Multiple solutions for a generalized Schr\"{o}dinger problem with 'concave-convex' nonlinearities

TL;DR

This paper investigates a class of generalized Schrödinger elliptic problems with concave-convex nonlinearities, providing an overview of the solution set across various parameter values.

Contribution

It offers a comprehensive analysis of the solution structure for Schrödinger problems with mixed nonlinearities, extending existing results to broader parameter ranges.

Findings

Characterization of solution sets for different parameter regimes

Identification of multiple solutions under certain conditions

Extension of known results to generalized nonlinearities

Abstract

A class of generalized Schr\"{o}dinger elliptic problems involving concave-convex and other types of nonlinearities is studied. A reasonable overview about the set of solutions is provided when the parameters involved in the equation assume different real values.

Equations

$$P_{\lambda ,\mu ,q,p}$$

$$i{\partial }_{t}z=-\Delta z+V(x)z-\rho (|z{|}^2)z-\Delta (l(|z{|}^2))l^{\prime }(|z{|}^2)z,x\in \mathrm{I}{\mathrm{R}}^N,$$

$$-\Delta u+V(x)u-\Delta(l(u^{2}))l^{\prime}(u^{2})u=\rho(u)\ \mbox{in ${\rm I}\hskip-2.41849pt{\rm R}^{N}$}.$$

$${\vartheta }_{\ast }(s)=1+2s^2,{\vartheta }_{\#}(s)=1+\frac{s^2}{2(1+s^2)}+s^2\text{mbox}and{\vartheta }_{†}(s)=1+\ln (1+e^{s^2}),$$

$$P_{\lambda ,\mu ,q,p}$$

$$P_{\lambda ,\mu ,q,p}^{\prime }$$

$$f^{\prime }(s)=\frac{1}{\vartheta (f(s){)}^{1/2}}\text{mbox}fors>0\text{mbox}andf(0)=0,$$

$$\underset{|s|\to \infty }{\lim }\frac{|f(s)|}{\sqrt{|s|}}={\left(\frac{8}{{\alpha }^2}\right)}^{1/4}\text{mbox}and\underset{|s|\to \infty }{\lim }\frac{f(s)}{s}=0,$$

$${\rm Y}(t)={\int }_0^t\vartheta (r{)}^{1/2}dr.$$

$$\underset{s\to 0}{\lim }\frac{f(s)}{s}=\underset{s\to 0}{\lim }{f}^{\prime }(s)=\frac{1}{\vartheta (0{)}^{1/2}}.$$

$$r_1(s)=f(s)\vartheta (f(s){)}^{1/2}-s.$$

$$r_1^{\prime }(s)={\vartheta }^{\prime }(f(s))f(s)/2\vartheta (f(s))>0.$$

$$r_2(s)=2s-f(s)\vartheta (f(s){)}^{1/2}.$$

$$r_2^{\prime }(s)=1-{\vartheta }^{\prime }(f(s))f(s)/2\vartheta (f(s))\ge 0,$$

$${\left(\frac{f(s)}{\sqrt{s}}\right)}^{\prime }=\frac{2f^{\prime }(s)s-f(s)}{2s\sqrt{s}}\ge 0,\forall s>0,$$

$$\lim_{|s|\to\infty}\frac{|f(s)|}{\sqrt{|s|}}=l,\ \mbox{with $l\in(0,\infty]$}.$$

$$\underset{s\to \infty }{\lim }f(s)/\sqrt{s}=\infty .$$

$$\underset{s\to \infty }{\lim }\frac{f(s)}{\sqrt{s}}$$

$$\underset{s\to \infty }{\lim }f(s)/\sqrt{s}=0.$$

$$l=2\sqrt{\frac{\underset{s\to \infty }{\lim }{\left(\sqrt{s}/f(s)\right)}^2}{\underset{s\to \infty }{\lim }\vartheta (f(s))/f(s{)}^2}}=2\sqrt{\frac{1/l^2}{({\alpha }^2/2)}}.$$

$$l={\left(\frac{8}{{\alpha }^2}\right)}^{1/4}.$$

$$\underset{s\to \infty }{\lim }\frac{f(s)}{s}=\underset{s\to \infty }{\lim }\frac{f(s)}{\sqrt{s}}\frac{1}{\sqrt{s}}={\left(\frac{8}{{\alpha }^2}\right)}^{1/4}\times 0=0.$$

$${\int }_{\Omega }\nabla u\nabla vdx=\lambda {\int }_{\Omega }{f}^{\prime }(u){\left|f(u)\right|}^{q-2}f(u)vdx+\mu {\int }_{\Omega }{f}^{\prime }(u){\left|f(u)\right|}^{p-2}f(u)vdx,$$

$$J_{\lambda ,\mu }(u)=\frac{1}{2}\Vert u{\Vert }^2-\frac{\lambda }{q}{\int }_{\Omega }{\left|f(u)\right|}^{q}dx-\frac{\mu }{p}{\int }_{\Omega }{\left|f(u)\right|}^{p}dx.$$

$$\chi_{\left[1<\liminf\limits_{n\to\infty}u_{n}\right]}(x)\leq\liminf\limits_{n\to\infty}\chi_{[1<u_{n}]}(x)\ \mbox{in $\Omega$}.$$

$$g(x)=\underset{n\to \infty }{\mathrm{liminf}}{\chi }_{[1<u_n]}(x).$$

$${\chi }_{[1<u_{n_k}]}(x)=0,\forall k\in \mathrm{I}\mathrm{N}.$$

$$u_{n_k}(x)\le 1,\forall k\in \mathrm{I}\mathrm{N}.$$

$$u(x)=\underset{n\to \infty }{\mathrm{liminf}}{u}_n(x)\le \underset{k\to \infty }{\mathrm{liminf}}{u}_{n_k}(x)\le 1,$$

$${\chi }_{[1<u]}(x)=0.$$

Full text

$\ast$ Corresponding author: João R. Santos Júnior was partially supported by CNPq-Proc. 302698/2015-9 and CAPES-Proc. 88881.120045/2016-01, Brazil.

1. Introduction

We are interested in investigating the following classes of stationary generalized Schrödinger problems

[TABLE]

where $\Omega\subset{\rm I}\hskip-2.41849pt{\rm R}^{N}$, $N\geq 3$, is a bounded smooth domain, $1<q<4$, $\max\{2,q\}<p<22^{\ast}$, $\lambda$ and $\mu$ are real parameters and $\vartheta:{\rm I}\hskip-2.41849pt{\rm R}\to[1,\infty)$ is a general even $C^{1}$-function whose hypothesis will be posteriorly mentioned.

When $\Omega={\rm I}\hskip-2.41849pt{\rm R}^{N}$, equation ($P_{\lambda,\mu,q,p}$) is related to the existence of solitary wave solutions for the parabolic quasilinear Schrödinger equation

[TABLE]

where $z:{\rm I}\hskip-2.41849pt{\rm R}\times{\rm I}\hskip-2.41849pt{\rm R}^{N}\to\mathds{C}$, $V:{\rm I}\hskip-2.41849pt{\rm R}^{N}\to{\rm I}\hskip-2.41849pt{\rm R}$ is a given potential and $l,\rho$ are real functions. Equation (1.1) appears naturally as a model for several physical phenomena, depending on the type of function $l$ considered. In fact, if $l(s)=s$, (1.1) describes the behavior of a superfluid film in plasma physics, see [10]. For $l(s)=(1+s)^{1/2}$, (1.1) models the self-channeling of a high-power ultrashort laser in matter, see [1, 2, 3, 11]. Furthermore, (1.1) also appears in plasma physics and fluid mechanics [12], in dissipative quantum mechanics [9], in the theory of Heisenberg ferromagnetism and magnons [16] and in condensed matter theory [14].

If we take $z(t,x)=e^{-iEt}u(x)$ in (1.1), we get the corresponding steady state equation

[TABLE]

In the case that $\rho(s)=\lambda|s|^{q-2}s+\mu|s|^{p-2}s$ and ${\rm I}\hskip-2.41849pt{\rm R}^{N}$ is replaced by $\Omega$, problem (1.2) can be obtained from ($P_{\lambda,\mu,q,p}$), simply by choosing $\vartheta(s)=1+(l(s^{2})^{\prime})^{2}/2$, for some $C^{2}$-function $l$.

Many authors have studied stationary Schrödinger problems like ($P_{\lambda,\mu,q,p}$) under different nonlinearities and functions $\vartheta$, when $\Omega={\rm I}\hskip-2.41849pt{\rm R}^{N}$. Without any intention to provide a complete overview about the matter, we just refer the reader to some seminal contributions: In the case $\vartheta(s)=1+2s^{2}$, see [4, 7, 8, 6, 13, 15, 20, 22]. In the case $\vartheta(s)=1+s^{2}/2(1+s^{2})$, see [5, 18, 19].

The main goal of the present paper is provide a reasonable outline about the existence of multiple solutions for problem ($P_{\lambda,\mu,q,p}$), when the parameters involved assume different values and function $\vartheta$ satisfies general conditions which cover some of the cases previously mentioned. More specifically, we are assuming that:

1. $(\vartheta_{1})$

$s\mapsto\vartheta(s)$ is decreasing in $(-\infty,0)$ and increasing in $(0,\infty)$; 2. $(\vartheta_{2})$

$s\mapsto\vartheta(s)/s^{2}$ nondecreasing in $(-\infty,0)$ and nonincreasing in $(0,\infty)$; 3. $(\vartheta_{3})$

$\lim_{|s|\to\infty}\vartheta(s)/s^{2}=\alpha^{2}/2$, for some $\alpha>0$.

Some examples of functions satisfying $(\vartheta_{1})-(\vartheta_{3})$ can be given by:

[TABLE]

other examples can be found in [17], where the authors consider the problem ($P_{\lambda,\mu,q,p}$) with power type nonlinearities.

Due to the nature of the generalized Schrödinger operator, some interesting phenomena can be observed when one compares ($P_{\lambda,\mu,q,p}$) to the classical concave-convex problem involving the laplacian operator. For example, results of existence of infinitely many solutions with “high energy”, commonly influenced by convex part of the nonlinearity, are just occurring when $p>4$. Moreover, multiplicity of solutions with “low energy” has been obtained for values of $q$ that are not in the interval $(1,2)$. More specifically, what is noticed is the existence of a “grey zone”, namely, $2\leq q<p\leq 4$, where the set of solutions has an intermediate behaviour, presenting simultaneously influence of both powers as well as of the length of $\lambda$ and $\mu$, see Theorem 1.2. Our main results are as follows:

Theorem 1.1**.**

The following claims hold:

1. $(i)$

If $\lambda,\mu\leq 0$, then ($P_{\lambda,\mu,q,p}$) does not have any nontrivial solution; 2. $(ii)$

Suppose that $\vartheta$ satisfies $(\vartheta_{1})-(\vartheta_{2})$, $1<q\leq 2$ and $p\geq 4$ hold. If $\lambda<0$, then ($P_{\lambda,\mu,q,p}$) does not have solutions $u$ satisfying $J_{\lambda,\mu}(f^{-1}(u))\leq 0$. Analogously, if $\mu<0$, then ($P_{\lambda,\mu,q,p}$) does not have solutions $u$ satisfying $J_{\lambda,\mu}(f^{-1}(u))\geq 0$; 3. $(iii)$

Suppose that $\vartheta$ satisfies $(\vartheta_{1})-(\vartheta_{3})$. If $\max\{2,q\}<p\leq 4$ and $\lambda<0$, then there exists $\mu_{\ast}>0$ such that ($P_{\lambda,\mu,q,p}$) does not have any nontrivial solution, whatever $\mu\in(0,\mu_{\ast})$. Moreover, if $1<q<2<p\leq 4$ and $\lambda>0$, then there exists $s_{\ast}>0$ such that ($P_{\lambda,\mu,q,p}$) does not have solutions $u$ satisfying $J_{\lambda,\mu}(f^{-1}(u))\geq 0$, whatever $\mu\in(-s_{\ast},s_{\ast})$. 4. $(iv)$

Suppose that $\vartheta$ satisfies $(\vartheta_{1})-(\vartheta_{3})$. If $2\leq q<4$ and $\mu<0$, then there exists $\lambda_{\ast}>0$ such that ($P_{\lambda,\mu,q,p}$) does not have any nontrivial solution, whatever $\lambda\in(0,\lambda_{\ast})$. Moreover, if $2\leq q<p\leq 4$ and $\mu>0$, then there exists $t_{\ast}>0$ such that ($P_{\lambda,\mu,q,p}$) does not have solutions $u$ satisfying $J_{\lambda,\mu}(f^{-1}(u))\leq 0$, whatever $\lambda\in(-t_{\ast},t_{\ast})$. 5. $(v)$

Suppose that $\vartheta$ satisfies $(\vartheta_{1})-(\vartheta_{3})$. If $2\leq q<p\leq 4$, then there exist $r_{\ast}>0$ such that ($P_{\lambda,\mu,q,p}$) does not have any nontrivial solution, whatever $\lambda,\mu\in(-r_{\ast},r_{\ast})$.

Theorem 1.2**.**

Suppose that $\vartheta$ satisfies $(\vartheta_{1})-(\vartheta_{3})$. The following claims hold:

1. $(i)$

Let $\lambda\in{\rm I}\hskip-2.41849pt{\rm R}$, $\mu>0$ and $1<q<4$. If $4<p<22^{\ast}$, then ($P_{\lambda,\mu,q,p}$) has a sequence of solutions $\{u_{n}\}$ with $J_{\lambda,\mu}(f^{-1}(u_{n}))\to\infty$. Furthermore, if $\max\{q,2\}<p<4$, then for each $k\in{\rm I}\hskip-2.41849pt{\rm N}$ there exists $\mu_{k}>0$ such that $\eqref{P}$ has at least $k$ pairs of nontrivial solutions $u_{k}$ with $J_{\lambda,\mu}(f^{-1}(u_{k}))>0$, provided that $\mu\in(\mu_{k},\infty)$; 2. $(ii)$

Let $\lambda>0$, $\mu\in{\rm I}\hskip-2.41849pt{\rm R}$ and $p\neq 4$. If $1<q<2$, then ($P_{\lambda,\mu,q,p}$) has a sequence of solutions $\{u_{n}\}$ with $J_{\lambda,\mu}(f^{-1}(u_{n}))<0$ and $J_{\lambda,\mu}(f^{-1}(u_{n}))\to 0$. Furthermore, if $2\leq q<4$, then for each $k\in{\rm I}\hskip-2.41849pt{\rm N}$ there exists $\lambda_{k}>0$ such that $\eqref{P}$ has at least $k$ pairs of nontrivial solutions $u_{k}$ with $J_{\lambda,\mu}(f^{-1}(u_{k}))<0$, provided that $\lambda\in(\lambda_{k},\infty)$. 3. $(iii)$

Let $\lambda>0$, $\mu<\lambda_{1}\alpha^{2}/4$ and $p=4$. Then, for each $k\in{\rm I}\hskip-2.41849pt{\rm N}$ there exists $\lambda_{k}>0$ such that $\eqref{P}$ has at least $k$ pairs of nontrivial solutions $u_{k}$ with $J_{\lambda,\mu}(f^{-1}(u_{k}))<0$, provided that $\lambda\in(\lambda_{k},\infty)$, where $\alpha$ is defined in $(\vartheta_{3})$.

Throughout the paper $|A|$ denotes the Lebesgue measure of a measurable set $A\subset{\rm I}\hskip-2.41849pt{\rm R}^{N}$, $[1<u]:=\{x\in\Omega:1<u(x)\}$, $\lambda_{1}$ is the first eigenvalue of laplacian operator with homogeneous Dirichlet boundary condition and $C,C_{0},C_{1},C_{2}$ stand for positive constants whose exact value is not relevant for our purpose.

The paper is organized as follows.

In Section 2 we study a suitable change of variable which becomes problem ($P_{\lambda,\mu,q,p}$) in a more manageable one. In Section 3 we prove nonexistence results. In Section 4 we prove existence results.

2. Preliminaries

Our approach consists in switching the task to look for solutions of the general semilinear problem

[TABLE]

by task to find solutions of

[TABLE]

where $f\in C^{2}({\rm I}\hskip-2.41849pt{\rm R})$ is a solution of the ordinary differential equation

[TABLE]

with $f(s)=-f(-s)$ for $s\in(-\infty,0)$. Since $f$ is odd and $\vartheta$ is even, equation (ODE) is yet true for negative values. It is well known that $v$ is a weak solution of ($P_{\lambda,\mu,q,p}^{\prime}$) if, and only if, $u=f(v)$ is a weak solution of ($P_{\lambda,\mu,q,p}$), see [17] or [18].

Despite the proof of next lemma can also be found in [17], for the reader’s convenience and by its relevant role throughout the paper, we provide it here.

Lemma 2.1**.**

Let $\vartheta\in C^{1}({\rm I}\hskip-2.41849pt{\rm R})$ and $f$ a solution of (ODE). The following claims hold:

1. $(i)$

$f$* is uniquely defined and it is an increasing $C^{2}$-diffeomorphism, with $f^{\prime\prime}(s)=-\vartheta^{\prime}(f(s))/2\vartheta(f(s))^{2}$, for all $s>0$;* 2. $(ii)$

$0<f^{\prime}(s)\leq 1$, for all $s\in{\rm I}\hskip-2.41849pt{\rm R}$; 3. $(iii)$

$\lim_{s\to 0}f(s)/s=1/\vartheta(0)^{1/2}$; 4. $(iv)$

$|f(s)|\leq|s|$, for all $s\in{\rm I}\hskip-2.41849pt{\rm R}$; 5. $(v)$

Suppose $(\vartheta_{1})-(\vartheta_{2})$ hold. Then, $|f(s)|/2\leq f^{\prime}(s)|s|<|f(s)|$, for all $s\in{\rm I}\hskip-2.41849pt{\rm R}\backslash\{0\}$, and the map $s\mapsto|f(s)|/\sqrt{|s|}$ is nonincreasing in $(-\infty,0)$ and nondecreasing in $(0,\infty)$; 6. $(vi)$

Suppose that $(\vartheta_{1})-(\vartheta_{3})$ hold. Then,

[TABLE]

where $\alpha$ is given in $(\vartheta_{3})$.

Proof.

$(i)$-$(ii)$ Existence, uniqueness, regularity, monotonicity and $(ii)$ follow directly from (ODE). To see that $f({\rm I}\hskip-2.41849pt{\rm R})={\rm I}\hskip-2.41849pt{\rm R}$, observe that $f(s)=(\Upsilon^{-1})(s)$, where

[TABLE]

Since $\vartheta\geq 1$, $|\Upsilon(t)|\geq|t|$ for all $t\in{\rm I}\hskip-2.41849pt{\rm R}$. Consequently, $\lim_{|t|\to\infty}|\Upsilon(t)|=\infty$. Thence, $\lim_{|s|\to\infty}|f(s)|=\infty$.

$(iii)$ Notice that, by L’Hôspital rule, we get

[TABLE]

$(iv)$ It follows from $(ii)$. $(v)$ Since $f$ is odd and $\vartheta$ is even, it is sufficient to prove the inequalities for $s>0$. For that, let $r_{1}:[0,\infty)\to{\rm I}\hskip-2.41849pt{\rm R}$ defined by

[TABLE]

Notice that $r_{1}(0)=0$ and, by (ODE) and $(\vartheta_{1})$, we have

[TABLE]

Therefore, the second inequality in $(v)$ follows. Now, to prove the first inequality in $(v)$, let $r_{2}:[0,\infty)\to{\rm I}\hskip-2.41849pt{\rm R}$ be defined by

[TABLE]

We have that $r_{2}(0)=0$ and, by (ODE) and $(\vartheta_{2})$,

[TABLE]

showing that the inequality in $(v)$ holds. Moreover, since

[TABLE]

the second part of $(v)$ follows.

$(vi)$ Observe that from $(v)$, we have

[TABLE]

Again, since $f$ is odd and $\vartheta$ is even, it is sufficient to consider the case $s\to\infty$. Suppose that

[TABLE]

If this is the case then, by $(i)$, we get $f(s)\to\infty$ as $s\to\infty$. By applying the L’Hôspital rule and using $(\vartheta_{3})$, we conclude from (2.1), that

[TABLE]

Showing that

[TABLE]

Since (2.2) contradicts (2.1), it follows that $0<\lim_{s\to\infty}f(s)/\sqrt{s}=l<\infty$. Applying one more time the L’Hôspital rule, we have

[TABLE]

Or equivalently,

[TABLE]

On the other hand, from (2.3),

[TABLE]

∎

Naturally, a weak solution of ($P_{\lambda,\mu,q,p}^{\prime}$) is a function $u\in H_{0}^{1}(\Omega)$ satisfying

[TABLE]

for all $v\in H_{0}^{1}(\Omega)$. Moreover, the energy functional $J_{\lambda,\mu}:H_{0}^{1}(\Omega)\to{\rm I}\hskip-2.41849pt{\rm R}$ associated to ($P_{\lambda,\mu,q,p}^{\prime}$) is

[TABLE]

Lemma 2.1 assures that the previous notion of weak solution makes sense, as well as ensures that functional $J_{\lambda,\mu}$ is well defined and is $C^{1}$. Before finishing this section, we are going to introduce two technical lemmas which will be very helpful later on.

Lemma 2.2**.**

Let $\{u_{n}\}$ be a sequence of measurable functions $u_{n}:\Omega\to{\rm I}\hskip-2.41849pt{\rm R}$. Then,

[TABLE]

Proof.

Let us define $u:=\liminf\limits_{n\to\infty}u_{n}$ and $g:\Omega\to\{0,1\}$ by

[TABLE]

If $g\equiv 1$, there is nothing to be proven. Otherwise, it is sufficient to prove that if $g(x)=0$, then $\chi_{[1<u]}(x)=0$. Indeed, observe that if $g(x)=0$ then there exists a subsequence $u_{n_{k}}$ where $\{n_{k}\}\subset{\rm I}\hskip-2.41849pt{\rm N}$ depends on $x$, such that

[TABLE]

Equivalently,

[TABLE]

Passing to the lower limit as $k$ goes to infinity, we obtain

[TABLE]

or yet

[TABLE]

∎

Now on, let us agree that, $\{e_{j}\}$ stands for a Hilbertian basis of $H_{0}^{1}(\Omega)$ composed by functions in $L^{\infty}(\Omega)$ (for example the basis composed by eigenfunctions of laplacian operator with Dirichlet boundary condition),

[TABLE]

Since $|f(s)|$ behaves like $|s|$ near the origin and like $|s|^{1/2}$ at infinity, next lemma will be very helpful to get some important estimates for the existence results.

Lemma 2.3**.**

Let $S_{k}$ be the unit sphere of $Y_{k}$. There exist positive constants $\beta_{k},\beta_{k}(r),\alpha_{k},\tau_{k}$ such that:

1. $(i)$

[TABLE]

for all $u\in S_{k}$ and $s>\alpha_{k}$, and

[TABLE]

for all $u\in S_{k}$ and $0<s<\tau_{k}$. 2. $(ii)$

for each $r\in[1,2^{\ast}]$,

[TABLE]

*for all $u\in S_{k}$ and $s>\alpha_{k}$. *

Proof.

$(i)$ First, we are going to prove that (2.6) holds. Indeed, suppose that there exist $\{s_{n}\}\subset(0,\infty)$ and $\{u_{n}\}\subset S_{k}$ with $s_{n}\to\infty$ and

[TABLE]

Since $Y_{k}$ has finite dimension, there exists

[TABLE]

such that, up to a subsequence, $u_{n}\to u$ in $H_{0}^{1}(\Omega)$ and

[TABLE]

Therefore,

[TABLE]

It follows from (2.10), (2.11), Lemma 2.2$(i)$, Fatou Lemma and (2.9) that

[TABLE]

A clear contradiction. Therefore (2.6) holds. Now, in order to prove (2.7), observe that if $u\in S_{k}$ then, by Cauchy-Schwarz inequality

[TABLE]

where $M:=\max_{j=0}^{k}|e_{j}|_{\infty}$. Consequently, choosing $\tau_{k}:=1/(k+1)M^{2}$ the result follows.

$(ii)$ By Fatou Lemma, Lemma 2.2 and since $Y_{k}$ has finite dimension, we have

[TABLE]

for all $u\in S_{k}$ and some $\zeta_{k}(r)>0$. Choosing $0<\beta_{k}(r)<\zeta_{k}(r)$, the result is proven.

∎

3. Nonexistence results

**Proof of Theorem 1.1:

**

$(i)$ Indeed, by $f(0)=0$ and Lemma 2.1$(ii)$ we have $f(s)s\geq 0$ for all $s\in{\rm I}\hskip-2.41849pt{\rm R}$. Thus, if $u$ is a solution, then

[TABLE]

Therefore $u=0$.

$(ii)$ Suppose that $\lambda<0$ and $u$ is a nontrivial weak solution of ($P_{\lambda,\mu,q,p}^{\prime}$). By previous item, we have $\mu>0$. By Lemma 2.1$(v)$,

[TABLE]

If $J_{\lambda,\mu}(u)\leq 0$, then

[TABLE]

Thus,

[TABLE]

By comparing (3.1) and (3.2), we get

[TABLE]

whenever $1<q\leq 2$ and $p\geq 4$. A clear contradiction.

Now, let $\mu<0$ and $u$ be a weak solution of ($P_{\lambda,\mu,q,p}^{\prime}$). Again, by item $(i)$, we have $\lambda>0$. By Lemma 2.1$(v)$,

[TABLE]

If $J_{\lambda,\mu}(u)\geq 0$, then

[TABLE]

Thus,

[TABLE]

Comparing (3.3) and (3.4), we get

[TABLE]

for all $1<q\leq 2$ and $p\geq 4$. The result follows.

$(iii)$ If $\max\{2,q\}<p\leq 4$, $\lambda<0$ and $u$ is a nontrivial weak solution of ($P_{\lambda,\mu,q,p}^{\prime}$), then, by $f(0)=0$ and Lemma 2.1$(ii)$, $f(s)s\geq 0$ for all $s\in{\rm I}\hskip-2.41849pt{\rm R}$. Moreover, by item $(i)$, we have $\mu>0$. Thence,

[TABLE]

By Lemma 2.1$(v)$,

[TABLE]

It follows from items $(v)$ and $(vi)$ of Lemma 2.1 that

[TABLE]

for all $|s|>1$. Thus, by Lemma 2.1$(iv)$ and since $2\leq p\leq 4$,

[TABLE]

By (3.5), (3.6) and Sobolev embeddings,

[TABLE]

Since $u$ is a nontrivial solution, we obtain

[TABLE]

To prove the second part, suppose that $\lambda>0$ and $u$ is a nontrivial solution with $J_{\lambda,\mu}(u)\geq 0$. It follows from Lemma 2.1$(v)$ that

[TABLE]

Consequently,

[TABLE]

As $2\leq p\leq 4$, by (3.6),

[TABLE]

Since $1<q<2$, we have

[TABLE]

The result is proven.

$(iv)$ Let $2\leq q<4$, $\mu<0$ and $u$ be a nontrivial weak solution of ($P_{\lambda,\mu,q,p}^{\prime}$), by Lemma 2.1$(v)$

[TABLE]

By item $(i)$, (3.6) and Sobolev embeddings,

[TABLE]

Since $u$ is a nontrivial solution, we obtain

[TABLE]

Finally, suppose that $\mu>0$ and $u$ is a nontrivial solution with $J_{\lambda,\mu}(u)\leq 0$. It follows from Lemma 2.1$(v)$ that

[TABLE]

Since $p<4$,

[TABLE]

Since $2\leq q<4$, by (3.6)

[TABLE]

Therefore

[TABLE]

$(v)$ Let $2\leq q<p\leq 4$ and $u$ be a nontrivial weak solution of ($P_{\lambda,\mu,q,p}^{\prime}$). By Lemma 2.1$(v)$ and (3.6),

[TABLE]

Since $u$ is nontrivial, the result follows. $\square$

4. Multiplicity of solutions

The proof of the existence results will be divided in several propositions. Before, we need to introduce some definitions. We say that $J_{\lambda,\mu}$ satisfies the $(PS)_{c}^{\ast}$ condition, with respect to $\{Y_{n}\}$, if any sequence $\{u_{n}\}\subset H_{0}^{1}(\Omega)$, such that

[TABLE]

contains a subsequence converging to a critical point of $J_{\lambda,\mu}$. Any sequence $\{u_{n}\}\subset H_{0}^{1}(\Omega)$ satisfying (4.1) is said to be a $(PS)_{c}^{\ast}$ for $J_{\lambda,\mu}$. It is well known that the $(PS)_{c}^{\ast}$ condition implies the classical $(PS)_{c}$ condition, see [21].

Proposition 4.1**.**

Suppose $(\vartheta_{1})-(\vartheta_{3})$ hold.

1. $(i)$

If $p=4$, then $J_{\lambda,\mu}$ satisfies the $(PS)_{c}^{\ast}$ condition, for all $1<q<4$, $\lambda\in{\rm I}\hskip-2.41849pt{\rm R}$ and $\mu<\lambda_{1}\alpha^{2}/4$; 2. $(ii)$

If $p\neq 4$, then $J_{\lambda,\mu}$ satisfies the $(PS)_{c}^{\ast}$ condition, for all $1<q<\min\{4,p\}$ and $\lambda,\mu\in{\rm I}\hskip-2.41849pt{\rm R}$.

Proof.

$(i)$ Let $p=4$ and $\{u_{n}\}$ be a $(PS)_{c}^{\ast}$ sequence for $J_{\lambda,\mu}$, i.e., (4.1) holds. If $\lambda>0$ and $\mu\leq 0$, it follows by Lemma 2.1$(v)$ that

[TABLE]

Now, we have to consider two cases: if $1<q\leq 2$, we conclude from Lemma 2.1$(iv)$ and Sobolev embedding that

[TABLE]

Before consider the case $2<q<4$, observe that, we cannot use the Lemma 2.1$(iv)$ in the same way as previously because $|u|^{q}$ might not be integrable. To overcome this difficulty, we note that, by items $(v)$ and $(vi)$ of Lemma 2.1

[TABLE]

for all $s\in{\rm I}\hskip-2.41849pt{\rm R}$. By Lemma 2.1$(iv)$, for each $2\leq r\leq 22^{\ast}$,

[TABLE]

Thus, if $2<q<4$, it follows from (4.4) and Sobolev embedding that

[TABLE]

By (4.2) and (4.5), $\{u_{n}\}$ is bounded in $H_{0}^{1}(\Omega)$. If $\lambda,\mu>0$, by Lemma 2.1$(v)$, (4.4) and Sobolev embedding, we have

[TABLE]

Hence $\{u_{n}\}$ is bounded in $H_{0}^{1}(\Omega)$, if $\mu<\lambda_{1}\alpha^{2}/4$.

On the other hand, if $\lambda,\mu\leq 0$ we get

[TABLE]

showing that $\{u_{n}\}$ is bounded in $H_{0}^{1}(\Omega)$. If $\lambda\leq 0$ and $\mu>0$,

[TABLE]

Therefore $\{u_{n}\}$ is again bounded in $H_{0}^{1}(\Omega)$, if $\mu<\lambda_{1}\alpha^{2}/4$. Thence, up to a subsequence, we have

[TABLE]

[TABLE]

and

[TABLE]

Defining $v_{n}:=P_{Y_{n}}u$ as been the orthogonal projection of $u$ onto $Y_{n}$, we have

[TABLE]

Since $u_{n}-v_{n}\in Y_{n}$ and $\{u_{n}-v_{n}\}$ is bounded in $H_{0}^{1}(\Omega)$, we conclude that

[TABLE]

Thence,

[TABLE]

By (4.6), (4.7), (4.8) and (4.9), we conclude that

[TABLE]

The result follows now from (4.6) and (4.9).

$(ii)$ Let $p\neq 4$ and $\{u_{n}\}$ be a $(PS)_{c}^{\ast}$ sequence for $J_{\lambda,\mu}$. If $\lambda>0$ and $\mu\leq 0$ we can reason exactly like in the case $p=4$. On the other hand, if $\lambda,\mu>0$ we have to consider separately two cases: if $p<4$, it follows by Lemma 2.1$(v)$, (4.4) and Sobolev embedding that

[TABLE]

By estimating the last installment as (4.2) and (4.5) we conclude that $\{u_{n}\}$ is bounded in $H_{0}^{1}(\Omega)$. In the case $p>4$, it is sufficient to note that, by Lemma 2.1$(v)$

[TABLE]

Once more time the boundedness of $\{u_{n}\}$ in $H_{0}^{1}(\Omega)$ follows from a reasoning similar to (4.2) and (4.5).

Finally, if $\lambda,\mu\leq 0$, we argue exactly like in the case $p=4$ and, if $\lambda\leq 0$ and $\mu>0$, we have

[TABLE]

when $p<4$, and

[TABLE]

when $p>4$. In all cases we can conclude that $\{u_{n}\}$ is bounded in $H_{0}^{1}(\Omega)$. Now the result follows exactly equal to the case $p=4$.

∎

Proposition 4.2**.**

Suppose $(\vartheta_{1})-(\vartheta_{3})$, $4<p<22^{\ast}$ and $\mu>0$. Then there exist $0<r_{k}<\rho_{k}$ such that:

[TABLE]

and

[TABLE]

Proof.

To prove (4.11), observe that by Lemma 2.1$(v)$

[TABLE]

Thus, for each $u\in S_{k}$ and $\rho>0$

[TABLE]

By Lemma 2.3$(ii)$, there exist positive constants $\alpha_{k},\beta_{k}(p/2)$ such that, for every $u\in S_{k}$ and $\rho>\alpha_{k}$, we get

[TABLE]

Now, we are going to consider two cases: If $1<q\leq 2$, it follows from Lemma 2.1$(iv)$ and Sobolev embedding that

[TABLE]

Since $p>4$, choosing $\rho_{k}>\max\{1,[p(1/2+|\lambda|C_{1}/q)/\mu f(1)^{p}\beta_{k}(p/2)]^{2/(p-4)}\}$, we have

[TABLE]

for all $u\in S_{k}$. On the hand, if $2<q<4$, by (4.13), (4.4) and Sobolev embedding, we have

[TABLE]

Therefore, choosing $\rho_{k}>\max\{1,[p(1/2+|\lambda|(8/\alpha^{2})^{q/4}C_{1}/q)/\mu f(1)^{p}\beta_{k}(p/2)]^{2/(p-4)}\}$, we have

[TABLE]

for all $u\in S_{k}$. This proves (4.11).

To prove (4.12), note that for any $1\leq r<2^{\ast}$, we can define

[TABLE]

It is a straightforward consequence of compact Sobolev embeddings that

[TABLE]

see Lemma 3.8 in [21]. If $1<q<2$, by Lemma 2.1$(iv)$ and (4.4)

[TABLE]

By Sobolev embeddings and (4.14),

[TABLE]

for all $u\in Z_{k}$. Since $1<q<2$, for $\left\|u\right\|\geqslant R_{\ast}$ with $R_{\ast}>0$ large enough,

[TABLE]

for some $r>2p/(p-2)$. Thus, for $\left\|u\right\|\geqslant R_{\ast}$, we get

[TABLE]

It follows from (4.15) that, by choosing $r_{k}=1/[\mu(8/\alpha^{2})^{p/4}\theta_{p/2,k}^{p/2}]^{2/(p-4)}$, there exists $k_{0}\in{\rm I}\hskip-2.41849pt{\rm N}$ such that $r_{k}\geqslant R_{\ast}$ for all $k\geqslant k_{0}$. Therefore,

[TABLE]

for all $u\in Z_{k}$ with $\left\|u\right\|=r_{k}$ and $k\geqslant k_{0}$. Since $r_{k}\to\infty$ as $k\to\infty$, the result follows. If $2\leq q<4$, it follows from (4.4) that

[TABLE]

By Sobolev embeddings and (4.14),

[TABLE]

Now, since $1\leq q/2<2$, we can proceed in an analogous way to the case $1<q<2$ for the choice of $r_{k}$. Since we can choose $\rho_{k}$ even greater, in order to have $\rho_{k}>r_{k}$, the result follows.

∎

Proposition 4.3**.**

Suppose that $\vartheta$ satisfies $(\vartheta_{1})-(\vartheta_{3})$, $1<q<2$ and $\lambda>0$ hold. Then, there exists $0<r_{k}<\rho_{k}$ such

1. $(i)$

$\inf_{u\in Z_{k},\left\|u\right\|=\rho_{k}}J_{\lambda,\mu}(u)\geq 0$; 2. $(ii)$

$\max_{u\in Y_{k},\|u\|=r_{k}}J_{\lambda,\mu}(u)<0$; 3. $(iii)$

$\inf_{u\in Z_{k},\|u\|\leq\rho_{k}}J_{\lambda,\mu}(u)\to 0$* as $k\to\infty$.*

Proof.

$(i)$ Let us consider $p\geq 4$. Since $1<q<2$, by Lemma 2.1$(iv)$, (4.4) and (4.14), we get

[TABLE]

for all $u\in Z_{k}$. If $p\geq 4$, there exists $\delta>0$ small enough, such that

[TABLE]

for all $u\in Z_{k}$ with $\|u\|\leq\delta$ (and $k$ large enough if $p=4$). Thus, by choosing

[TABLE]

we have $(1/4)\rho_{k}^{2}=(\lambda/q)\theta_{q,k}^{q}\rho_{k}^{q}$. Consequently, $\rho_{k}\to 0$ as $k\to\infty$ and, therefore, there exists $k_{0}>0$ satisfying $\rho_{k}\leq\delta$ for all $k\geq k_{0}$. Finally, by (4.19)

[TABLE]

for all $u\in Z_{k}$, $k\geqslant k_{0}$, with $\left\|u\right\|=\rho_{k}$. On the other hand, if $2<p<4$, we conclude from (4.18) that

[TABLE]

for all $u\in Z_{k}$ with $\|u\|<1$, $1<\gamma:=\min\{q,p/2\}<2$, $\eta_{k}:=\max\{\theta_{q,k},\theta_{p/2,k}\}$ and $k\geq k_{0}$. Thus, by choosing

[TABLE]

with $k\geq k_{0}$, the result follows.

$(ii)$ By Lemma 2.1$(iii)$, there exists $\varepsilon>0$ such that

[TABLE]

for all $|s|\leq 1$. Thus,

[TABLE]

By the second part of Lemma 2.3$(i)$ and Lemma 2.1$(iv)$, we have

[TABLE]

for all $u\in S_{k}$ and $0<r<\tau_{k}$. Since $Y_{k}$ has finite dimension, there exists $\zeta_{k}(q)>0$ such that

[TABLE]

for all $u\in S_{k}$ and $0<r<\tau_{k}$, where in the last installment we use the fact that $p>2$. By Sobolev embeddings

[TABLE]

Thence,

[TABLE]

for all $0<r<\min\{1,\rho_{k},\tau_{k}\}$. Since $1<q<2$, by choosing

[TABLE]

the item is proven.

$(iii)$ By (4.20) and (4.21), we conclude that

[TABLE]

where, $o_{k}(1)\to 0$ as $k\to\infty$. Consequently, $b_{k}\to 0$ as $k\to\infty$. ∎

$\square$

Proof of Theorem 1.2$(i)$:

Since $J_{\lambda,\mu}$ is an even functional, the first part of Theorem 1.2$(i)$ is a direct consequence of Fountain Theorem in [21] and Propositions 4.1$(ii)$ and 4.2. To prove the second part, observe that if $1<q<2$, it follows from $\mu>0$, Lemma 2.1$(iv)$, (4.4) and Sobolev embeddings, that

[TABLE]

for all $u\in Z_{m}$. On the other hand, if $2\leq q<4$, it follows from $\mu>0$, (4.4) and Sobolev embeddings, that

[TABLE]

for all $u\in Z_{m}$. Consequently,

[TABLE]

where $\alpha:(1,4)\to[1,2)$ is give by $\alpha(s)=s$ if $1<s<2$ and $\alpha(s)=s/2$ if $2\leq s<4$. Thence, there exists $R_{\ast}$ large enough such that

[TABLE]

for all $u\in Z_{m}$ with $\|u\|\geq R_{\ast}$. Since $p<4$,

[TABLE]

for all $u\in Z_{m}$ with $\|u\|\geq\max\{R_{\ast},1\}$. Observe that there exists $m_{0}>0$ such that

[TABLE]

for all $m\geq m_{0}$. By choosing $r_{m}=\max\{R_{\ast},m\}$, we have

[TABLE]

Finally, by items $(iv)$ and $(v)$ of Lemma 2.1 and (4.4), there exists $C>0$ such that

[TABLE]

for all $u\in S_{m}$. It follows from Lemma 2.3$(ii)$ and Sobolev embedding that there exists $\alpha_{m},\beta_{m}(p/2)>0$ such that

[TABLE]

for some $\rho_{m}>\max\{\alpha_{m},r_{m}\}$ and for all $u\in S_{m}$. Therefore, there exists $\mu_{m}>0$ such that

[TABLE]

for all $\mu>\mu_{m}$. To finish the proof, let us define

[TABLE]

[TABLE]

and

[TABLE]

By definition of $c_{m}$ and Lemma 3.4 in [21], we have

[TABLE]

for all $m$. On the other hand, by (4.22), we conclude that

[TABLE]

for all $m\geq m_{0}$. It is also a consequence of (4.22) and (4.24) that given $k\in{\rm I}\hskip-2.41849pt{\rm N}$, there exists $m(k)>m_{0}$ with $k\leq m(k)-m_{0}$, such that we have at least $k$ different numbers $c_{j}$ when $m_{0}\leq j\leq m(k)$. Thus, by (4.23) and Theorem 3.5 in [21], there exist $\mu_{k}:=\mu_{m(k)}>0$ and a $(PS)_{c_{j}}$-sequence for $J_{\lambda,\mu}$, for each $m_{0}\leq j\leq m(k)$, whenever $\mu>\mu_{k}$. Finally, by Proposition 4.1$(ii)$, follows that the numbers $c_{j}$ are critical points of $J_{\lambda,\mu}$ as $\mu>\mu_{k}$. $\square$

Proof of Theorem 1.2$(ii)$:

Since $J_{\lambda,\mu}$ is an even functional, the proof of first part of Theorem 1.2$(ii)$ follows from Dual Fountain Theorem in [21] and Propositions 4.1$(ii)$ and 4.3. To prove the second part, note that, since $2\leq q<4$ and $\lambda>0$, it follows by (4.4) and Sobolev embeddings, that

[TABLE]

for all $u\in Z_{m}$. Thus, for $m$ large enough, we have $0<\eta_{m}:=\max\{\theta_{q/2,m},\theta_{p/2,m}\}<1$ and

[TABLE]

for all $u\in Z_{m}$ with $\|u\|<1$. By choosing $\rho_{m}=\left[2\left(\lambda/q+|\mu|/p\right)(8/\alpha^{2})^{q/4}\eta_{m}^{q/2}\right]^{2/(4-q)}$, it follows that for $m\geq m_{\ast}$, with $m_{\ast}$ large enough

[TABLE]

On the other hand, by Lemma 2.1$(iii)$ and (4.4)

[TABLE]

for all $u\in S_{m}$. It follows from Lemma 2.3$(i)$ that there exists $\tau_{m}>0$ such that

[TABLE]

for some $0<r_{m}<\min\{\tau_{m},\rho_{m}\}$ fixed and for all $u\in S_{m}$. Despite $q$ can be greater than $2^{\ast}$ when the dimension $N$ is large enough, it is a consequence of definition of $Y_{m}$ that $Y_{m}\subset L^{\infty}(\Omega)$ and, therefore, $|.|_{q}$ defines a norm in $Y_{m}$. Since $Y_{m}$ has finite dimension,

[TABLE]

for some $\zeta_{m}(q)>0$. Therefore, there exists $\lambda_{m}>0$ such that

[TABLE]

for all $\lambda>\lambda_{m}$.

Finally, by (4.25), we conclude that

[TABLE]

where, $o_{m}(1)\to 0$ as $m\to\infty$. Showing that

[TABLE]

To finish the proof, for each $t\geq m\geq m_{\ast}$, we are going to apply the Theorem 3.5 in [21] to the functional $-J_{\lambda,\mu}$ on $Y_{t}$, for this, let us define:

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

By definition of $c_{m}^{t}$ and Lemma 3.4 in [21], we have

[TABLE]

for all $t\geq m\geq m_{\ast}$. Therefore, up to a subsequence, there exists

[TABLE]

such that

[TABLE]

From (4.27), (4.28) and (4.30), given $k\in{\rm I}\hskip-2.41849pt{\rm N}$, there exist $m(k)$ with $k<m(k)-m_{\ast}$ and $\lambda_{k}:=\lambda_{m(k)}>0$ such that we have at least $k$ different numbers $c_{m}$ as $m_{\ast}\leq m\leq m(k)$, whenever $\lambda>\lambda_{k}$. Thus, by Theorem 3.5 in [21], for each $m_{\ast}\leq m\leq m(k)$, there exists $u_{t}\in Y_{t}$ such that

[TABLE]

whenever $\lambda>\lambda_{k}$. Consequently, by (4.31) and (4.32), up to a subsequence, $\{u_{t}\}$ is a $(PS)^{\ast}_{c_{m}}$ sequence. By Proposition 4.1$(ii)$, $c_{m}$ is a critical point of $J_{\lambda,\mu}$ for all $m_{\ast}\leq m\leq m(k)$. The result follows. $\square$

Proof of Theorem 1.2$(iii)$: It is sufficient to argue exactly like in the proof of the second part of Theorem 1.2$(ii)$ and use Proposition 4.1$(i)$ instead of Proposition 4.1$(ii)$.

$\square$