1 Introduction

In this paper, we are concerned with the existence and multiplicity of nontrivial weak solutions for the non-uniformly elliptic equations of N-Laplacian type of the form

\begin{aligned} -\text {div}(a(x,\nabla u))+V(x)\vert u\vert ^{N-2} u=\lambda \Bigg (\exp \Bigg (\alpha \vert u\vert ^{\frac{N}{N-1}}\Bigg )+f(x,u)\Bigg ) ~~~ \text {in} \; ~\mathbb {R}^N, \end{aligned}
(1.1)

where $$N\ge 2$$, $$\alpha$$ is some positive constant and $$\lambda$$ is some positive parameter, $$V:\mathbb {R}^N\rightarrow ]0,+\infty [$$ is a continuous function such that $$V(x)\ge V_0$$ for some positive constant $$V_0$$ and $$V^{-1}\in L^{\frac{1}{N-1}}(\mathbb {R}^N)$$ or $$\lim V(x)\rightarrow \infty$$ as $$|x|\rightarrow \infty$$, i.e., $$\text {meas}\lbrace x\in \mathbb {R}^N : V(x)\le M\rbrace <\infty$$ for every $$M>0$$. Motivated by [6], we assume that A be a measurable function on $$\mathbb {R}^N\times \mathbb {R}$$ such that $$A(x,0)=0$$ and $$a(x,\tau )=\frac{\partial {A(x,\tau )}}{\partial \tau }$$ is a Carathéodory function on $$\mathbb {R}^N\times \mathbb {R}$$, and there are positive real numbers $$c_0, c_1, k_1$$ and two nonnegative measurable functions $$h_0, h_1$$ on $$\mathbb {R}^N$$ such that $$h_0\in L^{N/(N-1)}(\mathbb {R}^N)$$, $$h_1\in L_\mathrm{loc}^{\infty }(\mathbb {R}^N)$$ and $$h_1(x)\ge 1,$$ satisfying the properties:

($$A_1$$):

$$\vert a(x,\tau )\vert \le c_0(h_0(x)+h_1(x)\vert \tau \vert ^{N-1}),~~\forall \tau \in \mathbb {R}^N, \text {a.e.}~~x\in \mathbb {R}^N,$$

($$A_2$$):

$$0\le a(x,\tau ).\tau \le NA(x,\tau ) \quad \forall \tau \in \mathbb {R}^N, \text {a.e.} ~~x\in \mathbb {R}^N,$$

($$A_3$$):

$$A(x,\tau )\ge k_0 h_1(x)\vert \tau \vert ^N \quad \forall \tau \in \mathbb {R}^N, \text {a.e.} ~~x\in \mathbb {R}^N.$$

Then A verifies the growth condition:

\begin{aligned} \vert A(x,\tau )\vert \le c_0(h_0(x)\vert \tau \vert +h_1(x)\vert \tau \vert ^N), \quad \forall \tau \in \mathbb {R}^N, \; \text {a.e.~} x\in \mathbb {R}^N. \end{aligned}

Note that in the case of N-Laplacian, i.e., $$A(x,\tau )=\frac{1}{N}|\tau |^N$$, we choose

\begin{aligned} a(x,\tau )=\vert \tau \vert ^{N-2}\tau , \quad k_0=\frac{1}{N}, \; h_1(x)=1, \end{aligned}

which has been studied extensively, both in the case $$N=2$$ (i.e., Laplacian equation in $$\mathbb {R}^2$$) and in the case $$N>3$$ (i.e., N-Laplacian equation in $$\mathbb {R}^N$$).

The problems of this type are important in many fields of sciences, for example, in electromagnetism, astronomy and fluid dynamics. In fact, these models describe potentials of electric, gravitation and fluid, respectively.

In the case $$p<N$$, by the Sobolev embedding, the critical exponent is $$p^*=\frac{pN}{N-p}$$. When $$p=N$$, one has another maximal growth for any bounded domain $$\Omega$$, i.e.,

\begin{aligned} \sup _{u\in W_0^{1,N}(\Omega ), ||\nabla u||_{L^N}\le 1}\frac{1}{|\Omega |}\int _{\Omega }\exp (\alpha _N\vert u\vert ^{N/(N-1)})\mathrm{d}x<\infty , \end{aligned}

where $$\alpha _N=NW_{N-1}^{\frac{1}{N-1}}$$ and $$W_{N-1}$$ is the surface area of the unit sphere in $$\mathbb {R}^N$$. In this situation, the constant $$\alpha _N$$ is sharp.([1, 4, 7])

In the case of unbounded domain, if we replace $$||\nabla u||_{L^N}$$ in the supremum by the standard Sobolev norm, then the following expression (Trudinger–Moser inequality) can be derived

\begin{aligned} \sup _{u\in W_0^{1,N}(\mathbb {R}^N), ||\nabla u||^N_{L^N}+|| u||^N_{L^N}\le 1}\int _{\mathbb {R}^N}(\exp (\alpha \vert u\vert ^{N/(N-1)}) -S_{N-2}(\alpha ,u))\mathrm{d}x=\Xi , \end{aligned}

where $$\Xi \le \infty$$ if $$\alpha \le \alpha _N$$ and $$\Xi =\infty$$ if $$\alpha >\alpha _N$$. In this expression, $$S_{N-2}(\alpha ,u)= \sum _{k=0}^{N-2}\frac{\alpha ^k}{k!}\vert u\vert ^{\frac{kN}{N-1}}$$ (see [5, 9]).

Motivated by the Trudinger–Moser inequality, we consider the following assumptions on f which allows us to treat the problem (1.1) variationally in a subspace of $$W^{1,N}(\mathbb {R}^N)$$. We assume that $$f:\mathbb {R}^N\times \mathbb {R}\rightarrow \mathbb {R}$$ be a Carathéodory function that $$\forall (x,s)\in \mathbb {R}^N\times [0,+\infty ], ~~f(x,s)\ge 0,$$ and

\begin{aligned} \vert f(x,u)\vert \le b_1\vert u\vert ^{\beta _1}+b_2\vert u\vert ^{\beta _2}(\exp (\alpha _0\vert u\vert ^{\frac{N}{N-1}})- S_{N-2}(\alpha _0,u)),~~~\forall (x,u)\in \mathbb {R}^N\times \mathbb {R}^+, \end{aligned}
(1.2)

for some $$b_1,b_2,\alpha _0>0$$ and $$\beta _1,\beta _2\ge 0$$.

Next, we introduce some notations

\begin{aligned} E=\bigg \{ u\in W_{0}^{1,N}(\mathbb {R}^N): \int _{\mathbb {R}^N}h_1(x)\vert \nabla u\vert ^N~\mathrm{d}x+\int _{\mathbb {R}^N} V(x)\vert u\vert ^N<\infty \bigg \}, \end{aligned}

which since V is positive and bounded from below, E is a reflexive Banach space when endowed with the norm

\begin{aligned} \parallel u\parallel _E=\left( \displaystyle \int _{\mathbb {R}^N}(h_1(x)\vert \nabla u\vert ^N+\frac{1}{k_0N}V(x)\vert u\vert ^N)~\mathrm{d}x\right) ^{1/N}. \end{aligned}

Then for $$q\in [N,+\infty [$$

\begin{aligned} E\hookrightarrow W^{1,N}(\mathbb {R}^N)\hookrightarrow L^q(\mathbb {R}^N), \end{aligned}

with continuous embedding. Thus, there exists a positive constant $$c_2$$ such that

\begin{aligned} \parallel u\parallel _{1,N}\le c_2\parallel u\parallel _E,~~\forall ~u\in E, \end{aligned}
(1.3)

where

\begin{aligned} \parallel u\parallel _{1,N}=\left( \displaystyle \int _{\mathbb {R}^N}(\vert \nabla u\vert ^N+\vert u\vert ^N\right) ~\mathrm{d}x)^{\frac{1}{N}} \end{aligned}

is the norm in the Sobolev space $$W^{1,N}(\mathbb {R}^N)$$.

Using the radial lemma which asserts

\begin{aligned} |u(x)|\le |x|^{-1}\left( \frac{N}{W_{N-1}}\right) ^{\frac{1}{N}}||u||_{N}~~\forall x\ne 0, \end{aligned}

for all $$u\in W^{1,N}(\mathbb {R}^N)$$ radially nonincreasing symmetric ([2, 5]), there exists a constant $$c_3>0$$ such that

\begin{aligned} \displaystyle \int _{\mathbb {R}^N}\left[ \exp \Bigg (\alpha \vert u\vert ^{\frac{N}{N-1}}\Bigg )-S_{N-2}(\alpha , u)\right] ~\mathrm{d}x\le c_3 \end{aligned}
(1.4)

provided that $$\parallel \nabla u\parallel _N\le 1,~\parallel u\parallel _N\le M<\infty$$ and $$\alpha <\alpha _N$$. Furthermore, there exists a constant $$c_4>0$$ such that

\begin{aligned} \displaystyle \int _{\mathbb {R}^N}|u|^q\left[ \exp \Bigg (\alpha \vert u\vert ^{\frac{N}{N-1}}\Bigg )-S_{N-2}(\alpha , u)\right] ~\mathrm{d}x\le c_4\parallel u\parallel _{1,N}^q \end{aligned}
(1.5)

provided that $$\parallel u\parallel _{1,N}\le M$$ which $$M<(\frac{\alpha _N}{\alpha })^{\frac{N-1}{N}}$$.

Inequalities (1.3) and (1.5) imply that there exists a constant $$c_5>0$$ such that

\begin{aligned} \displaystyle \int _{\mathbb {R}^N}|u|^q\left[ \exp \Bigg (\alpha \vert u\vert ^{\frac{N}{N-1}}\Bigg )-S_{N-2}(\alpha , u)\right] ~\mathrm{d}x\le c_5\parallel u\parallel _{E}^q \end{aligned}
(1.6)

provided that $$\parallel u\parallel _E\le M$$ with $$M<\frac{1}{c_2}(\frac{\alpha _N}{\alpha })^{\frac{N-1}{N}}$$.

Note that according to [9, Lemma 2.4], the assumptions $$V^{-1}\in L^1(\mathbb {R}^N)$$ or $$\lim V(x)\rightarrow \infty$$ as $$|x|\rightarrow \infty$$, on the potential V(x) lead us to have the compactness of the imbedding $$E\hookrightarrow L^q(\mathbb {R}^N),$$ for all $$1\le q<+\infty$$.

Now from (1.2), the imbedding $$E\hookrightarrow L^q(\mathbb {R}^N)$$ for $$q=\beta _1+1$$ and (1.5) and (1.6) for $$q=\beta _2+1$$, there exist $$c_6,c_7>0$$ such that for all $$(x,u)\in \mathbb {R}^N\times E,$$

\begin{aligned} \displaystyle \int _{\mathbb {R}^N} |F(x,u)|\mathrm{d}x&=\int _{\mathbb {R}^N}|\int _{0}^{u(x)}f(x,s)\mathrm{d}s|\mathrm{d}x\nonumber \\&\le \frac{b_1}{\beta _1+1}\int _{\mathbb {R}^N}|u|^{\beta _1+1}\mathrm{d}x\nonumber \\&\quad +\frac{b_2}{\beta _2+1}\int _{\mathbb {R}^N}|u|^{\beta _2+1}\Bigg (\exp \Bigg (\alpha \vert u\vert ^{\frac{N}{N-1}}\Bigg )-S_{N-2}(\beta ,u)\Bigg )\mathrm{d}x\nonumber \\&\le \frac{b_1}{\beta _1+1}c_6||u||_E^{\beta _1+1}+\frac{b_2}{\beta _2+1}c_7||u||_E^{\beta _2+1} \end{aligned}
(1.7)

provided that $$||u||_E\le \frac{1}{c_2}(\frac{\alpha _N}{\alpha })^{\frac{N-1}{N}}.$$

Thus, by (1.4), we have $$F(x,u)\in L^1({\mathbb {R}^N})$$ for all $$u\in E$$. Therefore, the functional $$I_\lambda :E\rightarrow \mathbb {R}$$ given by

\begin{aligned} I_\lambda (u)=\phi (u)-\lambda \psi (u), \quad \end{aligned}

for

\begin{aligned} \phi (u)&=\displaystyle \int _{\mathbb {R}^N}A(x,\nabla u)~\mathrm{d}x+\frac{1}{N}\displaystyle \int _{\mathbb {R}^N} V(x)\vert u\vert ^N~\mathrm{d}x\\&\ge \displaystyle \int _{\mathbb {R}^N}k_0h_1(x)\vert \nabla u\vert ^N+\frac{1}{N}\displaystyle \int _{\mathbb {R}^N}V(x)\vert u\vert _E^N\\&=k_0\parallel u\parallel ^N \end{aligned}

and

\begin{aligned} \psi (u)=\displaystyle \int _{\mathbb {R}^N} \left( \displaystyle \int _{0}^{u(x)}\exp \Bigg (\alpha \vert s\vert ^{\frac{N}{N-1}}\Bigg )~\mathrm{d}s+F(x,u) \right) ~\mathrm{d}x, \quad \end{aligned}

is well defined. Moreover, $$I_\lambda$$ is a $$C^1$$ functional on E and for $$u,v\in E$$

\begin{aligned} I'_\lambda (u)v= & {} \int _{\mathbb {R}^N}a(x,\nabla u)\nabla v~\mathrm{d}x+\int _{\mathbb {R}^N} V(x)\vert u\vert ^{N-2} uv~\mathrm{d}x\\&-\lambda \int _{\mathbb {R}^N}\exp \Bigg (\alpha \vert u\vert ^{\frac{N}{N-1}}\Bigg )v~\mathrm{d}x-\lambda \int _{\mathbb {R}^N} f(x,u)v~\mathrm{d}x. \end{aligned}

Consequently, the critical points of the functional $$I_\lambda$$ are precisely nontrivial weak solutions of the problem (1.1).

In the present paper, first we obtain the existence of a nontrivial weak solution for the problem (1.1), in a certain range of $$\lambda$$ (in Theorem 3.1). Then we prove the existence of the second weak solution for the problem (1.1) distinct from the first one (in Theorem 4.1).

2 Preliminaries

Our main tool is Theorem 2.2, consequence of a local minimum theorem [3, Theorem 5.1] which is inspired by Ricceri’s variational principle.

For a given non-empty set X, and two functionals $$\Phi ,\Psi :X\rightarrow \mathbb {R}$$, we define the following functions:

\begin{aligned} \varsigma (r_1,r_2)= & {} \inf _{v\in \Phi ^{-1}(]r_1,r_2[)}\frac{\sup _{u\in \Phi ^{-1}(]r_1,r_2[)}\Psi (u)-\Psi (v)}{r_2-\Phi (v)},\\ \rho _{1}(r_1,r_2)= & {} \sup _{v\in \Phi ^{-1}(]r_1,r_2[)}\frac{\Psi (v)- \sup _{u\in \Phi ^{-1}(]-\infty ,r_1[)}\Psi (u)}{\Phi (v)-r_1} \end{aligned}

for all $$r_1,r_2\in \mathbb {R},$$ $$r_1<r_2.$$

Definition 2.1

Let $$\phi _0$$ and $$\psi _0$$ be two continuously Gâteaux differentiable functionals defined on a real Banach space X and fix $$r_1,r_2\in [-\infty ,+\infty ]$$, with $$r_1<r_2$$. We say that the functional $$I_0=\phi _0-\psi _0$$ verifies the Palais–Smale condition cut off lower at $$r_1$$ and upper at $$r_2$$ (in short $$^{[r_1]}(PS)^{[r_2]}$$) if any sequence $$(u_n)\subset X$$ such that

(i):

$$(I_0(u_n))$$ is bounded,

(ii):

if $$I_{0}^\prime (u_n)\rightarrow 0$$ in $$X^*$$ (where $$X^*$$ denotes the topological dual of X),

(iii):

$$r_1<\phi _0(u_n)<r_2,~ \forall n\in \mathbb {N}$$

has a convergent subsequence. Clearly, if $$r_1=-\infty$$ and $$r_2=+\infty$$ it coincides with the classical (PS) condition. Moreover, if $$r_1=-\infty$$ and $$r_2\in \mathbb {R}$$ it is denoted by $$(PS)^{[r_2]}.$$

Theorem 2.2

([3] Theorem 5.1) Let X be a real Banach space; $$\Phi :X\rightarrow \mathbb {R}$$ be a sequentially weakly lower semicontinuous, coercive and continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on $$X^*$$, $$\Psi :X\rightarrow \mathbb {R}$$ be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact. Assume that there are $$r_1,r_2\in \mathbb {R}$$, $$r_1<r_2$$, such that

\begin{aligned} \varsigma (r_1,r_2)<\rho _{1}(r_1,r_2), \end{aligned}

and for each $$\lambda \in \Lambda =]\frac{1}{\rho _1(r_1,r_2)},\frac{1}{\varsigma (r_1,r_2)}[$$ the functional $$I_\lambda =\phi -\lambda \psi$$ satisfies $$^{[r_1]}(PS)^{[r_2]}$$ condition.

Then, for each $$\lambda \in \Lambda$$ there is $$u_{0,\lambda }\in \Phi ^{-1}(]r_1,r_2[)$$ such that $$I_{\lambda }(u_{0,\lambda })\le I_\lambda (u)$$ $$\forall u\in \Phi ^{-1}(]r_1,r_2[)$$ and $$I'_\lambda (u_{0,\lambda })=0.$$

Lemma 2.3

Assume that $$(A_1),(A_2), (A_3)$$ and $$(f_1)$$ hold. Then for each

\begin{aligned} {0}<r<k_0\inf \left( 1,\left( \frac{1}{2c_2}\right) ^N\left( \frac{(N-1)\alpha _{N}}{N\alpha _0}\right) \right) ^{N-1} \end{aligned}

and $$\lambda >0$$, the functional $$I_\lambda$$ satisfies $$(PS)^{[r]}$$.

Proof

Let $$(u_n)\subset E$$ be such that $$(I_\lambda (u_n))$$ is bounded, $$I_{\lambda }^\prime (u_n)\rightarrow 0$$ and $$\phi (u_n)<r,$$ for all $$n\in \mathbb {N}$$. Since $$k_0\parallel u_n\parallel _E^N\le \phi (u_n)$$, so $$\parallel u_n\parallel _E<(\frac{r}{k_0})^{\frac{1}{N}}$$. Therefore, there exists $$u\in E$$ such that $$u_n\rightharpoonup u$$ weakly in E. Thanks to Lemma 2.1. of [9] and using the assumption (1.2), one has

\begin{aligned}&\int _{\mathbb {R}^N}\vert f(x,u_n)\vert ^{\frac{N}{N-1}}~\mathrm{d}x\\&\quad \le b_3\left( \int _{\mathbb {R}^N}\vert u_n\vert ^{\frac{\beta _1N}{N-1}}~\mathrm{d}x+ \int _{\mathbb {R}^N}\vert u_n\vert ^{\frac{\beta _2N}{N-1}}\left( \exp \left( \frac{N}{N-1} \alpha _0\vert u\vert ^{\frac{N}{N-1}}\right) \right. \right. \\&\left. \left. \qquad -S_{N-2} \left( \frac{N\alpha _0}{N-1},u\right) \right) ~\mathrm{d}x\right) , \end{aligned}

where $$b_3=\max \{b_1,b_2\}$$. Since $$\parallel u_n\parallel _E\le (\frac{r}{k_0})^{\frac{1}{N}}<\frac{1}{2c_2} (\frac{\alpha _N}{\frac{N}{N-1} \alpha _0})^{\frac{N-1}{N}}$$, using (1.6) for $$q=\frac{\beta _2N}{N-1}$$, we deduce that

\begin{aligned} \sup _{n\in \mathbb {N}}\left( \displaystyle \int _{\mathbb {R}^N}\vert f(x,u_n)\vert ^{\frac{N}{N-1}}~\mathrm{d}x\right) <+\infty . \end{aligned}

This fact together with the compactness of the imbedding $$E\hookrightarrow L^N(\mathbb {R}^N)$$ implies

\begin{aligned} \lim _{n\rightarrow +\infty }\displaystyle \int _{\mathbb {R}^N} f(x,u_n)(u_n-u)~\mathrm{d}x=0. \end{aligned}
(2.1)

On the other hand, using the definition of $$S_{N-2}$$, one has

\begin{aligned} \int _{\mathbb {R}^N}\exp \Bigg (\alpha \vert u_n\vert ^{\frac{N}{N-1}}\Bigg )(u_n-u)~\mathrm{d}x&=\displaystyle \int _{\mathbb {R}^N} \exp \Bigg (\alpha \vert u_n\vert ^{\frac{N}{N-1}}-S_{N-2}(\alpha ,u_n) \Bigg )(u_n-u)~\mathrm{d}x\\&\quad +\displaystyle \int _{\mathbb {R}^N}S_{N-2}(\alpha ,u_n)(u_n-u)~\mathrm{d}x. \end{aligned}

Using the compact imbedding $$E\hookrightarrow L^1(\mathbb {R}^N)$$ together with (1.4) for the first part of the above inequality and $$E\hookrightarrow L^2(\mathbb {R}^N)$$ together with the fact that $$u_n$$ is converges and so bounded in $$L^q$$ for any $$q\ge 1$$, arguing as before, one has

\begin{aligned} \lim _{n\rightarrow +\infty }\displaystyle \int _{\mathbb {R}^N}\exp \Bigg (\alpha \vert u_n\vert ^{\frac{N}{N-1}}\Bigg )(u_n-u)~\mathrm{d}x=0. \end{aligned}
(2.2)

Now using $$(A_1)$$, it follows that

\begin{aligned} |\langle I_{\lambda }^\prime (u_n), u_n-u\rangle | =&\,|\int _{\mathbb {R}^N}(a(x,u_n)(u_n-u)+V(x)|u_n-u|^{N-2}(u_n-u))\mathrm{d}x\\&-\lambda \int _{\mathbb {R}^N}\Bigg (\exp \Bigg (\alpha \vert u_n\vert ^{\frac{N}{N-1}}\Bigg )(u_n-u)+f(x,u_n)(u_n-u)\Bigg )~\mathrm{d}x|\\ \le&\int _{\mathbb {R}^N}(|a(x,u_n)||u_n-u|+V(x)|u_n|^{N-2}|u_n-u|)\mathrm{d}x\\&+\lambda \int _{\mathbb {R}^N}\Bigg (\Bigg |\exp \Bigg (\alpha \vert u_n\vert ^{\frac{N}{N-1}}\Bigg )\Bigg ||u_n-u|+|f(x,u_n)(u_n-u)|~\mathrm{d}x\\ \le&\, c_0\int _{\mathbb {R}^N}((h_0(x)|u_n-u|+h_1(x)\vert u_n\vert ^{N-1}|u_n-u| \Bigg )\\&+V(x)|u_n|^{N-2}|u_n-u|)\mathrm{d}x\\&+\lambda \int _{\mathbb {R}^N}(|\exp \Bigg (\alpha \vert u_n\vert ^{\frac{N}{N-1}}\Bigg )||u_n-u|+|f(x,u_n)(u_n-u)|~\mathrm{d}x. \end{aligned}

This inequality together with $$h_0\in L^{N/(N-1)}(\mathbb {R}^N)$$ and $$h_1\in L_\mathrm{loc}^{\infty }(\mathbb {R}^N)$$ ($$u_n$$ is converges to u in $$L^1$$ and $$L^N$$) and using the fact

\begin{aligned} \langle I_{\lambda }^\prime (u_n), u_n-u\rangle \rightarrow 0 \quad \text {as}~~n\rightarrow +\infty , \end{aligned}

it can be derived that $$(u_n)$$ is strongly convergent to u in E. $$\square$$

3 Existence of the First Solution

Chosen large enough $$R>0$$ such that

\begin{aligned}&(4)^{\frac{1+N}{N}}c_0\left( \int _{R\le \vert x\vert \le R+1}h_0\mathrm{d}x\right) \left( C^*+\frac{b_1}{\beta _1+1}c_6+\frac{b_2}{\beta _2+1}c_7\right) \nonumber \\&\quad < k_0^{\frac{1}{N}}B\left( k_0\int _{R\le \vert x\vert \le R+1}h_1(x)\mathrm{d}x+\frac{1}{N}\Gamma _V\right) ^{\frac{N-1}{N}}, \end{aligned}
(3.1)

where

\begin{aligned} \Gamma _V= \int _{R\le \vert x\vert \le R+1} V(x)\mathrm{d}x+\int _{R\le \vert x\vert \le R+1} V(x)(R+1-\vert x\vert )^N~\mathrm{d}x \end{aligned}

and

\begin{aligned} B= W_{N-1}\frac{R^N}{N} \end{aligned}

and the constant $$c_0,c_6,c_7$$ introduced in $$(A_1)$$, (1.6) and (1.7).

Now we are ready to present our first main result

Theorem 3.1

Assume that $$(A_1),(A_2), (A_3)$$ and $$(f_1)$$ hold. Then there exists $$\lambda ^*=\lambda ^*(r)>0$$ such that for all $$0<\lambda <\lambda ^*$$ the functional $$I_\lambda$$ admits a nontrivial critical point $$u_1$$ satisfying

\begin{aligned} 0<\phi (u_1)<r\quad \text {and}\quad I_\lambda (u_1)\le I_\lambda (w)\quad \text {for all}\quad w\in \phi ^{-1}(]0,r[), \end{aligned}

for each $$r>0$$ with the property

\begin{aligned} r\le \frac{k_0}{4}\inf \left( 1,\left( \frac{1}{c_2}\right) ^N\left( \frac{(N-1)\alpha _{N}}{N\alpha _0}\right) ^{N-1}, \left( \frac{1}{c_2}\right) ^N\left( \frac{\alpha _{N}}{\alpha }\right) ^{N-1}\right) . \end{aligned}
(3.2)

Proof

For $$\lambda >0$$ and $$R>0$$ as in (3.1), define the function

\begin{aligned} \theta _\lambda =\left\{ \begin{array}{ll}\eta _\lambda &{}\quad \mathrm{if}~~\vert x\vert <R,\\ \eta _\lambda (R+1-\vert x\vert )&{}\quad \mathrm{if}~~R\le \vert x\vert \le R+1,\\ 0&{}\quad \mathrm{if}~~\vert x\vert >R+1,\end{array}\right. \end{aligned}

with $$\eta _\lambda$$ is a real number satisfying

\begin{aligned} 0<\eta _\lambda <\inf \bigg (Z_\lambda ,T_r\bigg ), \end{aligned}
(3.3)

where

\begin{aligned} Z_\lambda =\left( \frac{\lambda B-c_0\int _{R\le \vert x\vert \le R+1}h_0}{c_0\int _{R\le \vert x\vert \le R+1}h_1+\frac{1}{N}\Gamma _V}\right) ^{\frac{1}{N-1}} \end{aligned}

and

\begin{aligned} T_r= \left( \frac{r}{k_0\int _{R\le \vert x\vert \le R+1}h_1(x)+\frac{1}{N}\Gamma _V}\right) ^{\frac{1}{N}}. \end{aligned}

First we note that $$\theta _\lambda \in E$$.

Owing to our assumptions, one has

\begin{aligned} \phi (\theta _\lambda )=&\displaystyle \int _{\mathbb {R}^N}A(x,\nabla \theta _\lambda )\mathrm{d}x +\frac{1}{N} \displaystyle \int _{\mathbb {R}^N} V(x)\vert \theta _\lambda \vert ^N\mathrm{d}x\nonumber \\ =&\displaystyle \int _{R\le \vert x\vert \le R+1}A(x,\nabla \eta _\lambda (R+1-\vert x\vert ))\mathrm{d}x+\frac{1}{N} \displaystyle \int _{\mathbb {R}^N} V(x)\vert \theta _\lambda \vert ^N\mathrm{d}x\nonumber \\ \le&\displaystyle \int _{R\le \vert x\vert \le R+1}c_0(h_0(x)\vert \eta _\lambda \vert +h_1(x)\vert \eta _\lambda \vert ^N)\mathrm{d}x +\frac{1}{N} \displaystyle \int _{\mathbb {R}^N} V(x)\vert \theta _\lambda \vert ^N\mathrm{d}x\nonumber \\ =&c_0\eta _\lambda \displaystyle \int _{R\le \vert x\vert \le R+1}h_0(x)\mathrm{d}x+c_0\delta _{\lambda }^N\displaystyle \int _{R\le \vert x\vert \le R+1}h_1(x)\mathrm{d}x+\frac{1}{N}\delta _{\lambda }^N\Gamma _V:=G. \end{aligned}
(3.4)

So using the fact that $$F(x,\theta _\lambda )\ge 0$$, and $$\exp (\alpha \vert s\vert ^{\frac{N}{N-1}})\ge 1$$, it follows that

\begin{aligned} \psi (\theta _\lambda )\ge \displaystyle \int _{\vert x\vert<R}\left( \displaystyle \int _{0}^{\theta _\lambda } \exp \Bigg (\alpha \vert s\vert ^{\frac{N}{N-1}}\Bigg )~\mathrm{d}s\right) ~\mathrm{d}x\ge \displaystyle \int _{\vert x\vert <R}\theta _\lambda \mathrm{d}x= B\eta _\lambda . \end{aligned}

Thanks to (3.2) and (3.3), we get

\begin{aligned} \frac{\psi (\theta _\lambda )}{\phi (\theta _\lambda )}\ge \frac{B\delta _{\lambda }}{G} \ge \frac{1}{\lambda }. \end{aligned}
(3.5)

On the other hand, if $$u\in E$$ has the property $$\phi (u)<r$$, then

\begin{aligned} k_0\displaystyle \int _{R\le \vert x\vert \le R+1}h_1(x)\vert \nabla u\vert ^N\mathrm{d}x&+\frac{1}{N}\displaystyle \int _{\mathbb {R}^N}V(x)\vert u\vert ^N\mathrm{d}x<\phi (u)\\&=\int _{\mathbb {R}^N} A(x,\nabla u)+\frac{1}{N} \displaystyle \int _{\mathbb {R}^N}V(x)\vert u\vert ^N\mathrm{d}x<r, \end{aligned}

and so

\begin{aligned} k_0\parallel u\parallel _{E}^N\le \phi (u)<r, \end{aligned}

i.e.,

\begin{aligned} \parallel u\parallel <\left( \frac{r}{k_0}\right) ^{\frac{1}{N}}. \end{aligned}

By (3.1) and (1.7), there exists $$C^*>0$$ such that

\begin{aligned} \psi (u)&=\int _{\mathbb {R}^N}\left( \displaystyle \int _{0}^{u(x)}\exp \Bigg (\alpha \vert s\vert ^{\frac{N}{N-1}}\Bigg )~\mathrm{d}s+F(x,u)\right) ~\mathrm{d}x\nonumber \\&\le \int _{\mathbb {R}^N}\left( \exp \Bigg (\alpha \vert u\vert ^{\frac{N}{N-1}}\Bigg )|u|\mathrm{d}s+F(x,u)\right) ~\mathrm{d}x\nonumber \\&\le \int _{\mathbb {R}^N}\exp \Bigg (\alpha \vert u\vert ^{\frac{N}{N-1}}\Bigg )-S_{N-2}(a,u))|u|\mathrm{d}x\nonumber \\&\quad +\int _{\mathbb {R}^N}S_{N-2}(a,u)|u|\mathrm{d}x+ \int _{\mathbb {R}^N}F(x,u)\mathrm{d}x\nonumber \\&\le C^*||u||_E+\bigg (\frac{b_1}{\beta _1+1}c_6+\frac{b_2}{\beta _2+1}c_7\bigg )||u||_E\nonumber \\&\le \bigg (C^*+\frac{b_1}{\beta _1+1}c_6+\frac{b_2}{\beta _2+1}c_7\bigg )\left( \frac{r}{k_0} \right) ^{\frac{1}{N}}. \end{aligned}
(3.6)

Notice that in (3.5) we use the property $$||u||\le 1$$ .

Set

\begin{aligned} \lambda ^*=\frac{r^{\frac{N-1}{N}}}{(\frac{4}{k_0})^{\frac{1}{N}} \bigg (C^*+\frac{b_1}{\beta _1+1}c_6+\frac{b_2}{\beta _2+1}c_7\bigg )}. \end{aligned}
(3.7)

Using the definition of $$T_r$$ and (3.4), we have $$0<\phi (\theta _\lambda )<r$$. Therefore, one has

\begin{aligned} \varsigma (0,r)&=\inf _{v\in \Phi ^{-1}(]0,r[)}\frac{\sup _{u\in \Phi ^{-1}(]0,r[)}\Psi (u)-\Psi (v)}{r-\Phi (v)}\\&\le \frac{\sup _{u\in \Phi ^{-1}(]0,r[)}\Psi (u)-\Psi (\theta _\lambda )}{r-\Phi (\theta _\lambda )}\\&\le \frac{\sup _{u\in \Phi ^{-1}(]0,r[)}\Psi (u)}{r}\le \frac{1}{\lambda ^*}<\frac{1}{\lambda }<\frac{\psi (\theta _\lambda )}{\phi (\theta _\lambda )}\nonumber \\&<\sup _{v\in \Phi ^{-1}(]0,r[)}\frac{\Psi (v)- \sup _{u\in \Phi ^{-1}(]-\infty ,0[)}\Psi (u)}{\Phi (v)-0}\\&=\rho _{1}(0,r), \end{aligned}

for all $$0<\lambda <\lambda ^*$$. (Note that $$\Psi (0)=0$$)

Hence, employing Theorem 2.2, for each $$0<\lambda <\lambda ^*$$, the functional $$\Phi -\lambda \Psi$$ admits at least one critical point $$u_1\in E$$ such that $$0<\phi (u_1)<r$$, that is $$\parallel u\parallel _{E}<(\frac{r}{k_0})^{1/N},$$ and $$I_\lambda (u_1)\le I_\lambda (w),$$ for any $$w\in \phi ^{-1}(]0,r[)$$. This completes the proof. $$\square$$

4 Existence of the Second Solution

Now, in this section, we prove the existence of the second local minimum distinct from the first one. For this purpose, we verify the hypotheses of Theorem 2.2. for another range of r.

Theorem 4.1

Assume that $$(A_1),(A_2), (A_3)$$ and $$(f_1)$$ hold. Then there exists $$\lambda _*\in ]0,\lambda ^*[$$ such that for $$\lambda _*<\lambda <\lambda ^*$$, the functional $$I_\lambda$$ admits a critical point $$u_2$$ which satisfies

\begin{aligned} r<\phi (u_2)<2r,\quad \text {and}\quad I_\lambda (u_2)\le I_\lambda (w),~~~\forall ~w\in \phi ^{-1}(]r,2r[). \end{aligned}

Proof

First set

\begin{aligned} \lambda _*=\left( \frac{2r}{k_0\int _{R\le \vert x\vert \le R+1}h_1(x)+\frac{1}{N}\Gamma _V}\right) ^{\frac{N-1}{N}} \left( \frac{2c_0\int _{R\le \vert x\vert \le R+1}h_0}{B}\right) . \end{aligned}

From assumption (3.1), we get

\begin{aligned} \lambda _*<\lambda ^*. \end{aligned}

For $$\lambda _*<\lambda <\lambda ^*$$, we take the function $$\theta _\lambda$$ as before but in the following different condition:

\begin{aligned} T_r<\eta _\lambda <\inf \{T_{2r},Z_{\lambda /2}\}. \end{aligned}
(4.1)

Therefore,

\begin{aligned} r<k_0\displaystyle \int _{R\le \vert x\vert \le R+1}h_1(x)\vert \nabla \theta _\lambda \vert ^N\mathrm{d}x+\frac{1}{N}\displaystyle \int _{\mathbb {R}^N} V(x)\vert \theta _\lambda \vert ^N \le \phi (\theta _\lambda )\le G<2r, \end{aligned}

so 4.1 yields that

\begin{aligned} \frac{\psi (\theta _\lambda )}{2\phi (\theta _\lambda )}>\frac{B\eta _\lambda }{2(c_0\eta _\lambda \int _{R\le \vert x\vert \le R+1}h_0+c_0\delta _{\lambda }^N\int _{R\le \vert x\vert \le R+1}h_1+\frac{1}{N}\delta _{\lambda }^N\Gamma _V)}> \frac{1}{\lambda }. \end{aligned}

Similar to the arguments in the last section, by replacing r by 2r, we have

\begin{aligned} \varsigma (r,2r)&=\inf _{v\in \Phi ^{-1}(]r,2r[)}\frac{\sup _{u\in \Phi ^{-1}(]r,2r[)}\Psi (u)-\Psi (v)}{2r-\Phi (v)}\\&\le \frac{\sup _{u\in \Phi ^{-1}(]r,2r[)}\Psi (u)-\Psi (\theta _\lambda )}{2r-\Phi (\theta _\lambda )}\\&\le \frac{\sup _{u\in \Phi ^{-1}(]r,2r[)}\Psi (u)}{2r}\\&\le \left( \frac{1}{k_0}\right) ^{\frac{1}{N}}\left( C^*+\frac{b_1}{\beta _1+1}c_6 +\frac{b_2}{\beta _2+1}c_7\right) (2r)^{-\frac{N-1}{N}}\\&\le \left( \frac{1}{k_0}\right) ^{\frac{1}{N}}\left( C^*+\frac{b_1}{\beta _1+1}c_6 +\frac{b_2}{\beta _2+1}c_7\right) r^{-\frac{N-1}{N}}\\&<\frac{1}{\lambda ^*}<\frac{1}{\lambda }<\frac{\psi (\theta _\lambda ) -\sup _{u\in \Phi ^{-1}(]-\infty ,0[)}\Psi (u)}{\phi (\theta _\lambda )-r}\\&<\sup _{v\in \Phi ^{-1}(]r,2r[)}\frac{\Psi (v)-\sup _{u\in \Phi ^{-1}(] -\infty ,r[)}\Psi (u)}{\Phi (v)-r}\\&=\rho _{1}(r,2r), \end{aligned}

for all $$\lambda _*<\lambda <\lambda ^*$$. Note that in the second inequality we use the fact that if $$\frac{t_1}{t_2}<s<\frac{t_3}{t_4}$$, then $$\frac{t_1-t_3}{t_2-t_4}<s$$, and the following inequalities:

\begin{aligned} \frac{\psi (\theta _\lambda )}{2\phi (\theta _\lambda )}>\frac{1}{\lambda }> \frac{\sup _{\phi (u)<2r}\psi (u) }{2r}>\frac{\sup _{r<\phi (u)<2r}\psi (u) }{2r} \end{aligned}

for all $$\lambda _*<\lambda <\lambda ^*$$.

Since

\begin{aligned} 0<2r<k_0\inf \left( 1,\left( \frac{1}{2c_2}\right) ^N\left( \frac{\alpha _N}{\frac{N}{N-1} \alpha _0}\right) ^{N-1}\right) , \end{aligned}

then by Lemma 2.3, the functional $$I_\lambda$$ satisfies $$^{[r]}(PS)^{[2r]}$$. Hence, employing Theorem 2.2, we conclude that the functional $$I_\lambda$$ admits a critical point $$u_2$$ satisfying

\begin{aligned} r<\phi (u_2)<2r,\quad \text {and}\quad I_\lambda (u_2)\le I_\lambda (w),\quad \forall ~~w\in \phi ^{-1}(]r,2r[), \end{aligned}

for each $$\lambda _*<\lambda <\lambda ^*$$. Since $$\phi (u_1)<r$$, then $$u_1\ne u_2$$. $$\square$$

Remark 4.2

To prove the existence of another local minimum distinct from the first and second ones, we can assume an algebraic condition on f with the classical Ambrosetti–Rabinowitz condition: there exist $$\nu >N$$ and $$R'>0$$ such that

\begin{aligned} 0<\nu F(x,t)\le tf(x,t) \quad \text {~for all~} \; |t|>R', x\in \mathbb {R}^N. \end{aligned}

Clearly, the functional $$\Phi -\lambda \Psi$$ is of class $$C^1$$ and $$(\Phi -\lambda \Psi )(0)=0$$.

The first part of proof guarantees that $$u_2\in E$$ is a local nontrivial local minimum for $$\Phi -\lambda \Psi$$ in E. Therefore, there is $$s>0$$ such that

\begin{aligned} \inf _{||u-u_2||=s}(\Phi -\lambda \Psi )(u)>(\Phi -\lambda \Psi )(u_2). \end{aligned}

So the condition [8, Theorem 2.2, $$(I_1)$$] is verified. Now choosing $$u\ne 0$$, from Ambrosetti–Rabinowitz condition, one has

\begin{aligned} (\Phi -\lambda \Psi )(tu)\le & {} c_0t\int _{\mathbb {R}^N}h_0(x)|\nabla u|\mathrm{d}x+c_0t^N\displaystyle \int _{\mathbb {R}^N}h_1(x)|\nabla u|^N\mathrm{d}x\\&+\frac{t^N}{N} \int _{\mathbb {R}^N}V(x)|u|^N\mathrm{d}x\\&-\lambda \int _{\mathbb {R}^N}\left( \int _{0}^{tu}\exp \Bigg (\alpha \vert s\vert ^{\frac{N}{N-1}}\Bigg )\mathrm{d}s+F(x,tu)\right) \mathrm{d}x\\\le & {} c_0t\displaystyle \int _{\mathbb {R}^N}h_0(x)|\nabla u|\mathrm{d}x+c_0t^N\displaystyle \int _{\mathbb {R}^N}h_1(x)|\nabla u|^N\mathrm{d}x\\&+\frac{t^N}{N} \int _{\mathbb {R}^N}V(x)|u|^N\\&-\lambda t\int _{\mathbb {R}^N}|u|\mathrm{d}x-\lambda a_3t^\nu \int _{\mathbb {R}^N}|u|^\nu \mathrm{d}x+\lambda a_4\longrightarrow -\infty \end{aligned}

as $$t\rightarrow \infty$$, since $$\nu >N$$. So the condition [8, Theorem 2.2, $$(I_1)$$] is fulfilled.

Moreover, by standard computations, the functional $$\Phi -\lambda \Psi$$ satisfies (PS) condition. Hence, the classical theorem of Ambrosetti and Rabinowitz gives a critical point $$u_3$$ of $$\Phi -\lambda \Psi$$ such that $$(\Phi -\lambda \Psi )(u_3)>(\Phi -\lambda \Psi )(u_2)$$. So $$u_1$$, $$u_2$$ and $$u_3$$ are distinct weak solutions of the problem (1.1).