Quasilinear equations involving indefinite nonlinearities and exponential critical growth in ${\mathbb {R}}^N$

Abstract

In this work, we establish the existence of nonzero solutions for a class of quasilinear elliptic equations involving indefinite nonlinearities with exponential critical growth of Trudinger–Moser type. Our proofs rely on variational arguments in a Orlicz–Sobolev space with a version of the Trudinger–Moser inequality.

1 Introduction and main results

In this paper, we consider a class of quasilinear elliptic equations involving a sign-changing weight function and a nonlinearity with exponential critical growth. More precisely, we study the existence of nonzero solutions for the equation

$$\begin{aligned} -\Delta _\Phi u +\Phi '(|u|)\frac{u}{|u|} = W(x)f(u)\ \ \ \text {in}\ \ \ {\mathbb {R}}^N, \end{aligned}$$

(1.1)

where $\Delta _\Phi u={\mathrm {div}}\left( \Phi '(|\nabla u|)\nabla u/|\nabla u|\right)$, $N\ge 2$, $W:{\mathbb {R}}^N \rightarrow {\mathbb {R}}$ is a continuous function changing sign, $f\in C({\mathbb {R}})$ has exponential critical growth and $\Phi :{\mathbb {R}}\rightarrow {\mathbb {R}}_+$ is a function satisfying some appropriate conditions.

Elliptic problems with indefinite nonlinearities have been intensively studied in the last years. We would like to mention that existence of solutions for indefinite elliptic problems of the type

$$\begin{aligned} \left\{ \begin{aligned} -\Delta u&=\lambda u+W(x)f(u)\quad&\text{ in }&\quad \Omega ,\\ u&=0\quad&\text{ on }&\quad \partial \Omega ,\\ \end{aligned} \right. \end{aligned}$$

(1.2)

where $\Omega \subset {\mathbb {R}}^N$ is a smooth bounded domain and $\lambda >0$, has already been established in various contexts when the dimension $N\ge 3$. If the domain $\Omega$ is a compact manifold of dimension $N\ge 3$, the critical exponent case $f(s)=|s|^{2^*-2}s$, where $2^*=2N/(N-2)$, arises in the prescribed scalar curvature problem (see [36]). For manifolds carrying scalar flat metrics, sufficient conditions for the existence of positive solutions were given in [29]. Results for more general nonlinearities were obtained by Alama–Tarantello [5]. After that, many authors have studied indefinite semilinear elliptic problems when the nonlinear term f(s) has polynomial growth (see [4, 6, 11, 20, 22, 23, 28, 40, 44] and references therein). Indefinite problems of type (1.2) involving critical growth in the Sobolev case were treated by various authors; see, for instance, [19, 33, 34].

We quote that there are few results involving indefinite nonlinearity with exponential critical growth. In the paper [41], the authors establish a version for dimension two of the main result in [5] (see also [40]) when the nonlinearity f(s) has exponential critical growth. In [1], the authors consider an indefinite problem having exponential subcritical growth in all ${\mathbb {R}}^2$ with the nonlinearity being of the form $f(s)=\phi (s)e^s$ and $\phi (s)$ between two powers. We also emphasize that Alves et al. [8] studied the existence of solution for the problem

$$\begin{aligned} \left\{ \begin{aligned} -{\mathrm {div}}\left( |\nabla u|^{N-2}\nabla u\right) +|u|^{N-2}u&=W(x)f(u)\quad&\text{ in }&\quad \Omega ,\\ u&=0\quad&\text{ on }&\quad \partial \Omega ,\\ \end{aligned} \right. \end{aligned}$$

(1.3)

where $\Omega$ is an exterior domain of ${\mathbb {R}}^N (N\ge 2)$ and f(s) has exponential critical growth.

Motivated by the previous works, our main purpose here is to study Eq. (1.1) by considering the maximal growth on the nonlinear term f(s) which allows us to treat the problem variationally in the Orlicz–Sobolev space $W^{1,\Phi }({\mathbb {R}}^N)$. Furthermore, W(x) is a weight function changing sign and having a thick zero set. In what follows, $\Phi :{\mathbb {R}}\rightarrow {\mathbb {R}}_+$ is a N-function of class $C^1$ fulfilling the conditions

$(\Phi _1)$:

There exists $C>0$ such that $t^N/C\le \Phi (t)\le C t^N$ for all $t\in [0,1/C)$;

$(\Phi _2)$:

$\displaystyle \lim _{t\rightarrow +\infty }\frac{\Phi (t)}{t^N\log ^\alpha t}=1$, for some $\alpha \in [0,N-1)$;

and $W^{1,\Phi }({\mathbb {R}}^N)$ is the Orlicz–Sobolev space that consists of functions in $L_\Phi ({\mathbb {R}}^N)$ (the Orlicz space associated with the N-function $\Phi$) such that its weak derivatives exist and belong to $L_\Phi ({\mathbb {R}}^N)$. For the definition and information about N-functions, see Sect. 2. We regard $W^{1,\Phi }({\mathbb {R}}^N)$ endowed with the norm

$$\begin{aligned} \Vert u\Vert _{1,\Phi }=|\nabla u|_{\Phi }+|u|_\Phi , \end{aligned}$$

where $|\cdot |_\Phi$ denotes the Luxemburg norm associated with $L_\Phi ({\mathbb {R}}^N)$. It is known that the Lorentz–Zygmund space $L^{N,N,\tau }({\mathbb {R}}^N)$, $\tau =\alpha /N$, reproduces (up to equivalent norms) the Orlicz spaces $L_\Phi ({\mathbb {R}}^N)$ (see [10, 43]). Thus, $W^{1,\Phi }({\mathbb {R}}^N)$ is equivalent to $W^{1}L^{N,N,\tau }({\mathbb {R}}^N)$.

When $\alpha =0$, $\Phi (t)=|t|^N$ satisfies conditions $(\Phi _1)$ and $(\Phi _2)$, and therefore, (1.1) becomes an elliptic equation involving the N-Laplacian operator, namely

$$\begin{aligned} -\Delta _Nu+|u|^{N-2}u={\hat{g}}(x,u)\quad \text {in}\quad {\mathbb {R}}^N, \end{aligned}$$

(1.4)

where ${\hat{g}}(x,u)=W(x)f(u)/N$. Equations of type (1.4), with ${\hat{g}}(x,u)$ having definite sign and critical exponential growth with respect to the Trudinger–Moser inequality, have been intensively investigated by many authors; see, for example, [2, 15, 24, 26, 27, 39, 42].

Throughout this paper, we assume that f(t) can behave like $\exp (b|t|^\gamma )$ as $t\rightarrow +\infty$; more precisely, we suppose the following growth condition on the nonlinearity f(t):

$(f_1)$:

there exist constants $C>0$ and $b>0$ such that

$$\begin{aligned} f(t)\le Ct^{N-1}+C\left[ \exp (bt^{\gamma })-S_{N,\alpha }(bt^{\gamma })\right] \end{aligned}$$

for all $t\in {\mathbb {R}}$, where $\gamma =N/(N-1-\alpha )$ and

$$\begin{aligned} S_{N,\alpha }(bt^{\gamma })=\displaystyle \sum _{0\le j < \frac{N}{\gamma }}\frac{(bt^{\gamma })^j}{j!}. \end{aligned}$$

This growth on f(t) is motivated by a version of the Trudinger–Moser inequality in the space $W^{1,\Phi }({\mathbb {R}}^N)$ (see Lemma 3.3) which was proved in [18, 21].

Next, let us to obtain some properties of the function $\Phi$. As a consequence of $(\Phi _1)$, it follows that

$$\begin{aligned} \liminf _{t\rightarrow 0^+}\frac{\Phi (2t)}{\Phi (t)}\ge \frac{2^N}{C^2}. \end{aligned}$$

Moreover, by $(\Phi _2)$ we have

$$\begin{aligned} \lim _{t\rightarrow +\infty }\frac{\Phi (2t)}{\Phi (t)}=2^N. \end{aligned}$$

Thus, there exists $K_1\ge 2^N$ such that $\Phi (2t)\le K_1\Phi (t)$ for all $t\ge 0$; that is, $\Phi$ satisfies the $\Delta _2$-condition. Moreover, since $\Phi$ is convex, we reach $(K_1-1)\Phi (t)\ge \Phi (2t)-\Phi (t)\ge \Phi '(t)t$, and therefore,

$$\begin{aligned} c_{\alpha }:=\sup _{t>0}\frac{\Phi '(t)t}{\Phi (t)}\le K_1-1. \end{aligned}$$

(1.5)

On the other hand, given any $t_0>0$, by the mean value theorem there exists $s\in (0,t_0)$ verifying $\Phi (t_0)=\Phi '(s)t_0$, and since $\Phi '(t)$ is nondecreasing for $t> 0$, it follows that $\Phi (t_0)\le \Phi '(t_0)t_0$, and thus,

$$\begin{aligned} 1\le m_{\alpha }:=\inf _{t>0}\frac{\Phi '(t)t}{\Phi (t)}. \end{aligned}$$

(1.6)

By using $(\Phi _1)$ and $(\Phi _2)$, we shall show that $m_\alpha >1$ (see Proposition 2.7). Moreover, by deriving $\Phi (t)/t^{c_\alpha }$ we have that (1.5) implies that $\Phi (t)/t^{c_\alpha }$ is nonincreasing for $t>0$. Similarly, from (1.6) we deduce that $\Phi (t)/t^{m_\alpha }$ is nondecreasing for $t>0$. Hence,

$$\begin{aligned} \Phi (1)t^{c_\alpha }\le \Phi (t)\le \Phi (1)t^{m_\alpha },\ \text{ for } \text{ all } \ t\in [0,1]. \end{aligned}$$

Combining these inequalities with condition $(\Phi _1)$, for all $t\in [0,\min \{1,1/C\})$ we reach

$$\begin{aligned} \Phi (1)t^{c_\alpha }\le \Phi (t)\le Ct^N\quad \text {and}\quad \frac{1}{C}t^N\le \Phi (t)\le \Phi (1)t^{m_\alpha }, \end{aligned}$$

which imply that $m_\alpha \le N\le c_\alpha$.

With respect to the term W(x), we require the following conditions:

$(W_1)$:

$W:{\mathbb {R}}^N \rightarrow {\mathbb {R}}$ is a continuous function changing sign and $W\in L^r({\mathbb {R}}^N)$ for some $r\in [N,+\infty )$;

$(W_2)$:

$\Omega ^{+}$ is a bounded set and $\overline{\Omega ^{+}}\cap \overline{\Omega ^{-}}=\emptyset$, where

$$\begin{aligned} \Omega ^{+} = \left\{ x \in {\mathbb {R}}^N: W(x)> 0\right\} \quad \text{ and }\quad \Omega ^{-} = \left\{ x \in {\mathbb {R}}^N: W(x)< 0\right\} . \end{aligned}$$

Since $\overline{\Omega ^{+}}$ is compact and $\overline{\Omega ^{+}}\subset {\mathbb {R}}^N\backslash \overline{\Omega ^{-}}$, there exists a function $\zeta \in C^{\infty }({\mathbb {R}}^N)$ such that $0 \le \zeta (x) \le 1$ for all $x \in {\mathbb {R}}^N$, $\zeta (x) = 1$ for $x \in \Omega ^{+}$ and $\zeta (x) = 0$ for $x \in \Omega ^{-}$. From now on, we set

$$\begin{aligned} K := \sup _{{\mathbb {R}}^N}\left| \nabla \zeta \right| >0. \end{aligned}$$

(1.7)

Besides the condition $(f_1)$ on the nonlinearity f(t), setting $F(t) = \int ^{t}_{0} f(s) {{\mathrm{{d}}}}s$, we also consider the following assumptions:

$(f_2)$:

There exists $\sigma > (1+K)c_\alpha$ such that

$$\begin{aligned}0 < \sigma F(t) \le f(t)t,\,\,\text{ for } \text{ all }\ t\ne 0; \end{aligned}$$

$(f_3)$:

There exist $\theta > m_\alpha$ and $\mu >0$ such that $F(t)\ge \mu t^{\theta }$ for all $t\in [0,1]$.

We observe that by $(f_2)$ and deriving the quotient $F(t)/t^\sigma$ we deduce that $F(t)/t^\sigma$ is nondecreasing for $t>0$. Thus, $F(t)\le F(1)t^\sigma = C_1 t^\sigma$ for all $t\in [0,1]$. Consequently, in view of $(\Phi _1)$, $F(t)/\Phi (t)\le C_1C t^{\sigma -N}$ for all $t\in (0,\min \{1,1/C\})$, and therefore, we obtain

$$\begin{aligned} \lim _{t\rightarrow 0^+}\frac{F(t)}{\Phi (t)}=0. \end{aligned}$$

(1.8)

Assumption $(f_3)$ is used to estimate the minimax level of the energy functional associated with (1.1). Note that we require this condition only for $t\in [0,1]$.

We say that $u:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$ is a weak solution of problem (1.1) if $u\in W^{1,\Phi }({\mathbb {R}}^N)$ and it holds

$$\begin{aligned} \int _{{\mathbb {R}}^N}\left[ \Phi '(|\nabla u|)\frac{\nabla u}{|\nabla u|}\nabla v+\Phi '(|u|)\frac{u}{|u|}v\right] {\mathrm{{d}}}x-\int _{{\mathbb {R}}^N}W(x)f(u)v{\mathrm{{d}}}x=0,\quad \text{ for } \text{ all }\ v\in W^{1,\Phi }({\mathbb {R}}^N). \end{aligned}$$

In order to state our main result, let us introduce some notations. Without loss of generality, we can assume that $0 \in \Omega ^+$. Let $\delta >0$ be such that $B_\delta :=B_{\delta }(0) \subset \subset \Omega ^+$. For $A\subset {\mathbb {R}}^N$ measurable, from now on we will denote its Lebesgue measure by |A| and we introduce the number

$$\begin{aligned} \mu ^*:= \max \left\{ \mu _1,\left[ \frac{a_1|B_\delta | \frac{\theta -m_\alpha }{m_\alpha }\left( \frac{\mu _1 m_\alpha }{\theta }\right) ^{\frac{\theta }{\theta -m_\alpha }}}{\left( 1-\frac{c_\alpha }{ \sigma } -\frac{Kc_\alpha }{\sigma }\right) \xi _0\left( \frac{K_{N,\alpha }^{1/\gamma }}{(r'b)^{1/\gamma }}\right) }\right] ^\frac{\theta -m_\alpha }{m_\alpha }\right\} , \end{aligned}$$

(1.9)

where $\xi _0(t)=\min \{t^{m_\alpha },t^{c_\alpha }\}$ for $t\ge 0$, $a_1:= \max \{ W(x) ; \,x \in \overline{B_\delta }\}>0$, $r'=r/(r-1)$ (r is given in $(W_1)$),

$$\begin{aligned} \mu _1:=\frac{2\Phi (1)|B_\delta |}{a_0|B_{\delta /2}|}, \end{aligned}$$

$a_0:= \min \{ W(x) ; \,x \in \overline{B_\delta }\}>0$ and $K_{N,\alpha }=B^{1/B}N\omega _{N-1}^{\gamma /N}$, $B=1-\alpha /(N-1)$ and $\omega _{N-1}$ is the measure of the unit sphere in ${\mathbb {R}}^N$.

Now, we are ready to present our main result:

Theorem 1.1

Assume that $\Phi$ is a N-function verifying $(\Phi _1)-(\Phi _2)$. Moreover, suppose that $(W_1)-(W_2)$, $(f_1)-(f_3)$ are satisfied with $\mu \ge \mu ^*$ in $(f_3)$. Then, problem (1.1) has a nonzero weak solution.

The main features of this class of problems, considered in this paper, are that it is defined in the whole ${\mathbb {R}}^N$ and involves exponential critical growth (according to Lemma 3.3) and the operator inhomogeneous $\Delta _\Phi u=\mathrm {div}\left( \Phi '(|\nabla u|)\nabla u/|\nabla u|\right)$. We will show that the functional energy associated with (1.1) verifies the Palais–Smale compactness condition in certain energy levels. By applying the mountain pass theorem, we will establish the existence of nonzero solution for Eq. (1.1). Here, we improve and complement some previously cited works. As far as we know, there are no papers which deal with Eq. (1.1) in the Orlicz context, where the nonlinearities have exponential critical growth and changes sign. Besides, to prove the existence of nonzero solution, we do not assume the conditions $(\Phi _3)$, (1.7), (1.12), (1.13), (1.16) and (1.17) in [18] (see also similar assumptions in [16]). We also mention that we do not impose an specific hypothesis on F(t) at the origin and we do not assume the condition

$$\begin{aligned} \text{ there } \text{ exist } \ M>0,\, t_0>0\,\,\ \text {such that}\ \,\, F(t)\le M|f(t)|,\ \,\text{ for } \text{ all }\ |t|\ge t_0, \end{aligned}$$

which is often used in semilinear and quasilinear problems involving exponential critical growth. In this direction, our paper improves and complements, for example, the works [16, 18, 24,25,26,27, 39, 46].

Example 1.2

Notice that, for $0\le \alpha <N-1$, the hypotheses of Theorem 1.1 are, for example, satisfied by $\Phi (t)$ and f(t) given by:

• (i) $\Phi (t)=|t|^N+N\int _0^{|t|}s^{N-1}\mathrm {arc}\sinh ^{\alpha }s\,{\mathrm{{d}}}s$;

• (ii) $f(t)= F'(t)$ with $F(t)=\mu t^p \exp (bt^{\gamma })$, where $\mu \ge \mu ^*$, $p>c_\alpha$, $b>0$ and $\gamma =N/(N-1-\alpha )$.

Notice that in this example we have $m_\alpha =N$ and $c_\alpha =N+\alpha$.

Remark 1.3

We emphasize that the approach used in this paper can be adapted with slight modifications to deal with a more general condition than $(\Phi _2)$, namely

$$\begin{aligned} \lim _{t\rightarrow +\infty }\frac{\Phi (t)}{t^N\left( \prod _{j=1}^{l-1}\log ^\alpha _{[j]}(t)\right) \log ^\alpha _{[l]}(t)}=1,\quad l\in {\mathbb {N}}\quad \mathrm {and}\quad \alpha \in [0,N-1), \end{aligned}$$

where $\log _{[k]}(t)=\log (\log _{[k-1]}(t))$ and $\log _{[1]}(t)=\log (t)$, which was considered in [16,17,18]. For the sake of simplicity, we prefer to treat only the case $l=1$.

Remark 1.4

Equations involving the operator $\Delta _\Phi$ appear in several physical contexts, as observed in [30] (and references [6, 7, 8] therein). They are related to concrete examples from fluid mechanics and plasticity theory. Moreover, if $\Omega$ is a domain of ${\mathbb {R}}^N$, with $N=2$ or $N=3$, and

$$\begin{aligned} \Phi (t)=|t|^N+N\int _0^{|t|}s^{N-1}\mathrm {arc}\sinh ^{\alpha }s\,{\mathrm{{d}}}s, \ \ t\in {\mathbb {R}}, \end{aligned}$$

(1.10)

$\alpha \in (0,N-1)$, the slow steady-state motion of a fluid of Prandtl–Eyring type in $\Omega$ can be modeled by the following set of equations:

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm {div}\left( \Phi '(|D u|)D u/|D u|\right) + (\text {potential term})=0\quad \hbox { in}\quad \Omega ,\\ \mathrm {div}\,u=0\quad \hbox {in}\quad \Omega ,\\ u=0\quad \hbox {on}\quad \partial \Omega ,\\ \end{array} \right. \end{aligned}$$

where $u: \Omega \rightarrow {\mathbb {R}}^N$ denotes the velocity field of a incompressible fluid and $Du:=\frac{1}{2}(\nabla u+\nabla u^{\perp })$ is the symmetric gradient of u (for more details see [13] and [31]).

Remark 1.5

In the papers [3, 12, 35], the authors establish integrability estimates for N-Laplace equation with an external force. In short, they consider the problem

$$\begin{aligned} \left\{ \begin{aligned} -\Delta _N u&=f(x)&in\quad \Omega ,\\ u&=0&on\quad \partial \Omega , \end{aligned} \right. \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^N$ is a bounded smooth domain and f belongs to $L^1(\Omega )$ or to a better space, and they show that if $u\in W^{1,N}_0(\Omega )$ is a weak solution (or an entropy solution) then u satisfies an integrability estimate of type

$$\begin{aligned} \int _{\Omega }\exp \left[ \frac{(N\omega _N^{1/(N-1)}-\delta )|u(x)|}{\Vert f\Vert _1^{1/(N-1)}}\right] {\mathrm{{d}}}x\le \frac{N\omega _N^{1/(N-1)}}{\delta }|\Omega |,\quad \text {for all}\ \ \delta \in (0,N\omega _N^{1/(N-1)}), \end{aligned}$$

where $\omega _N$ is the measure of the unit sphere in ${\mathbb {R}}^N$ and $|\Omega |$ is the measure of $\Omega$. Since the function $\Phi$ satisfies condition $(\Phi _1)-(\Phi _2)$, we believe that the approach used in [3, 12, 35] seems to lead to similar results in the case of the problem

$$\begin{aligned} \left\{ \begin{aligned} -\Delta _\Phi u&=f(x)&\mathrm{{in}}\quad \Omega ,\\ u&=0&\mathrm{{on}}\quad \partial \Omega , \end{aligned} \right. \end{aligned}$$

which can enable us to get regularity results to the solutions of (1.1).

This paper is organized as follows: In Sect. 2, we present some preliminary results about Orlicz spaces which are used in the work. In Sect. 3, we establish the variational framework for our problem and we obtain some embedding results involving our working space. Section 4 shows that the energy functional has the geometric structure of the mountain pass theorem, and in Sect. 5, we prove that this functional satisfies the Palais–Smale condition in certain energy levels. Finally, in Sect. 6, we prove Theorem 1.1.

Throughout this paper, $W^{1,N}({\mathbb {R}}^N)$ denotes the Sobolev space endowed with the norm

$$\begin{aligned} \Vert u\Vert _{1,N} =\left( \int _{{\mathbb {R}}^N} (|\nabla u|^N+|u|^N) {\mathrm{{d}}}x\right) ^{1/N}, \quad u\in W^{1,N}({\mathbb {R}}^N). \end{aligned}$$

We use $|\cdot |_p$ to denote the norm of the Lebesgue space $L^p({\mathbb {R}}^N)$, $1\le p\le \infty$. We denote by $B_R$ the ball centered at the origin with radius $R>0$ and the symbols $C, C_i$, $i = 0,1,2,\ldots$ will denote different (possibly) positive constants.

2 Preliminaries

In order to facilitate the understanding of the paper, in this section we present briefly some results about Orlicz spaces. For the proofs and more details, see, for instance, [7, 37, 45].

A function $A: {\mathbb {R}} \rightarrow [0, +\infty )$ is called N-function if it is convex, even, $A(t) =0$ if and only if $t=0$, ${A(t)}/{t} \rightarrow 0$ as $t\rightarrow 0$ and ${A(t)}/{t} \rightarrow +\infty$ as $t\rightarrow +\infty$. In particular, we have $A'(0)=0$, and if A is differentiable, then $A'(t)$ is nondecreasing for $t\ge 0$, which implies that A(t) is increasing for $t>0$. For a N-function A and an open set $\Omega \subset {\mathbb {R}}^N$, the Orlicz class is the set defined by

$$\begin{aligned} K_{A,\mu }(\Omega ) = \left\{ u:\Omega \rightarrow {\mathbb {R}}; \text{ u } \text{ is } \text{ measurable } \text{ and } \int _{\Omega } A(|u(x)|)\, d\mu < \infty \right\} . \end{aligned}$$

The linear space $L_{A,\mu }(\Omega )$ generated by $K_{A,\mu }(\Omega )$ is called Orlicz space. If $\mu$ is the Lebesgue measure, then we denote $K_{A,\mu }(\Omega )$ and $L_{A,\mu }(\Omega )$ by $K_{A}(\Omega )$ and $L_{A}(\Omega )$, respectively. When A satisfies the $\Delta _2$-condition, namely, there exists a constant $k>0$ such that

$$\begin{aligned} A(2t) \le kA(t),\quad \text{ for } \text{ all }\ t\ge 0, \end{aligned}$$

the Orlicz class $K_{A,\mu }(\Omega )$ is a linear space and hence equal to $L_{A,\mu }(\Omega )$. We consider the following norm (called of Luxemburg’s norm) on $L_{A,\mu }(\Omega )$:

$$\begin{aligned} |u|_{A,\Omega }= \inf \left\{ \lambda >0; \int _{\Omega } A\left( \frac{|u(x)|}{\lambda }\right) \, d\mu \le 1 \right\} . \end{aligned}$$

It can be shown that $\left( L_{A,\mu }(\Omega ), |\cdot |_{A, \Omega }\right)$ is a Banach space (see [45, Theorem 10, p. 67]). In the case $\Omega = {\mathbb {R}}^N$, we denote $|\cdot |_{A, {\mathbb {R}}^N}$ by $|\cdot |_{A}$. The complement N-function of A is defined by

$$\begin{aligned} {\tilde{A}}(t) = \sup _{s>0}\left\{ ts - A(s)\right\} . \end{aligned}$$

It is not difficult to verify that $\tilde{{\tilde{A}}}=A$. In the spaces $L_{A,\mu }(\Omega )$ and $L_{{\tilde{A}},\mu }(\Omega )$, an extension of the Hölder inequality holds, namely

$$\begin{aligned} \left| \int _\Omega u(x)v(x)\, d\mu \right| \le 2|u|_{A,\Omega }|v|_{{\tilde{A}},\Omega },\quad \text{ for } \text{ all } \, u\in L_{A,\mu }(\Omega ),\ v\in L_{{\tilde{A}},\mu }(\Omega ). \end{aligned}$$

(2.1)

As a consequence, for every ${\tilde{u}} \in L_{{\tilde{A}},\mu }(\Omega )$ there corresponds a continuous linear functional $f_{{\tilde{u}}} \in (L_{A,\mu }(\Omega ))^{'}$ given by $f_{{\tilde{u}}}(v) = \int _\Omega {\tilde{u}}(x)v(x)\, d\mu$, $v\in L_{A,\mu }(\Omega )$. Thus, we can define

$$\begin{aligned} \Vert {\tilde{u}}\Vert _{{\tilde{A}},\Omega }= \sup _{|v|_{A, \Omega }\le 1}\int _\Omega {\tilde{u}}(x)v(x)\, d\mu \end{aligned}$$

and $\Vert \cdot \Vert _{{\tilde{A}},\Omega }$ is called the Orlicz norm on the space $L_{{\tilde{A}},\mu }(\Omega )$. Similarly, we can define the Orlicz norm $\Vert \cdot \Vert _{{A},\Omega }$ on $L_{A,\mu }(\Omega )$. The norms $|\cdot |_{{A},\Omega }$ and $\Vert \cdot \Vert _{{A},\Omega }$ are equivalent and satisfy the inequalities

$$\begin{aligned} | u |_{{A},\Omega } \le \Vert u \Vert _{{A},\Omega } \le 2| u |_{{A},\Omega },\quad \text{ for } \text{ all }\ u\in L_{A}(\Omega ). \end{aligned}$$

We define the Orlicz–Sobolev space $W^{1,A}(\Omega )$ as follows

$$\begin{aligned} W^{1,A}(\Omega ) = \left\{ u:\Omega \rightarrow {\mathbb {R}}: \text{ u } \text{ is } \text{ measurable } \text{ and } u, |\nabla u| \in L_A(\Omega )\right\} \end{aligned}$$

equipped with the norm

$$\begin{aligned} \Vert u\Vert _{W^{1,A}(\Omega )} := |u|_{A,\Omega } + |\nabla u|_{A,\Omega }, \end{aligned}$$

where $\nabla u$ is the gradient of u, and we are using its Euclidean norm in ${\mathbb {R}}^N$. An important property is that if A and ${\tilde{A}}$ verify the $\Delta _2$-condition, then the spaces $L_{A}(\Omega )$ and $W^{1,A}(\Omega )$ are reflexive, separable and

$$\begin{aligned} \left( L_{{A}}(\Omega ),|\cdot |_{{A},\Omega }\right) ' = \left( L_{{\tilde{A}}}(\Omega ),\Vert \cdot \Vert _{{\tilde{A}},\Omega }\right) \ \text{ and } \ \left( L_{{\tilde{A}}}(\Omega ),|\cdot |_{{\tilde{A}},\Omega }\right) ' = \left( L_{{A}}(\Omega ),\Vert \cdot \Vert _{{A},\Omega }\right) . \end{aligned}$$

Proposition 2.1

Let A be a N-function of class $C^1$ . Then ${\tilde{A}}$ is a function $C^1$ and verifies the conditions below:

(i):

$A({\tilde{A}}'(t)) = {\tilde{A}}'(t)t - {\tilde{A}}(t), \quad \text{ for } \text{ all }\ t \ge 0$;

(ii):

$A'({\tilde{A}}'(t)) = t, \quad \text{ for } \text{ all }\ t \ge 0.$

Proof

See Lemma A.2 and Lemma 2.5 in [32]. $\square$

Definition 2.2

We say that a N-function A satisfies the $\nabla _2$ condition (we denote by $A\in \nabla _2$), if there exists $\gamma >1$ such that

$$\begin{aligned} A(t) \le \dfrac{1}{2\gamma }A(\gamma t), \quad \text{ for } \text{ all } \ t \ge 0. \end{aligned}$$

Proposition 2.3

Let A and ${\tilde{A}}$ be a pair of differentiable complementary N -functions. Then, the following assertions are equivalent:

(a) :

$A \in \Delta _2$;

(b) :

there exists $\alpha \in (1, \infty )$ such that $A'(t)t/A(t) < \alpha$ for all $t \ge 0$;

(c) :

there exists $\beta \in (1, \infty )$ such that ${\tilde{A}}'(s)s/{\tilde{A}}(s)>\beta$ for all $s \ge 0$;

(d) :

${\tilde{A}} \in \nabla _2$.

Proof

See Theorem 3 (p. 22) in [45]. $\square$

Next, we state two lemmas due to Fukagai et al. [32, Lemma 2.1, Lemma 2.5] and Lieberman [38, Lemma 1.1 (e)] which will be used in our arguments.

Lemma 2.4

Suppose that A is a N -function and let ${\tilde{A}}$ be the complement N -function of A . Then,

(a) :

${\tilde{A}} (A'(t)) \le A(2t)$ for all $t\ge 0$;

(b) :

${\tilde{A}} \left( \frac{A(t)}{t}\right) \le A(t)$ for all $t>0$;

(c) :

$A'(t)s\le A'(t)t+A'(s)s$ for all $t,s\ge 0$.

Lemma 2.5

Suppose that A is a differentiable N-function satisfying

$$\begin{aligned} m\le \frac{A'(t)t}{A(t)}\le M,\ \ \text{ for } \text{ all }\ t>0, \end{aligned}$$

(2.2)

for some $M\ge m>0$. Defining, for $t\ge 0$, $\xi _0(t) = \min \{t^{m}, t^{M}\}$ and  $\xi _1(t) = \max \{t^{m}, t^{M}\}$, one has

$$\begin{aligned} \xi _0(\rho )A(t) \le A(\rho t) \le \xi _1(\rho )A(t)\quad \text{ for }\quad \rho , t \ge 0 \end{aligned}$$

and

$$\begin{aligned} \xi _0(|u|_{A, \Omega } )\le \int _{\Omega } A(|u(x )|)\,d\mu \le \xi _1(|u|_{A, \Omega } )\quad \text{ for }\quad u \in L_{A,\mu }(\Omega ). \end{aligned}$$

Lemma 2.6

Let $\Phi$ be a N-function satisfying conditions $(\Phi _1)$ and $(\Phi _2)$. Then, $\Phi$ satisfies the condition $\nabla _2$.

Proof

In order to show that $\Phi \in \nabla _2$, we will verify that there exists a constant $C>0$ such that $C\Phi (t) \rho ^N \le \Phi (\rho t)$ for each $\rho > 1$ and $t\ge 0$. In fact, setting $M_0:= \inf _{t\in [0,1]}\Phi (t)/t^N$ and $M_1:= \sup _{t\in [0,1]}\Phi (t)/t^N$, by ($\Phi _1$) one has $0<M_0\le M_1<\infty$. Therefore, $M_0 t^N \rho ^N \le \Phi (\rho t) \le M_1 \rho ^N t^N$ if $\rho t\le 1$. Thus,

$$\begin{aligned} \Phi (\rho t)\ge \frac{M_0}{M_1}\rho ^N\Phi (t)\quad \mathrm {for}\quad \rho t\le 1 . \end{aligned}$$

(2.3)

On the other hand, setting

$$\begin{aligned} M_2:= \inf _{t\ge 1}\frac{\Phi (t)}{\log ^\alpha (t+1)}\quad \mathrm {and}\quad M_3:= \sup _{t\ge 1}\frac{\Phi (t)}{\log ^\alpha (t+1)}, \end{aligned}$$

from condition $(\Phi _2)$ we have $0<M_2\le M_3<\infty$ and

$$\begin{aligned} M_3 t^N\log ^\alpha (t+1)\ge \Phi (t)\ge M_2 t^N\log ^\alpha (t+1),\quad \text{ for } \text{ all } \ t\ge 1. \end{aligned}$$

Therefore, if $\rho t>1$ then

$$\begin{aligned} \Phi (\rho t)\ge M_2t^N\rho ^N\log ^\alpha (\rho t+1)\ge M_2t^N\rho ^N\log ^\alpha ( t+1)\ge \frac{M_2}{M_3}\rho ^N\Phi (t). \end{aligned}$$

(2.4)

Hence, taking $C=\min \left\{ M_0/M_1,M_2/M_3\right\}$, by (2.3) and (2.4) we get $\Phi (\rho t) \ge C \rho ^N \Phi (t)$ for each $\rho > 1$ and $t\ge 0$. Finally, taking $\gamma =\rho > (2/C)^{1/(N-1)}$ we obtain $\Phi (t) \le \Phi (\gamma t)/(2\gamma )$ for $t \ge 0$, which shows that $\Phi \in \nabla _2$ and the proof is finished. $\square$

Proposition 2.7

If $\Phi$ is a N-function satisfying $(\Phi _1)-(\Phi _2)$, then $m_\alpha >1$, where $m_\alpha$ was defined in (1.6).

Proof

The proof follows combining Proposition 2.3 and Lemma 2.6. $\square$

Remark 2.8

As we saw in Introduction, our N-function $\Phi$ satisfies $\Delta _2$-condition and assumption (2.2) with $m=m_\alpha$ and $M=c_\alpha$.Combining Proposition 2.3 with Lemma 2.6, ${\tilde{\Phi }}$ satisfies $\Delta _2$-condition. Therefore, it can be shown that

$$\begin{aligned} u_{n}\rightarrow 0\ \ \mathrm {in}\ \ L_{\Phi ,\mu }({\mathbb {R}}^N) \Longleftrightarrow \int _{{\mathbb {R}}^N}\Phi (|u_{n}|)\ d\mu \rightarrow 0 \end{aligned}$$

and $(u_n)$ is bounded in $L_{\Phi ,\mu }({\mathbb {R}}^N)$ if and only if $(\int _{{\mathbb {R}}^N}\Phi (|u_n|)d\mu )$ is bounded. Moreover, as observed above $\left( L_{\Phi ,\mu }({\mathbb {R}}^N), |\cdot |_\Phi \right)$ is a separable and reflexive Banach space as well as $\left( W^{1,\Phi }({\mathbb {R}}^N), \Vert \cdot \Vert \right)$.

3 Variational framework

The next lemma presents some embeddings which will be used in our arguments.

Lemma 3.1

If $(\Phi _1)$ and $(\Phi _2)$ are satisfied, then the following embeddings are continuous:

(a) :

$L_{\Phi }({\mathbb {R}}^N)\hookrightarrow L^N({\mathbb {R}}^N)$;

(b) :

$W^{1,\Phi }({\mathbb {R}}^N)\hookrightarrow W^{1,N}({\mathbb {R}}^N)$;

(c) :

$W^{1,\Phi }({\mathbb {R}}^N)\hookrightarrow L^r({\mathbb {R}}^N)$ for any $r\in [N,\infty )$.

Proof

We observe that by $(\Phi _1)-(\Phi _2)$ there exists $C_1>0$ such that $t^N\le C_1\Phi (t)$ for all $t\ge 0$. Thus, if $u_n\rightarrow 0$ in $L_{\Phi }({\mathbb {R}}^N)$ then

$$\begin{aligned} \int _{{\mathbb {R}}^N}|u_n|^N {\mathrm{{d}}}x\le C_1\int _{{\mathbb {R}}^N}\Phi (|u_n|){\mathrm{{d}}}x\rightarrow 0 \end{aligned}$$

and items a) and b) are proved. The proof of item c) follows directly from b) and by the continuous embedding from $W^{1,N}({\mathbb {R}}^N)$ into $L^r({\mathbb {R}}^N)$ for any $r\in [N,\infty )$. $\square$

Now, we prove a result of convergence, which will be crucial in the sequel.

Proposition 3.2

Under conditions $(W_1)$ and $(\Phi _1)-(\Phi _2)$ , if $u_n\rightharpoonup u$ in $W^{1,\Phi }({\mathbb {R}}^N)$ then

$$\begin{aligned} \displaystyle \lim _{n\rightarrow +\infty }\left( \int _{\mathbb {R^N}}|W(x)||u_n-u|^s\,{\mathrm{{d}}}x\right) ^{1/s}=0,\quad \text{ for } \text{ all }\ s\in [N,\infty ). \end{aligned}$$

Proof

In view of $(W_1)$ and by the Hölder inequality, we have

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N}|W(x)||u_n-u|^s\,{\mathrm{{d}}}x&=\int _{B_R}|W(x)| |u_n-u|^s\,{\mathrm{{d}}}x+\int _{B_R^c}|W(x)| |u_n-u|^s\,{\mathrm{{d}}}x\\&\le |W|_{\infty ,B_R}\int _{B_R} |u_n-u|^s\,{\mathrm{{d}}}x\\&\quad +\left( \int _{B_R^c}|W(x)|^r\,{\mathrm{{d}}}x\right) ^{1/r} \left( \int _{B_R^c}|u_n-u|^{sr'}\,{\mathrm{{d}}}x\right) ^{1/r'}. \end{aligned} \end{aligned}$$

(3.1)

On the other hand, since the embedding $W^{1,N}({\mathbb {R}}^N) \hookrightarrow L^{s}(B_{R})$ is compact and $W^{1,\Phi }({\mathbb {R}}^N)\hookrightarrow L^{sr'}({\mathbb {R}}^N)$ is continuous, given $\varepsilon >0$ there exists $n_0\in {\mathbb {N}}$ such that

$$\begin{aligned} \int _{B_{R}}|u_n-u|^s\,{\mathrm{{d}}}x<\frac{\varepsilon }{2 |W|_{\infty ,B_R}}\ \ \text{ for } \text{ all }\ n>n_0 \end{aligned}$$

(3.2)

and there exists $C>0$ verifying

$$\begin{aligned} \left( \int _{B_R^c}|u_n-u|^{sr'}\,{\mathrm{{d}}}x\right) ^{\frac{1}{r'}}\le C, \ \ \text{ for } \text{ all }\ n\in {\mathbb {N}}. \end{aligned}$$

(3.3)

Moreover, by $(W_1)$ there exists $R_0>0$ sufficiently large such that

$$\begin{aligned} \left( \int _{B_R^c}|W(x)|^r\,{\mathrm{{d}}}x \right) ^{\frac{1}{r}}<\frac{\varepsilon }{2C}, \ \ \text{ for } \text{ all }\ R\ge R_0. \end{aligned}$$

(3.4)

Hence, from (3.1)–(3.4),

$$\begin{aligned} \int _{{\mathbb {R}}^N}|W(x)||u_n-u|^s\,{\mathrm{{d}}}x<\varepsilon ,\quad \text{ for } \text{ all }\ n>n_0, \end{aligned}$$

and the proof is complete. $\square$

The next lemma presents a version of the Trudinger–Moser inequality for functions in $W^{1,\Phi }({\mathbb {R}}^N)$, which was proved by Cerný in [18] (see also [21]). It is necessary to use variational methods to find solutions for problem (1.1) with nonlinearities f(t) satisfying the condition growth $(f_1)$.

Lemma 3.3

If $N\ge 2$, ${\widetilde{K}}>0$, $\alpha \in [0,N-1)$, $\Phi$ is a N-function verifying $(\Phi _1)-(\Phi _2)$ and $u\in W^{1,\Phi }({\mathbb {R}}^N)$, then

$$\begin{aligned} \int _{{\mathbb {R}}^N}\left[ \exp ({\widetilde{K}}|u|^\gamma )-S_{N,\alpha }({\widetilde{K}}|u|^\gamma )\right] {\mathrm{{d}}}x<\infty . \end{aligned}$$

Furthermore, if $|\nabla u|_\Phi \le 1$, $|u|_\Phi \le M<\infty$ and ${\widetilde{K}}<K_{N,\alpha }$ then there exists a constant $C=C(N,\alpha , M, \Phi ,{\widetilde{K}})>0$, which depends only $N,\alpha , M,\Phi$ and ${\widetilde{K}}$ such that

$$\begin{aligned} \int _{{\mathbb {R}}^N}\left[ \exp ({\widetilde{K}}|u|^\gamma )-S_{N,\alpha }({\widetilde{K}}|u|^\gamma )\right] {\mathrm{{d}}}x\le C, \end{aligned}$$

where $K_{N,\alpha }=B^{1/B}N\omega _{N-1}^{\gamma /N}$, $B=1-\alpha /(N-1)$ and $\omega _{N-1}$ is the measure of the unit sphere in ${\mathbb {R}}^N$.

To finalize this section, we get two technical lemmas that will be necessary to show the regularity of the energy functional associated with our problem.

Lemma 3.4

For each $p\ge 1$, there exists $C=C(p)>0$ such that

$$\begin{aligned} \left[ \exp (t)-S_{N,\alpha }(t)\right] ^p\le C\left[ \exp (pt)-S_{N,\alpha }(pt)\right] , \ \text{ for } \text{ all }\ t\ge 0. \end{aligned}$$

Proof

It suffices to proof that the limits

$$\begin{aligned} \lim _{t\rightarrow 0}\frac{\left[ \exp (t)-S_{N,\alpha }(t)\right] ^p}{\left[ \exp (pt)-S_{N,\alpha }(pt)\right] }\quad \text {and}\quad \lim _{t\rightarrow +\infty }\frac{\left[ \exp (t)-S_{N,\alpha }(t)\right] ^p}{\left[ \exp (pt)-S_{N,\alpha }(pt)\right] } \end{aligned}$$

are finite, which is a direct consequence of the L’Hospital rule. $\square$

Lemma 3.5

Let $(u_n)$ be a sequence in $W^{1,\Phi }({\mathbb {R}}^N)$ strongly convergent. Then there exist a subsequence $(u_{n_k})$ of $(u_n)$ and $v\in W^{1,\Phi }({\mathbb {R}}^N)$ such that $u_{n_k}(x)\le v(x)$ almost everywhere in $x\in {\mathbb {R}}^N$.

Proof

The arguments used to show this lemma follows the same lines of the proof of Proposition 1 of [26] with slight modifications and we omit it. $\square$

The energy functional associated with problem (1.1) is given by

$$\begin{aligned} J(u)=\int _{{\mathbb {R}}^N}[\Phi (|\nabla u|)+\Phi (|u|)]\,{\mathrm{{d}}}x -\int _{{\mathbb {R}}^N} W(x)F(u)\,{\mathrm{{d}}}x. \end{aligned}$$

Notice that by $(f_1)$, Lemmas 3.3, 3.5 and Proposition 3.2, J is well defined on $W^{1,\Phi }({\mathbb {R}}^N)$, and moreover, by using standard computations (see [18, Proposition 4.1]), we can see that $J\in C^1(W^{1,\Phi }({\mathbb {R}}^N),{\mathbb {R}})$ and its derivative is given by

$$\begin{aligned} \langle J'(u),v\rangle =\int _{{\mathbb {R}}^N}\left[ \Phi '(|\nabla u|)\frac{\nabla u}{|\nabla u|}\nabla v+\Phi '(|u|)\frac{u}{|u|}v\right] {\mathrm{{d}}}x-\int _{{\mathbb {R}}^N}W(x)f(u)v{\mathrm{{d}}}x \end{aligned}$$

for $u,v\in W^{1,\Phi }({\mathbb {R}}^N)$. Consequently, critical points of J are precisely the weak solutions of (1.1).

4 Mountain pass structure

In order to get Theorem 1.1, we shall use the mountain pass theorem due to Ambrosetti and Rabinowitz [9]:

Theorem 4.1

Let X be a Banach space and $J \in C^1(X;{\mathbb {R}})$ with $J(0)=0$. Suppose that there exist $\rho ,\tau >0$ and $e\in X$, with $\Vert e\Vert >\rho$, such that

$$\begin{aligned} \inf _{\Vert u\Vert =\rho }J(u)\ge \tau \quad \text {and}\quad J(e) < 0. \end{aligned}$$

(4.1)

Then, J possesses a Palais–Smale sequence at level c characterized as

$$\begin{aligned} c:=\inf _{\gamma \in \Gamma }\max _{t\in [0,1]}J(\gamma (t))\ge \tau , \end{aligned}$$

where $\Gamma = \left\{ \gamma \in C([0,1]; X):\gamma (0)=0\;\ \text {and} \;\ \gamma (1)=e \right\}$. Moreover, if J satisfies the Palais–Smale condition at level c then J has a critical point $u_0$ such that $J(u_0)=c$.

The number c is called mountain pass level or minimax level of the functional J.

In the sequel, we show that the functional J has the mountain pass geometry, condition (4.1). This is proved in the next lemmas.

Lemma 4.2

Assume $(f_1)$ and $(f_2)$ . Then, there exist $\rho , \beta >0$ such that

$$\begin{aligned} J(u)\ge \beta , \ \ \text{ for } \text{ all }\ \Vert u\Vert =\rho . \end{aligned}$$

Proof

From (1.8), given $\epsilon >0$, there exists $\delta >0$ verifying $F(t)\le \varepsilon \Phi (t)$ for all $|t|\le \delta$. On the other hand, by using $(f_1)$ and taking $p>c_\alpha$ we have

$$\begin{aligned} F(t)\le C|t|^p[\exp (b|t|^{\gamma })-S_{N,\alpha }(b|t|^{\gamma })], \ \ \text{ for } \text{ all }\ |t|> \delta , \end{aligned}$$

for some $C=C(\delta ,p)>0$. Therefore,

$$\begin{aligned} F(t)\le \varepsilon \Phi (t)+ C|t|^p[\exp (b|t|^{\gamma })-S_{N,\alpha }(b|t|^{\gamma })], \ \ \text{ for } \text{ all }\ t\ge 0. \end{aligned}$$

(4.2)

From $(W_2)$, we obtain

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N} W(x)F(u){\mathrm{{d}}}x&\le \int _{\Omega ^{+}} W(x)F(u){\mathrm{{d}}}x \\&\le \varepsilon C_1\int _{\Omega ^{+}} \Phi (u) {\mathrm{{d}}}x+ C_2\int _{\Omega ^{+}} |u|^{p}[\exp (b|u|^{\gamma })-S_{N,\alpha }(b|u|^{\gamma })]{\mathrm{{d}}}x. \end{aligned} \end{aligned}$$

(4.3)

Fixing $\varepsilon = 1/2C_1$, using Hölder inequality, Lemmas 3.1 and 3.4(c), we reach

$$\begin{aligned} \begin{aligned} J(u)&\ge \left( 1-\varepsilon C_1 \right) \int _{{\mathbb {R}}^N}[\Phi (|\nabla u|)+\Phi (u)]{\mathrm{{d}}}x\\&\quad -C_2\left\{ \int _{{\mathbb {R}}^N}[\exp (2b|u|^{\gamma })-S_{N,\alpha }(2b|u|^{\gamma })]{\mathrm{{d}}}x\right\} ^{\frac{1}{2}}|u|_{2p}^p\\&\ge \frac{1}{2}\int _{{\mathbb {R}}^N}[\Phi (|\nabla u|)+\Phi (|u|)]{\mathrm{{d}}}x\\&\quad -C_3\left\{ \int _{{\mathbb {R}}^N}\left[ \exp \left( 2b|\nabla u|_{\Phi }^{\gamma }\left( \frac{|u|}{|\nabla u|_{\Phi }}\right) ^{\gamma }\right) -S_{N,\alpha }\left( 2b|\nabla u|_{\Phi }^{\gamma }\left( \frac{|u|}{|\nabla u|_{\Phi }}\right) ^{\gamma }\right) \right] {\mathrm{{d}}}x\right\} ^{\frac{1}{2}}\Vert u\Vert ^p. \end{aligned} \end{aligned}$$

Now, if $0< \rho < 1$ is such that $2b\rho ^{\gamma }<K_{N,\alpha }$ then for $\Vert u\Vert =\rho$ we obtain $2b|\nabla u|_{\Phi }^{\gamma }\le 2b\rho ^{\gamma }<K_{N,\alpha }$. Thus, by Lemma 3.3

$$\begin{aligned} \left\{ \int _{{\mathbb {R}}^N}\left[ \exp \left( 2b|\nabla u|_{\Phi }^{\gamma }\left( \frac{|u|}{|\nabla u|_{\Phi }}\right) ^{\gamma }\right) -S_{N,\alpha }\left( 2b|\nabla u|_{\Phi }^{\gamma }\left( \frac{|u|}{|\nabla u|_{\Phi }}\right) ^{\gamma }\right) \right] {\mathrm{{d}}}x\right\} ^{\frac{1}{2}}\le C. \end{aligned}$$

Consequently, from Lemma 2.5, for $\Vert u\Vert =\rho$ we get

$$\begin{aligned} J(u)\ge \frac{1}{2}\left( |\nabla u|_{\Phi }^{c_\alpha }+|u|_{\Phi }^{c_\alpha }\right) -C_4\rho ^p\ge 2^{-(c_\alpha +1)}\rho ^{c_\alpha }-C_4\rho ^p. \end{aligned}$$

Choosing $\rho >0$ sufficiently small so that $2^{-(c_\alpha +1)}\rho ^{c_\alpha }-C_4\rho ^p=:\beta >0$, we conclude $J(u)\ge \beta$ for all $\Vert u\Vert =\rho$. $\square$

Lemma 4.3

There exists $v_0\in W^{1,\Phi }({\mathbb {R}}^N)$ with $\Vert v_0\Vert >\rho$ such that $J(v_0)<0$.

Proof

By $(f_2)$, there exist constants $C_1,C_2>0$ such that

$$\begin{aligned} F(t) \ge C_1 t^\sigma - C_2, \ \ \text{ for } \text{ all }\ t\ge 0. \end{aligned}$$

Since $\Omega ^+ \ne \emptyset$, taking $0\le \varphi \in C_0^{\infty }(\Omega ^+)$ such that ${\mathcal {K}}=supp(\varphi )$, we have

$$\begin{aligned} \begin{aligned} J(t\varphi )&\le \xi _1(t)\int _{\Omega }[\Phi (|\nabla \varphi |)+\Phi (\varphi )]\,{\mathrm{{d}}}x -C_1t^\sigma \int _{{\mathcal {K}}}W(x)|\varphi |^\sigma \, {\mathrm{{d}}}x+C_2\int _{{\mathcal {K}}}W(x)\,{\mathrm{{d}}}x\\&=C_0t^{c_\alpha }-C_3t^\sigma +C_4, \quad \text {for all}\quad t > 1. \end{aligned} \end{aligned}$$

Since $\sigma >c_\alpha$, it follows that $J(t\varphi )\rightarrow -\infty$ as $t\rightarrow +\infty$. Thus, taking $v_0 :=t_0 \varphi$, with $t_0$ large enough, the proof is finished. $\square$

5 On Palais–Smale sequences

First, we recall that $(u_n)\subset W^{1,\Phi }({\mathbb {R}}^N)$ is a Palais–Smale ($(PS)_c$ for short) sequence at level $c\in {\mathbb {R}}$ for the functional J if $J(u_n)\rightarrow c$ and $J'(u_n)\rightarrow 0$ in the dual space $[W^{1,\Phi }({\mathbb {R}}^N)]'$. We say that J satisfies the $(PS)_c$ condition if any $(PS)_c$ sequence has a convergent subsequence. In this section, our main objective is to prove the $(PS)_c$ condition for J with c in a convenient interval.

Lemma 5.1

If $(u_n)\subset W^{1,\Phi }({\mathbb {R}}^N)$ is a $(PS)_c$ sequence associated with J, then $(u_n)$ is bounded in $W^{1,\Phi }({\mathbb {R}}^N)$.

Proof

Take $\zeta \in C^\infty ({\mathbb {R}}^N)$ given in (1.7). By condition (1.5), $(f_2)$ and Lemma 2.4, we have

$$\begin{aligned} \begin{aligned}&J(u_n)-\frac{1}{\sigma }\langle J'(u_n),u_n \zeta \rangle \\&\quad =\int _{{\mathbb {R}}^N} \left[ \Phi (|\nabla u_n|)+\Phi (|u_n|)\right] \,{\mathrm{{d}}}x-\int _{{\mathbb {R}}^N}W(x)F(u_n)\,{\mathrm{{d}}}x\\&\qquad -\frac{1}{\sigma }\int _{{\mathbb {R}}^N}\Phi '(|\nabla u_n|)|\nabla u_n|\zeta \,{\mathrm{{d}}}x -\frac{1}{\sigma }\int _{{\mathbb {R}}^N}\Phi '(|\nabla u_n|)\frac{\nabla u_n\nabla \zeta }{|\nabla u_n|}u_n\, {\mathrm{{d}}}x\\&\qquad -\frac{1}{\sigma }\int _{{\mathbb {R}}^N} \Phi '(|u_n|)|u_n| \zeta \, {\mathrm{{d}}}x+\frac{1}{\sigma }\int _{{\mathbb {R}}^N}W(x)f(u_n)u_n \zeta \, {\mathrm{{d}}}x\\&\quad \ge \left( 1-\frac{c_\alpha }{\sigma }\right) \int _{{\mathbb {R}}^N} \left[ \Phi (|\nabla u_n|)+\Phi (|u_n|)\right] \,{\mathrm{{d}}}x-\int _{\Omega ^+}W(x)F(u_n)\,{\mathrm{{d}}}x\\&\qquad -\frac{1}{\sigma }K\int _{{\mathbb {R}}^N}\Phi '(|\nabla u_n|) |u_n| \,{\mathrm{{d}}}x +\frac{1}{\sigma }\int _{\Omega ^+}W(x)f(u_n)u_n \, {\mathrm{{d}}}x\\&\quad \ge \left( 1-\frac{c_\alpha }{\sigma }\right) \int _{{\mathbb {R}}^N} \left[ \Phi (|\nabla u_n|)+\Phi (|u_n|)\right] \,{\mathrm{{d}}}x\\&\qquad - \frac{1}{\sigma }K \left( \int _{{\mathbb {R}}^N}\Phi '(|\nabla u_n|)|\nabla u_n|\,{\mathrm{{d}}}x + \int _{{\mathbb {R}}^N} \Phi '(|u_n|)|u_n|\, {\mathrm{{d}}}x \right) . \end{aligned} \end{aligned}$$

(5.1)

Thus,

$$\begin{aligned} \begin{aligned} J(u_n)-\frac{1}{\sigma }\langle J'(u_n),u_n \zeta \rangle \ge&\left( 1-\frac{c_\alpha }{ \sigma } -\frac{Kc_\alpha }{\sigma }\right) \int _{{\mathbb {R}}^N}[\Phi (|\nabla u_n|)+\Phi (|u_n|)]\,{\mathrm{{d}}}x. \end{aligned} \end{aligned}$$

(5.2)

On the other hand,

$$\begin{aligned} J(u_n)-\frac{1}{\sigma }\langle J'(u_n), u_n\zeta \rangle \le c+o_n(1)+o_n(1)\Vert u_n\zeta \Vert \le c+o_n(1)+o_n(1)\Vert u_n\Vert . \end{aligned}$$

(5.3)

Combining (5.2) and (5.3) and using Lemma 2.5, we obtain

$$\begin{aligned} c+o_n(1)+o_n(1)\Vert u_n \Vert \ge \left( 1-\frac{c_\alpha (1+K)}{ \sigma }\right) [\xi _0(|\nabla u_n|_\Phi )+\xi _0(| u_n|_{\Phi })]. \end{aligned}$$

(5.4)

Now, we argue by contradiction. Suppose that, up to a subsequence, $\Vert u_n\Vert \rightarrow \infty$. We have three possibilities to consider:

• (i) $|\nabla u_n|_\Phi \rightarrow \infty$ and $(|u_n|_{\Phi })$ is bounded;

• (ii) $(|\nabla u_n|_{\Phi })$ is bounded and $|u_n|_{\Phi }\rightarrow \infty$;

• (iii) $|\nabla u_n|_\Phi \rightarrow \infty$ and $|u_n|_{\Phi }\rightarrow \infty$.

If item (i) occurs, then there exists $n_1\in {\mathbb {N}}$ such that $|\nabla u_n|_\Phi > 1$ for all $n>n_1$. Thus, by the definition of $\xi _0$ and inequality (5.4) we get

$$\begin{aligned} c+o_n(1)+o_n(1)|\nabla u_n|_\Phi \ge \left( 1-\frac{c_\alpha (1+K)}{ \sigma }\right) |\nabla u_n|_\Phi ^{m_\alpha },\ \ \text{ for } \text{ all }\ n>n_1. \end{aligned}$$

(5.5)

Dividing this estimate by $|\nabla u_n|_\Phi ^{m_\alpha }$, we get a contradiction doing $n\rightarrow \infty$. Thus, (i) does not happen. Similarly, we can show that items (ii) and (iii) do not happen as well. Therefore, $(u_n)$ should be bounded in $W^{1,\Phi }({\mathbb {R}}^N)$ and the proof is finalized. $\square$

Corollary 5.2

If $(u_n)\subset W^{1,\Phi }({\mathbb {R}}^N)$ is a $(PS)_c$ sequence for J , then

$$\begin{aligned} \xi _0(|\nabla u_n|_\Phi )\le \left( 1-\frac{c_\alpha (1+K)}{ \sigma }\right) ^{-1} c +o_n(1). \end{aligned}$$

Proof

This estimate is a direct consequence of (5.4). $\square$

Before to show that the functional J satisfies the Palais–Smale condition in a convenient interval, we shall need of the following convergence result:

Lemma 5.3

Let $(u_n)$ be a Palais–Smale sequence for the functional J at any level $c\in {\mathbb {R}}$ such that

$$\begin{aligned} c<\left( 1-\frac{c_\alpha (1+K)}{ \sigma }\right) \xi _0\left( \frac{K_{N,\alpha }^{1/\gamma }}{(r'b)^{1/\gamma }}\right) . \end{aligned}$$

Then, there exists $u\in W^{1,\Phi }({\mathbb {R}}^N)$ verifying

$$\begin{aligned} \int _{{\mathbb {R}}^N}W(x)f(u_n)(u_n-u)\,{\mathrm{{d}}}x\rightarrow 0. \end{aligned}$$

Proof

By Corollary 5.2, we have

$$\begin{aligned} \xi _0(|\nabla u_n|_\Phi )\le \left( 1-\frac{c_\alpha (1+K)}{ \sigma }\right) ^{-1} c +o_n(1) < \xi _0\left( \frac{K_{N,\alpha }^{1/\gamma }}{(r'b)^{1/\gamma }}\right) +o_n(1). \end{aligned}$$

Since $\xi _0(t)$ is increasing for $t\ge 0$, we can obtain $n_1\in {\mathbb {N}}$ such that

$$\begin{aligned} |\nabla u_n|_\Phi \le \frac{K_{N,\alpha }^{1/\gamma }}{(r'b)^{1/\gamma }}-\delta ,\quad \text{ for } \text{ all }\ n>n_1, \end{aligned}$$

for some $\delta >0$ sufficiently small. Choosing still $s>1$ close to 1, we obtain

$$\begin{aligned} r'sb|\nabla u_n|_\Phi ^\gamma \le K_{N,\alpha }-\delta _1,\quad \text{ for } \text{ all }\ n>n_1, \end{aligned}$$

(5.6)

for some appropriate $\delta _1>0$. Now, by Lemma 5.1, there exists $u\in W^{1,\Phi }({\mathbb {R}}^N)$ such that, up to a subsequence, $u_n\rightharpoonup u$ in $W^{1,\Phi }({\mathbb {R}}^N)$. Next, setting $Q_n:=[\exp (b|u_n|^{\gamma })-S_{N,\alpha }(b|u_n|^\gamma )]$, by assumption $(f_1)$, Hölder inequality and Lemma 3.4, it follows that

$$\begin{aligned} \begin{aligned}&\Big |\int _{{\mathbb {R}}^N}W(x)f(u_n)(u_n-u)\,{\mathrm{{d}}}x\Big |\\&\quad \le C\int _{{\mathbb {R}}^N}|W(x)||u_n|^{N-1}|u_n-u|\,{\mathrm{{d}}}x+C\int _{{\mathbb {R}}^N} |W(x)||u_n-u||Q_n|\,{\mathrm{{d}}}x\\&\quad \le C \left( \int _{{\mathbb {R}}^N}|W(x)||u_n|^{N}{\mathrm{{d}}}x\right) ^{\frac{N-1}{N}}\left( \int _{{\mathbb {R}}^N}|W(x)||u_n-u|^N\,{\mathrm{{d}}}x\right) ^{\frac{1}{N}}\\&\qquad + C\left( \int _{{\mathbb {R}}^N}|W(x)||u_n-u|^{\frac{s}{s-1}}{\mathrm{{d}}}x\right) ^{\frac{s-1}{s}} \left( \int _{{\mathbb {R}}^N}|W(x)||Q_n|^{s}\right) ^{\frac{1}{s}} \\&\quad \le C |W|_{r}^{\frac{N-1}{N}}|u_n|_{r'N}^{N-1}\left( \int _{{\mathbb {R}}^N}|W(x)||u_n-u|^N\,{\mathrm{{d}}}x\right) ^{\frac{1}{N}}+\\&\qquad +C\left( \int _{{\mathbb {R}}^N}|W(x)||u_n-u|^{\frac{s}{s-1}}{\mathrm{{d}}}x\right) ^{\frac{s-1}{s}}|W|_{r}^{\frac{1}{s}}|Q_n|_{r's}. \end{aligned} \end{aligned}$$

By Proposition 3.2, we have

$$\begin{aligned} \int _{{\mathbb {R}}^N}|W(x)||u_n-u|^N\,{\mathrm{{d}}}x\rightarrow 0\quad \mathrm {and}\quad \int _{{\mathbb {R}}^N}|W(x)||u_n-u|^{\frac{s}{s-1}}{\mathrm{{d}}}x\rightarrow 0. \end{aligned}$$

Hence, to finalize the proof, just to justify that

$$\begin{aligned} \sup _{n\in {\mathbb {N}}}\int _{{\mathbb {R}}^N} [\exp (r'sb|u_n|^{\gamma })-S_{N,\alpha }(r'sb|u_n|^\gamma )]{\mathrm{{d}}}x<\infty . \end{aligned}$$

Indeed, we can write this integral as

$$\begin{aligned} \int _{{\mathbb {R}}^N}\left[ \exp \left( r'sb|\nabla u_n|_\Phi ^\gamma \left( \frac{|u_n|}{|\nabla u_n|_\Phi }\right) ^{\gamma }\right) -S_{N,\alpha }\left( r'sb|\nabla u_n|_\Phi ^\gamma \left( \frac{|u_n|}{|\nabla u_n|_\Phi }\right) ^{\gamma }\right) \right] {\mathrm{{d}}}x, \end{aligned}$$

and by (5.6), we have $r'sb|\nabla u_n|_\Phi ^\gamma< K_{N,\alpha }-\delta _1< K_{N,\alpha }$, for $n>n_1$. Therefore, invoking Lemma 3.3 we conclude that the above supreme is finite and the proof is complete. $\square$

Lemma 5.4

The functional J satisfies the $(PS)_c$ condition for all

$$\begin{aligned} c<\left( 1-\frac{c_\alpha (1+K)}{ \sigma }\right) \xi _0\left( \frac{K_{N,\alpha }^{1/\gamma }}{(r'b)^{1/\gamma }}\right) . \end{aligned}$$

(5.7)

Proof

Let $(u_n)$ be in $W^{1,\Phi }({\mathbb {R}}^N)$ such that $J(u_n)\rightarrow c$ and $J'(u_n)\rightarrow 0$ in $[W^{1,\Phi }({\mathbb {R}}^N)]'$ with c satisfying (5.7). By Lemma 5.1, $(u_n)$ is bounded in $W^{1,\Phi }({\mathbb {R}}^N)$, and therefore, up to a subsequence, $u_n\rightharpoonup u$ in $W^{1,\Phi }({\mathbb {R}}^N)$. Since the functional $I(u):= \int _{{\mathbb {R}}^N}[\Phi (|\nabla u|)+\Phi (|u|)]\,{\mathrm{{d}}}x$ is convex, we get

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^N}[\Phi (|\nabla u|)+\Phi (|u|)]\,{\mathrm{{d}}}x -\int _{{\mathbb {R}}^N}[\Phi (|\nabla u_n|)+\Phi (|u_n|)]\,{\mathrm{{d}}}x \\&\quad \ge \int _{{\mathbb {R}}^N}\Phi '(|\nabla u_n|)\frac{\nabla u_n}{|\nabla u_n|}(\nabla u-\nabla u_n)\,{\mathrm{{d}}}x \\&\qquad + \int _{{\mathbb {R}}^N}\Phi '(|u_n|)\frac{u_n}{|u_n|}(u-u_n)\,{\mathrm{{d}}}x \\&\quad = \langle J'(u_n),u-u_n\rangle + \int _{{\mathbb {R}}^N}W(x)f(u_n)(u-u_n)\,{\mathrm{{d}}}x . \end{aligned} \end{aligned}$$

(5.8)

According to Lemma 5.3, we know that $\int _{{\mathbb {R}}^N}W(x)f(u_n)(u-u_n)\,{\mathrm{{d}}}x\rightarrow 0$. Thus, by (5.8) one has

$$\begin{aligned} \int _{{\mathbb {R}}^N}[\Phi (|\nabla u|)+\Phi (|u|)]\,{\mathrm{{d}}}x \ge \int _{{\mathbb {R}}^N}[\Phi (|\nabla u_n|)+\Phi (|u_n|)]\,{\mathrm{{d}}}x+o_n(1), \end{aligned}$$

and consequently,

$$\begin{aligned} \displaystyle \limsup _{n\rightarrow \infty }\int _{{\mathbb {R}}^N}[\Phi (|\nabla u_n|) +\Phi (|u_n|)]\,{\mathrm{{d}}}x\le \int _{{\mathbb {R}}^N}[\Phi (|\nabla u|)+\Phi (|u|)]\,{\mathrm{{d}}}x. \end{aligned}$$

(5.9)

Since $I_1(u):=\int _{{\mathbb {R}}^N}\Phi (|\nabla u|)\,{\mathrm{{d}}}x$ is a sequentially weakly lower semicontinuous functional on $W^{1,\Phi }({\mathbb {R}}^N)$, the weak convergence $u_n\rightharpoonup u$ in $W^{1,\Phi }({\mathbb {R}}^N)$ implies that

$$\begin{aligned} \int _{{\mathbb {R}}^N}\Phi (|\nabla u|)\,{\mathrm{{d}}}x \le \displaystyle \liminf _{n\rightarrow \infty }\int _{{\mathbb {R}}^N}\Phi (|\nabla u_n|)\,{\mathrm{{d}}}x. \end{aligned}$$

(5.10)

The same reason shows that

$$\begin{aligned} \int _{{\mathbb {R}}^N}\Phi (| u|)\,{\mathrm{{d}}}x \le \displaystyle \liminf _{n\rightarrow \infty }\int _{{\mathbb {R}}^N}\Phi (|u_n|)\,{\mathrm{{d}}}x. \end{aligned}$$

(5.11)

By virtue of (5.9), we must have the equality in (5.10) and (5.11). Hence, up to subsequences,

$$\begin{aligned} \int _{{\mathbb {R}}^N}\Phi (|\nabla u_n|)\,{\mathrm{{d}}}x \rightarrow \int _{{\mathbb {R}}^N}\Phi (|\nabla u|)\,{\mathrm{{d}}}x\quad \text {and}\quad \int _{{\mathbb {R}}^N}\Phi (| u_n|)\,{\mathrm{{d}}}x\rightarrow \int _{{\mathbb {R}}^N}\Phi (| u|)\,{\mathrm{{d}}}x. \end{aligned}$$

Now, arguing as in [18, Lemma 6.2] we can see that, up to a subsequence, $\nabla u_n \rightarrow \nabla u$ almost everywhere in ${\mathbb {R}}^N$. Using a version of the Brezis–Lieb lemma (see [14, Theorem 2]), we conclude

$$\begin{aligned} \int _{{\mathbb {R}}^N}\Phi (|\nabla u_ n- \nabla u|)\,{\mathrm{{d}}}x\rightarrow 0\quad \text {and}\quad \int _{{\mathbb {R}}^N}\Phi (|u_n-u|)\,{\mathrm{{d}}}x \rightarrow 0, \end{aligned}$$

and according to Remark 2.8, it follows that $\Vert u_n -u\Vert =| \nabla u_n -\nabla u|_\Phi +| u_n -u|_{\Phi }\rightarrow 0$ and the proof is finalized. $\square$

6 Proof of Theorem 1.1

In order to apply Theorem 4.1 to find a nonzero critical point for J, we need to estimate the minimax level c of J, where

$$\begin{aligned} c^*:=\displaystyle \inf _{\gamma \in \Gamma }\max _{0\le t\le 1}J\left( \gamma (t)\right) \quad \mathrm {and}\quad \Gamma :=\left\{ \gamma \in C([0,1];W^{1,\Phi }({\mathbb {R}}^N));\gamma (0)=0 \ \text { and }\ J(\gamma (1))<0\right\} . \end{aligned}$$

Before to state our next result, we need to fix some notations. We recall that $B_\delta :=B_{\delta }(0) \subset \subset \Omega ^+$ for some $\delta >0$. We are going to consider a function $\varphi _0\in C_0^{\infty }(\Omega ^+)$ given by $\varphi _0(x)=1$ if $|x|\le \delta /2$, $\varphi _0(x)=0$ if $|x|\ge \delta$, $0\le \varphi _0(x)\le 1$ for all $x\in \Omega ^+$ and $|\nabla \varphi _0(x)|\le 1$ for all $x\in \Omega ^+$. Recalling that

$$\begin{aligned} \mu _1:=\frac{2\Phi (1)|B_\delta |}{a_0|B_{\delta /2}|}, \end{aligned}$$

where $a_0:= \min \{ W(x) :x \in {\overline{B}}_\delta \}$, by $(f_3)$ we infer that if $\mu \ge \mu _1$ then

$$\begin{aligned} \begin{aligned} J(\varphi _0)&\le \int _{B_\delta }[\Phi (|\nabla \varphi _0|)+\Phi (|\varphi _0|)]\,{\mathrm{{d}}}x-\mu _1 \int _{B_\delta }W(x)|\varphi _0|^\theta \,{\mathrm{{d}}}x\\&<2 \Phi (1)|B_\delta |-\mu _1 a_0 |B_{\delta /2}|=0. \end{aligned} \end{aligned}$$

In particular,

$$\begin{aligned} \int _{B_{\delta }}[\Phi (|\nabla \varphi _0|)+\Phi (|\varphi _0|)]\,{\mathrm{{d}}}x<\mu _1 \int _{B_{\delta }} W(x)|\varphi _0|^\theta \,{\mathrm{{d}}}x. \end{aligned}$$

(6.1)

Lemma 6.1

(Minimax Estimate). If condition $(f_3)$ holds with $\mu \ge \mu ^*$, where the number $\mu ^*$ was defined in (1.9), then

$$\begin{aligned} c*<\left( 1-\frac{c_\alpha (1+K)}{ \sigma }\right) \xi _0\left( \frac{K_{N,\alpha }^{1/\gamma }}{(r'b)^{1/\gamma }}\right) . \end{aligned}$$

Proof

By definition of $c^*$, $(f_3)$ and (6.1), one has

$$\begin{aligned} \begin{aligned} c^*&\le \max _{t\in [0,1]}J(t\varphi _0)\\&\le \max _{t\in [0,1]}\left[ t^{m_\alpha }\int _{B_{\delta }}[\Phi (|\nabla \varphi _0|)+\Phi (|\varphi _0|)]\,{\mathrm{{d}}}x-\mu t^\theta \int _{B_\delta }W(x)|\varphi _0|^\theta \,{\mathrm{{d}}}x\right] \\&\le \max _{t\in [0,1]}\left[ t^{m_\alpha }\mu _1 \int _{B_{\delta }} W(x)|\varphi _0|^\theta \,{\mathrm{{d}}}x-\mu t^\theta \int _{B_{\delta }}W(x)|\varphi _0|^\theta \, {\mathrm{{d}}}x\right] \\&\le \max _{t\ge 0}\left[ \mu _1 t^{m_\alpha }-\mu t^{\theta }\right] a_1 \int _{B_{\delta }}|\varphi _0|^\theta \, {\mathrm{{d}}}x. \end{aligned} \end{aligned}$$

A straightforward calculation shows that

$$\begin{aligned} \max _{t\ge 0}\left[ \mu _1 t^{m_\alpha }-\mu t^{\theta }\right] =\frac{1}{\mu ^\frac{m_\alpha }{\theta -m_\alpha }} \frac{\theta -m_\alpha }{m_\alpha }\left( \frac{\mu _1 m_\alpha }{\theta }\right) ^{\frac{\theta }{\theta -m_\alpha }}, \end{aligned}$$

and therefore,

$$\begin{aligned} c^*<\frac{a_1|B_{\delta }|}{\mu ^\frac{m_\alpha }{\theta -m_\alpha }} \frac{\theta -m_\alpha }{m_\alpha }\left( \frac{\mu _1 m_\alpha }{\theta }\right) ^{\frac{\theta }{\theta -m_\alpha }}. \end{aligned}$$

Thus, by using that $\mu \ge \mu ^*$, we reach the estimate

$$\begin{aligned} c^*<\left( 1-\frac{c_\alpha (1+K)}{ \sigma }\right) \xi _0\left( \frac{K_{N,\alpha }^{1/\gamma }}{(r'b)^{1/\gamma }}\right) . \end{aligned}$$

$\square$

Finalizing the proof of Theorem 1.1: According to Lemmas 5.4 and 6.1, J satisfies $(PS)_{c}$ condition. Moreover, since J has the mountain pass geometry, it follows by invoking mountain pass theorem that there exists a nonzero critical $u\in W^{1,\Phi }({\mathbb {R}}^N)$ for J such that $J(u)=c$ and the proof is finalized.