_{\lambda}(x)=0\}=\emptyset$. Hence we may apply Proposition 5.2 from the Appendix and thus $\overline{\eta}\in C([0,1],W)$ and for $t\in[0,1]$ we have

[TABLE]

∎

Corollary 4.1**.**

For all $\lambda\in(\lambda^{*},\Lambda)$ there holds

[TABLE]

Proof.

Let us start with the first inequality. Take any $\eta\in\Gamma_{\lambda}$. From Proposition 4.3, there is $t_{0}\in(0,1)$ such that $\eta(t_{0})\in\partial{\Theta}_{\mu_{0}}^{+}$, therefore by Proposition 4.2, $\max_{t\in[0,1]}\Phi_{\lambda}(\eta(t))\geq\Phi_{\lambda}(\eta(t_{0}))>\hat{J}_{\lambda}^{+}(\mu_{0})$ and consequently $\hat{J}_{\lambda}^{+}(\mu_{0})<c_{\lambda}$. Let $\overline{\eta}$ be given by Proposition 4.4. Then

[TABLE]

which implies that $c_{\lambda}<0$.

∎

Now we are able to find the second solution $\bar{u}_{\lambda}$.

Lemma 4.1**.**

For each $\lambda\in(\lambda^{*},\Lambda)$, $c_{\lambda}<0$ is a critical value $\overline{u}_{\lambda}$ of $\Phi_{\lambda}$ such that $\Phi_{\lambda}(s_{\lambda}^{+}(\overline{u}_{\lambda})=c_{\lambda}$, $\overline{u}_{\lambda}$ is a weak solution of (1.1) and $\overline{u}_{\lambda}>0$ in $\Omega$.

Proof.

Since $\Phi_{\lambda}(u)=\Phi_{\lambda}(|u|)$ for all $u\in W$, then by (4.2) there is a sequence of paths $\eta_{n}\geq 0$ in $\Omega$ such that

[TABLE]

Following [10], for each $\epsilon>0$ introduce

[TABLE]

where $K_{c_{\lambda},2\epsilon}=\{u\in W:|\Phi_{\lambda}(u)-c_{\lambda}|\leq 2\epsilon\}$. By Theorem E.5 from [10], there is a sequence $u_{n}\in W$ satisfying

[TABLE]

and

[TABLE]

By Corollary 4.1 we know that $c_{\lambda}<0$. Thus, by (4.4) we can apply Proposition 5.3 to conclude that $u_{n}\to\overline{u}_{\lambda}\in W\setminus 0$ so that $\Phi_{\lambda}(\overline{u}_{\lambda})=c_{\lambda}$ and $D_{u}\Phi_{\lambda}(\overline{u}_{\lambda})=0$. Moreover, once (4.5) is satisfied, we also have that $\overline{u}_{\lambda}\geq 0$. Now applying the Harnack inequality [15] we deduce that $\overline{u}_{\lambda}>0$ in $\Omega$.

∎

Conclusion of the proof of Theorem 1.1.

Let $\Lambda>\lambda^{*}$ be given by Lemma 3.2. Then Lemma 3.2 and Lemma 4.1 yield the existence of positive weak solutions $u_{\lambda},\overline{u}_{\lambda}$. Since $c_{\lambda}<0$, $\Phi_{\lambda}(\overline{u}_{\lambda})<0$. Thus in virtue that $\overline{u}_{\lambda}$ is a critical value of $\Phi_{\lambda}$, implies that $\Phi_{\lambda}^{\prime\prime}(\overline{u}_{\lambda})>0$. Corollary 4.1 and Lemma 4.1 imply that $\Phi_{\lambda}(u_{\lambda})<\Phi_{\lambda}(\overline{u}_{\lambda})<0$ for any $\lambda\in(\lambda^{*},\Lambda)$. Hence and by (li), Lemma 3.2 we get assertion (i) of the theorem. The proof of (ii) follows from (lii), Lemma 3.2.

∎

5 Appendix

Proposition 5.1**.**

For any $\lambda\geq\lambda^{*}$, the function $\mu\mapsto\hat{J}^{+}_{\lambda}(\mu)$ is continuous over the interval $(\lambda_{1},\lambda^{*})$.

Proof.

Let $\mu\in(\lambda_{1},\lambda^{*})$. Suppose, contrary to our claim, that there are $\mu_{n}\to\mu$ and $r>0$ such that $|\hat{J}^{+}_{\lambda}(\mu_{n})-\hat{J}^{+}_{\lambda}(\mu)|>r$ for all $n$, or equivalently

[TABLE]

for sufficiently large $n$. Suppose the first inequality is true, i.e., $\hat{J}^{+}_{\lambda}(\mu_{n})>\hat{J}_{\lambda}^{2}(\mu)+r$. From (3.1) this is possible only if $\mu_{n}<\mu$. Moreover, we can assume without loss of generality that $\mu_{n}$ is monotone increasing and consequently $\hat{J}_{\lambda}^{2}(\mu_{n})$ is decreasing. Thus $\hat{J}_{\lambda}^{+}(\mu_{n})\to I>\hat{J}_{\lambda}^{+}(\mu)$.

By Proposition 3.1, there is $v\in\mathcal{S}_{\lambda}(\mu)$ that is $J_{\lambda}^{2}(v)=\hat{J}_{\lambda}^{+}(\mu)$. Suppose $v\in\mathcal{S}_{\lambda}(\mu)\setminus\mathcal{S}^{\partial}_{\lambda}(\mu)$, then convergence $\mu_{n}\to\mu$ entails that there is $n$ such that $v\in\Theta_{\mu_{n}}^{2}$. However $J_{\lambda}^{+}(v)\geq\hat{J}_{\lambda}^{+}(\mu_{n})$ which contradicts to $J_{\lambda}^{+}(v)=\hat{J}_{\lambda}^{+}(\mu)<I\leq\hat{J}_{\lambda}^{+}(\mu_{n})$. Suppose now that $v\in\mathcal{S}^{\partial}_{\lambda}(\mu)$. Then, taking into account the continuity of $J_{\lambda}^{+}(u)$ on $\Theta_{\mu}^{+}$, we can choose $w\in\Theta_{\mu}^{+}$ such that $J_{\lambda}^{+}(v)\leq J_{\lambda}^{+}(w)<I$. However, there is $n$ such that $w\in\Theta_{\mu_{n}}^{+}$. This implies $\hat{J}^{+}_{\lambda}(\mu_{n})\leq J_{\lambda}^{+}(w)<I$ which is an absurd.

Now suppose the second inequality in (5.1) is true. Then $\mu<\mu_{n}$ and we may assume that $\mu_{n}$ is decreasing. Consequently $\hat{J}_{\lambda}^{+}(\mu_{n})$ is increasing and $\hat{J}_{\lambda}^{+}(\mu_{n})\to I<\hat{J}_{\lambda}^{+}(\mu)$. From Proposition 3.1, there is $v_{n}$ such that $v_{n}\in\mathcal{S}_{\lambda}(\mu_{n})$. If $v_{n}\in\Theta_{\mu}^{+}$ for some $n$ then $\hat{J}_{\lambda}^{+}(\mu)\leq J_{\lambda}^{+}(v_{n})=\hat{J}_{\lambda}^{+}(\mu_{n})$ which is contradicts to the assumption $\hat{J}_{\lambda}^{+}(\mu)>\hat{J}^{+}_{\lambda}(\mu_{n})$. Thus it is only possible that $v_{n}\in\overline{\Theta}^{+}_{\mu_{n}}\setminus\overline{\Theta_{\mu}^{+}}$ for all $n=1,2,...$.

By homogeneity of $J^{+}_{\lambda}(v)$ we may assume $||v_{n}||=1$. Hence by Proposition 2.2, $v_{n}\rightharpoonup v$ in $W$, $v_{n}\to v$ in $L^{p}(\Omega),L^{\gamma}(\Omega)$ for some $v\in W\setminus 0$. By the weak lower-semicontinuity we have that

[TABLE]

which implies that $v\in\overline{\Theta_{\mu}^{+}}$ and

[TABLE]

which is an absurd because $I<\hat{J}_{\lambda}^{+}(\mu)$.

∎

The next result can be found in [4]. We give a proof here for the reader’s convenience.

Proposition 5.2**.**

Let $u,v\in W\setminus 0$, $u,v\geq 0$ in $\Omega$ and define $\overline{\eta}(t)=[(1-t)u^{p}+tv^{p}]^{1/p}$ for $t\in[0,1]$. Suppose that the set $\{x\in\Omega:\ u(x)=v(x)=0\}$ has zero Lebesuge measure. Then

[TABLE]

and $\overline{\eta}\in C([0,1],W)$.

Proof.

First note that the weak derivative of $\eta(t)$ is given by $\nabla\eta(t)=[(1-t)u^{p}+tv^{p}]^{(1-p)/p}[(1-t)u^{p-1}\nabla u+tv^{p-1}\nabla v]$. Let $p^{\prime}$ be the conjugate exponent of $p$, i.e., $1/p+1/p^{\prime}=1$. From the Holder inequality, we have that

[TABLE]

for all $t\in[0,1]$. Once $\{x\in\Omega:\ u(x)=v(x)=0\}$ has zero Lebesuge measure, we have that $(1-t)u+tv>0$ a.e. in $\Omega$, for all $t\in(0,1)$ and therefore, from (5.2), we conclude that

[TABLE]

which implies

[TABLE]

Consequently $\overline{\eta}\in W$ for all $t\in[0,1]$. The continuity of $\eta$ follows by a standard application of the Lebesgue theorem.

∎

Proposition 5.3**.**

Suppose that $u_{n}\in W\setminus 0$ is a (P.-S.) sequence, i.e.

[TABLE]

Then $u_{n}$ has a strong convergent subsequence with non-zero limit point $u\in W\setminus 0$ satisfying $\Phi_{\lambda}(u)=c$ and $D_{u}\Phi(u)=0$.

Proof.

The assumption $D_{u}\Phi_{\lambda}(u_{n})\to 0$ entails $H_{\lambda}(u_{n})-F(u_{n})=o(1)$ and therefore

[TABLE]

This implies that $\|u_{n}\|$ is bounded, $||u_{n}||\geq\delta>0$ and $H_{\lambda}(u_{n})<0$ for sufficiently large $n$. Thus we may assume $u_{n}\rightharpoonup u$ in $W$, $u_{n}\to u$ in $L^{p}(\Omega)$ and $L^{\gamma}(\Omega)$ and $u\neq 0$. Hence and since $D_{u}\Phi(u_{n})\to 0$ as $n\to\infty$, we have

[TABLE]

Thus by $S^{+}$ property of the $p$-Laplacian operator (see [5]) we derive that $u_{n}\to u$ strongly in $W$.

∎