1 Introduction

In this article, we investigate a class of nonlinear problems in the Orlicz–Sobolev setting:

$$ \textstyle\begin{cases} -\operatorname{div}( a ( \vert \nabla u \vert ) \vert \nabla u \vert ) =\lambda g ( u ) +f ( x,u ), &\mbox{in }\varOmega,\\ u=0, &\mbox{on }\partial\varOmega, \end{cases} $$
(1.1)

where Ω is a bounded domain with smooth boundary ∂Ω, λ is a positive constant. \(a(t)\) is such that

$$\varphi ( t ):= \textstyle\begin{cases} a( \vert t \vert )t,&t\neq0,\\ 0,&t=0 \end{cases} $$

is an odd, increasing homeomorphism from \(\mathbb{R}\) to \(\mathbb{R}\). g is an odd, increasing homeomorphism from \(\mathbb{R}\) to \(\mathbb{R}\) with \(( \varphi_{0} -1 ) \) sublinear (see condition \(( g_{1} )\)), \(f \in \mathrm{C}( \overline{\varOmega}\times \mathbb{R},\mathbb{R})\), \(f ( x,0 ) =0 \) with \(( \varphi^{0} -1 ) \) superlinear near infinity (see condition \(( f_{3} )\)).

When \(a ( \vert t \vert ) t= | t|^{2} t\) with \(1< p <\infty\), problem (1.1) reads as follows:

$$ \textstyle\begin{cases} - \Delta_{p} u=f ( x,u ), &\text{in } \varOmega,\\ u=0, &\mbox{on }\partial\varOmega. \end{cases} $$
(1.2)

The key hypothesis imposed on f is the well-known Ambrosetti–Rabinowitz type condition (AR-condition for short) [1]: there exist \(\tau>p\), \(t_{0} >0\) such that

$$ 0< \tau F ({x,t} ) = \int_{0}^{{t}}f(x,s)\,ds \leq tf (x,t ), \quad \forall x\in \varOmega, \vert t \vert \geq t_{0}. $$
(1.3)

It is noted that the AR-condition ensures that f is \((p-1)\) superlinear at infinity.

However, the AR-condition is restrictive for many nonlinearities. Consequently, there have been many efforts to remove (1.3). In the case of \(p=2\), Miyagaki and Souto [2] introduced the following monotone condition: there is \(s_{0} >0\) such that

$$\begin{aligned}& \frac{{f(x,s)}}{s} \mbox{ is increasing in }s \geq s_{0}\mbox{ and decreasing in }s \leq- s_{0},\quad \forall x \in \varOmega. \end{aligned}$$
(1.4)

Li and Yang [3] developed (1.4) to the case of \(p>1\). Meanwhile, Li and Yang [3] proved that (1.4) implied the following weaker condition: there is \(C_{*} >0\) such that, for all \(s\in[0,1]\),

$$ \overline{F} ({x,st} )\leq \overline{F} ({x,t} )+ C_{*},\quad \forall ( x,t ) \in \varOmega\times \mathbb{R}, \overline{F} ({x,t} ) =tf ({x,t} ) - pF ( x,t ), $$
(1.5)

which is due to Jeanjean [4] and is used in [5, 6] and so on.

Ambrosetti, Brezis, and Cerami [7] initiated the study of semilinear elliptic problems with concave and convex nonlinearities. They investigated (1.1) with nonlinearities of the type \(\lambda_{u}^{p}+ u^{q}\), \(0< q<1 <p\) and obtained the existence of two positive solutions for small \(\lambda>0\) by using sub- and super-solutions. Wu [8] studied problem (1.1) in the case when nonlinear terms exhibit \(u^{p}+ \lambda f ( x )u^{q} \) with \(0< q<1 <p< 2^{*} \) and obtained two positive solutions by Nehari manifold. Later, Wu [9] considered semilinear problems (1.1) in \(H^{1} ( \mathbb{R}^{N} ) \) and established existence results. Papageorgiou and Rocha [10] considered a p-Laplacian problem with nonlinearities of the form \(m ({x} ) \vert u \vert ^{r-2}u+ f ( x,u ) \) with \(1< r < p<\infty\) when f is \(( p-1 )\) superlinear near infinity but does not satisfy the AR-condition. They employed variational approach and the Ekeland variational principle [11] to show the existence of two nontrivial solutions.

Divergence operators \(-\operatorname{div}( a ( \vert \nabla u \vert ) \vert \nabla u \vert ) \) involved in problem (1.1) are more general than p-Laplacian operators, please see [12,13,14,15,16,17,18,19,20,21,22]. Such operators have been intensively studied due to numerous and relevant applications in many fields such as plasticity [23], eletrorheological fluids [24], image processing [25]. When the nonlinear terms satisfy the AR-condition, problems of type (1.1) have been considered in [23, 26].

In the case of \(\lambda=0\), Chung [27], Carvalho et al. [28] studied problem (1.1) when f is \(( \varphi^{0} -1)\) superlinear near infinity without the AR-condition. By variational methods, Chung [27], Carvalho et al. [28] established existence results under different assumptions imposed on f.

In this paper, motivated by [12,13,14, 16,17,18], we investigate a class of quasilinear elliptic problems (1.1) with concave and convex nonlinearities which do not satisfy the AR-condition in Orlicz–Sobolev spaces. Using functional techniques and variational approach, combined with the Ekeland variational principle, we establish existence results of at least two nontrivial solutions for \(\lambda>0 \) small enough. We emphasize that the extension from p-Laplacian operators to \(-\operatorname{div}( a ( \vert \nabla u \vert ) \nabla u )\) is interesting and nontrivial, since the divergence operators \(-\operatorname{div}( a ( \vert \nabla u \vert ) \nabla u )\) involved in (1.1) have a more complicated structure, for example, they are non-homogeneous. In the case of \(\lambda=0\), problem (1.1) is studied in [27, 28], but their hypotheses do not apply when the concave terms are present. Furthermore, multiplicity results are given in this paper, while [27, 28] are concerned with existence of a nontrivial weak solution under our assumptions. Summarily, our results complement and extend previous studies such as [10, 27, 28].

2 Preliminaries

\(\varPhi: \mathbb{R}\rightarrow [0,\infty)\) is called an \(\mathcal{N}\)-function [29,30,31] provided that Φ is even, continuous, and convex with \(\varPhi ({t} ) >0\) for \(t>0\), \(\frac{ \varPhi ({t} )}{t} \rightarrow0\) as \(t\rightarrow0\), and \(\frac{ \varPhi ({t} )}{t} \rightarrow\infty\) as \(t\rightarrow\infty\). Its complementary function Φ̃ is defined as

$$\tilde{\varPhi} ({s} ):= \sup_{t>0} \bigl\{ t \vert s \vert - \varPhi ({t} ) \bigr\} ,\quad \forall s\in \mathbb{R}, $$

then Φ̃ is also an \(\mathcal{N}\)-function.

Young’s inequality holds true:

$$st \leq\varPhi ({t} )+ \tilde{\varPhi} ({s} ), \quad {s,t} \in \mathbb{R}. $$

If \(\varPhi_{1}\), \(\varPhi_{2}\) are two \(\mathcal{N}\)-functions, we say that \(\varPhi_{1} \) increases more slowly than \(\varPhi_{2}\) near infinity (in short, \(\varPhi_{1} \prec \varPhi_{2} (\infty)\)) if there exist two positive constants K, \(t_{0}\) such that \(\varPhi_{1} ( t ) \leq \varPhi_{2} ( K t )\), \(\forall t \geq t_{0}\). We say that \(\varPhi_{1}\) increases essentially more slowly than \(\varPhi_{2}\) near infinity (in short, \(\varPhi_{1} \prec \prec \varPhi_{2} (\infty)\)) provided \(\lim_{t\rightarrow\infty} \frac{ \varPhi_{1} (kt)}{ \varPhi_{2} (t)} =0\), \(\forall k>0\).

Φ is said to satisfy \(\Delta_{ 2}\)-condition near infinity (in short, \(\varPhi\in\Delta _{2} (\infty)\)) provided that there exist positive constants K, \(t_{0}\) such that

$$\varPhi ({2t} ) \leq K \varPhi ({t} ) \quad \forall t \geq t_{0}. $$

\(\varPhi\in\nabla_{2} ( \infty ) \) provided that \(\tilde{\varPhi}\in \Delta _{2} (\infty)\).

For a measurable function \(u : \varOmega\rightarrow \mathbb{R}\), denoted as \(u\in \tilde{L}\), we define Orlicz space \(L_{\varPhi} ( \varOmega ) \) by

$$L_{\varPhi} ( \varOmega )=\biggl\{ u\in \tilde{L}: \int_{\varOmega} \varPhi \bigl(\lambda u ({x} ) \bigr)\,dx < \infty \mbox{ for some } \lambda >0\biggr\} $$

endowed with Luxemburg norm

$$\Vert u \Vert _{ ( \varPhi )} = \inf \biggl\{ \lambda >0: \int_{\varOmega} \varPhi \biggl( \frac{u ({x} )}{\lambda} \biggr)\,dx \leq1 \biggr\} . $$

Then \((L_{\varPhi} ( \varOmega ), \Vert \cdot \Vert _{ ( \varPhi )})\) forms a Banach space.

In the sequel, we always assume that [30]

$$\int_{0}^{1} \frac{\varPhi^{-1} (t)}{t^{\frac{N+1}{N}}}\, dt< \infty, \qquad \int_{1}^{\infty} \frac{\varPhi^{-1} (t)}{t^{\frac{N+1}{N}}} \,dt=\infty. $$

The Sobolev conjugate \(\varPhi_{*} \) of Φ is defined by

$$\varPhi_{*}^{-1} ({t} ) = \int_{0}^{t} \frac{\varPhi^{-1} (s)}{ s^{\frac{N+1}{N}}}\, ds, \quad t \geq 0. $$

Let \(\varPhi_{*} ( -t ) = \varPhi_{*} ( t )\) for all \(t < 0\). Then \(\varPhi_{*}\) is an \(\mathcal{N}\)-function and \(\varPhi \prec\prec \varPhi_{*} (\infty)\) (see [30, 32]).

An Orlicz–Sobolev space \(W^{1, \varPhi} ( \varOmega )\) is defined by

$$W^{1, \varPhi} ( \varOmega ) =\bigl\{ u\in L_{\varPhi} ( \varOmega ): D^{\alpha} u\in L_{ \varPhi} ( \varOmega ), \vert \alpha \vert \leq1\bigr\} $$

endowed with

$$\Vert u \Vert _{W^{1, \varPhi}} = \Vert u \Vert _{( \varPhi )} + \Vert \nabla u \Vert _{( \varPhi )}. $$

Then \((W^{1, \varPhi} ( \varOmega ), \Vert \cdot \Vert _{W^{1, \varPhi}})\) forms a Banach space.

Let \(W_{0}^{1, \varPhi} ( \varOmega )\) be the closure of \(C_{c}^{\infty} (\varOmega)\) in \(W^{1, \varPhi} ( \varOmega )\). By Lemma 5.7 in [33], there exists a best positive constant \(\lambda_{ 1} \) such that

$$ \lambda _{1} \int_{\varOmega} \varPhi \bigl({u} ({x} ) \bigr)\,dx \leq \int_{\varOmega} \varPhi \bigl( \bigl\vert \nabla_{u} ({x} ) \bigr\vert \bigr)\,dx,\quad \forall u \in W_{0}^{1, \varPhi} (\varOmega). $$
(2.1)

Therefore, \(W_{0}^{1, \varPhi} ( \varOmega )\) can be reformed by an equivalent norm \(\Vert u \Vert := \Vert \nabla u \Vert _{( \varPhi )}\). If \(\varPhi\in\Delta_{ 2} ( \infty ) \cap \nabla_{2} ( \infty )\), then \(L_{ \varPhi} ( \varOmega )\)), \(W^{1, \varPhi} ( \varOmega )\), \(W_{0}^{1, \varPhi} ( \varOmega )\) are separable and reflexive Banach spaces (refer [30]).

In this paper, we always assume \(\varPhi ({t} )= \int_{0}^{t} \varphi ( s ) \,ds\), \(\forall t\in \mathbb{R}\), and

figure a

We note that (\(\varPhi _{1}\)) yields \(\varPhi\in\Delta_{ 2} ( \infty ) \cap \nabla_{2} ( \infty )\) (see [29]).

Lemma 2.1

([23])

For an \(\mathcal{N}\)-function Φ satisfying \(1\leq \varphi_{0} \leq \varphi^{0} <\infty \) for all \(t>0\) and for some \(\varphi_{0}\), \(\varphi^{0}\). Then

  1. (1)

    \(\Vert u \Vert _{( \varPhi )}^{\varphi_{0}} \leq \int_{\varOmega} \varPhi ({u} )\,dx\leq \Vert u \Vert _{( \varPhi )}^{\varphi^{0}}\) (\(\Vert u \Vert _{ ( \varPhi )} >1\)).

  2. (2)

    \(\Vert u \Vert _{( \varPhi )}^{\varphi^{0}} \leq \int_{\varOmega} \varPhi ({u} )\,dx\leq \Vert u \Vert _{( \varPhi )}^{\varphi_{0}}\) (\(0\leq \Vert u \Vert _{ ( \varPhi )} \leq1\)).

Lemma 2.2

([30])

Let Ω be an arbitrary domain. Then \(W_{0}^{1, \varPhi} (\varOmega) \hookrightarrow L_{ \varPhi_{*}} ( \varOmega ) \). Moreover, if \(\varOmega_{0}\) is a bounded subdomain of Ω, then the imbedding \(W_{0}^{1, \varPhi} (\varOmega) \hookrightarrow L_{ B} ( \varOmega_{0} ) \) exists and is compact for any \(\mathcal{N}\)-function B with \(B \prec\prec \varPhi_{*} (\infty)\).

Definition 2.1

([34])

Let \(( X, \Vert \cdot \Vert )\) be a real Banach space, \(J \in C^{1} (X, \mathbb{R} )\). We say J satisfies the \(C_{c}\) condition if any sequence \(\{ u_{n} \}\subset X\) such that \(J( u_{n} )\rightarrow c\) and \(\Vert J ' ( u_{n} ) \Vert _{*} (1+ \Vert u_{n} \Vert )\rightarrow0\) as \(n \rightarrow\infty\) has a convergent subsequence. \(\{ u_{n} \}\) is called a Cerami sequence at the level \(c\in \mathbb{R}\).

Lemma 2.3

([35])

Let \(( X, \Vert \cdot \Vert )\) be a real Banach space, \(J \in C^{1} (X, \mathbb{R} )\) satisfies the \(C_{c}\) condition for any \(c >0\), \(J ( \theta ) =0\), and the following conditions hold:

  1. (1)

    There exist two positive constants ρ, η such that \(J ( u ) \geq\eta\) for any \(u\in X\) with \(\Vert u \Vert =\rho\).

  2. (2)

    There exists a function \(\phi\in X\) such that \(\Vert \phi \Vert >\rho\) and \(J ( \phi ) <0\).

Then the functional J has a critical value \(c\geq\eta\), i.e., there exists \(u\in X\) such that \(J ' ( u ) =\theta\) and \(J ( u ) =c\).

We call \(u\in W_{0}^{1, \varPhi} ( \varOmega ) \) a weak solution of problem (1.1) if, for all \(v\in W_{0}^{1, \varPhi} (\varOmega)\),

$$\int_{\varOmega} a \bigl( \vert \nabla u \vert \bigr) \nabla u \cdot \nabla v\,dx- \lambda \int_{\varOmega} g ( u ) v\,dx- \int_{\varOmega} f ( x,u ) v\,dx=0. $$

Let \(( X, \Vert \cdot \Vert _{X} )\), \(( Y, \Vert \cdot \Vert _{Y} )\) be Banach spaces. \(X \hookrightarrow Y\) means \(( X, \Vert \cdot \Vert _{X} )\) is continuously imbedded in \(( Y, \Vert \cdot \Vert _{Y} )\). \(X \hookrightarrow\hookrightarrow Y \) means \(( X, \Vert \cdot \Vert _{X} )\) is compactly imbedded in \(( Y, \Vert \cdot \Vert _{Y} )\).

3 Main results

For convenience, we give some conditions.

\(( g_{1} )\) :

\(G\prec \varPhi ({\infty} )\), \(\lim_{t\rightarrow0} \frac{ \varPhi ({t} )}{ G ({t} )} =0\), where \(G ( t ) := \int_{0}^{t} g ( s )\,ds\), \(\forall t\in \mathbb{R}\).

\(( f_{1} )\) :

\(\vert f ( x,t ) \vert \leq C ( 1+ h ( \vert t \vert ) )\), \(\forall(x,t)\in\varOmega\times \mathbb{R}\),

where C is a positive constant, \(h: \mathbb{R} \rightarrow \mathbb{R}\) is an odd, increasing homeomorphism from \(\mathbb{R}\) to \(\mathbb{R}\), \(H ( t ) := \int_{0}^{t} h ( s )\,ds\) satisfies \(H \prec\prec \varPhi_{* }(\infty)\) and \(h_{0} := \inf_{t>0} \frac{th ( t )}{H ({t} )} > \varphi^{0}\).

\(( f_{2} )\) :

\(\mathop{\lim \sup}_{t\rightarrow0} \frac{ f ( x,t )}{ |\varphi ( t ) |} <\lambda_{ 1}\) uniformly for almost all \(x\in\varOmega\), where \(\lambda_{ 1}\) is defined in (2.1).

\(( f_{3} )\) :

\(\lim_{| t|\rightarrow\infty} \frac{ f ( x,t )}{ | t|^{\varphi^{0} -2} t} =+\infty \) uniformly for almost all \(x\in\varOmega\).

\(( f_{4} ) \) :

There exist \(D _{1} \geq1\) and \(\alpha(x) \in L^{1} (\varOmega)\) such that, for all \(s \in[0,1]\),

$$\overline{F} ({x,st} )\leq D _{1} \overline{F} ({x,t} )+ \alpha(x), \quad \forall ( x,t ) \in \varOmega\times \mathbb{R}, $$

where \(\overline{F} ({x,t} ):=tf ({x,t} ) - \varphi^{0} F ( x,t )\), \(F ({x,t} ) = \int_{0}^{t}f(x,s)\,ds\).

\(( \varPhi_{2} ) \) :

There exists \(\beta ( x ) \in L^{1} ( \varOmega )\) such that, for all \(s \in[0,1]\),

$$\overline{\varPhi} ({st} )\leq D_{ 1} \overline{\varPhi} ({t} )+ \beta ( x ),\quad \forall ( x,t ) \in \varOmega\times \mathbb{R}, $$

where \(\overline{\varPhi} ({t} ) = \varphi^{0} \varPhi ( t ) -t\varphi ( t )\).

The main result of this paper is given by the following theorem.

Theorem 3.1

Given Φ satisfies \(( \varPhi_{1} ) \) and \(( \varPhi_{2} )\), g satisfies \(( g_{1} )\), f satisfies \(( f_{1} )\)–\(( f_{4} )\). Then there exists \(\lambda _{*} >0\) such that, for each \(\lambda\in (0, \lambda_{ *} )\), problem (1.1) has two nontrivial weak solutions.

Remark 3.1

From \(( g_{1} )\) and \(( f_{4} )\), it follows that \(W_{0}^{1, \varPhi} (\varOmega) \hookrightarrow\hookrightarrow L^{1} (\varOmega)\), \(W_{0}^{1, \varPhi} (\varOmega) \hookrightarrow\hookrightarrow L_{G} (\varOmega)\), and \(W_{0}^{1, \varPhi} (\varOmega) \hookrightarrow\hookrightarrow L_{H} (\varOmega)\).

For any \(\lambda>0\), we define \(\mathcal{J}_{\lambda}: W_{0}^{1, \varPhi} (\varOmega)\rightarrow \mathbb{R}\) by

$$\mathcal{J}_{\lambda} ({u} )= \int_{\varOmega} \varPhi \bigl( \vert \nabla u \vert \bigr)\,dx- \lambda \int_{\varOmega} G ( u )\,dx- \int_{\varOmega} F ( x,u )\,dx. $$

Analogous to that in [32], we can deduce that \(\mathcal{J}_{\lambda} \in C^{1} ( W_{0}^{1, \varPhi} ( \varOmega ), \mathbb{R} )\), \(\mathcal{J}_{\lambda} ': W_{0}^{1, \varPhi} (\varOmega)\rightarrow ( W_{0}^{1, \varPhi} (\varOmega)^{*}\) and the derivative is given by, for all \(u,v \in W_{0}^{1, \varPhi} ( \varOmega )\),

$$\bigl\langle \mathcal{J}_{\lambda} ' (u ),v \bigr\rangle = \int_{\varOmega} a \bigl( \vert \nabla u \vert \bigr) \nabla u \cdot \nabla v\,dx- \lambda \int_{\varOmega} g ( u ) v\,dx- \int_{\varOmega} f ( x,u ) v\,dx. $$

So, critical points of the functional \(\mathcal{J}_{\lambda}\) are weak solutions of problem (1.1).

Lemma 3.2

Given that \(( \varPhi_{1} )\), \(( g_{1} )\), \(( f_{1} )\), and \(( f_{2} )\) hold, then there exist positive constants \(\lambda_{ *}\), ρ, η such that, for each \(\lambda\in (0, \lambda_{ *} )\), \(\mathcal{J}_{\lambda} ({u} )\geq \eta\) for any \(u \in W_{0}^{1, \varPhi} (\varOmega)\) with \(\Vert u \Vert =\rho\).

Proof

By conditions \(( f_{1} )\), \(( g_{1} )\) and Remark 3.1, there exists a positive constant \(C_{1}\) such that

$$ \Vert u \Vert _{( G )} \leq C_{1} \Vert u \Vert , \qquad \Vert u \Vert _{ ( H )} \leq C_{1} \Vert u \Vert , \quad \forall u \in W_{0}^{1, \varPhi} ( \varOmega ). $$
(3.1)

Let \(\rho\in ( 0, \min \{ 1, {1} / {C_{1}} \} )\) for each \(u\in S_{\rho} :=\{u \in W_{0}^{1, \varPhi} ( \varOmega ): \Vert u \Vert =\rho\}\), (3.1) implies that \(\Vert u \Vert _{( G )} <1\), \(\int_{\varOmega} G ( u(x) )\,dx<1\), and \(\Vert u \Vert _{( H )} <1\).

From condition \(( f_{2} )\), we deduce that there exist \(\varepsilon_{0} \in(0, \lambda_{1} ) \), \(\delta >0\) such that

$$ \bigl\vert F ( x,t ) \bigr\vert \leq(\lambda _{1} - \varepsilon_{0} ) \varPhi({t} )\quad \forall x \in \varOmega, \vert t \vert < \delta. $$
(3.2)

By \(( f_{1} )\), one has \(\vert F ( x,t ) \vert \leq C \vert t \vert +CH(t)\) for all \(x \in \varOmega\), \(\vert t \vert \geq\delta\). Since \(\frac{H(t)}{t}\) is increasing on \([ \delta,+\infty)\), we conclude \(\frac{ H(t)}{| t|} \geq \frac{H ( \delta )}{\delta} \) for \(\vert t \vert \geq\delta\). Combined with (3.2), we get

$$ \bigl\vert F ( x,t ) \bigr\vert \leq (\lambda_{ 1} - \varepsilon_{0} ) \varPhi ({t} ) +C_{ 2} H ( t ), \quad \forall x \in \varOmega, t\in \mathbb{R}. $$
(3.3)

By Lemma 2.1 and (3.3), for all \(u\in S_{\rho}\),

$$\begin{aligned} \mathcal{J}_{\lambda} ({u} ) =& \int_{\varOmega} \varPhi \bigl( \vert \nabla u \vert \bigr)\,dx- \lambda \int_{\varOmega} G ( u )\,dx- \int_{\varOmega} F ( x,u )\,dx \\ \geq& \int_{\varOmega} \varPhi \bigl( \vert \nabla u \vert \bigr)\,dx- \lambda \int_{\varOmega} G ( u )\,dx \\ &{}- (\lambda _{1} - \varepsilon_{0} ) \int_{\varOmega} \varPhi ({u} )\,dx -C_{ 2} \int_{\varOmega} H ( u )\,dx \\ \geq& \biggl(1- \frac{ (\lambda_{1} - \varepsilon_{0} )}{ \lambda_{ 1}} \biggr) \int_{\varOmega} \varPhi \bigl( \vert \nabla u \vert \bigr)\,dx- \lambda-C_{ 2} \Vert u \Vert _{( H )}^{h_{0}} \\ \geq& \frac{\varepsilon_{0}}{\lambda_{ 1}} \Vert u \Vert ^{\varphi^{0}}- \lambda-C_{ 3} \Vert u \Vert ^{h_{0}}. \end{aligned}$$
(3.4)

Denote \(m ( \rho ) = \frac{\varepsilon_{0}}{\lambda_{1}} - C_{3} \rho^{h_{0} - \varphi^{0}}\), by \(h_{0} > \varphi^{0}\), we have \(m ( \rho ) \rightarrow \frac{\varepsilon_{0}}{\lambda_{1}} >0\) as \(\rho \rightarrow 0^{+}\). Therefore, we can choose \(\rho >0\) small enough such that \(m ( \rho ) > \frac{\varepsilon_{0}}{2 \lambda_{1}}\). Set \(\lambda_{*}:= \frac{\varepsilon_{0} \rho^{\varphi^{0}}}{4 \lambda_{1}} >0\), \(\eta =: \frac{\varepsilon_{0} \rho^{\varphi^{0}}}{4 \lambda_{1}} >0\). For all \(\lambda \in ( 0, \lambda_{*} )\) and \(u \in S_{\rho}\), applying (3.4), we obtain

$$\mathcal{J}_{\lambda} ({u} )\geq \frac{\varepsilon_{0} \rho^{\varphi^{0}}}{4\lambda_{ 1}} =\eta>0. $$

 □

Lemma 3.3

Given that \(( \varPhi_{1} )\), \(( g_{1} )\), and \(( f_{3} )\) hold. Then, for any \(\lambda >0\), \(\rho >0\), there exists a function \(u_{\lambda} \in W_{0}^{1, \varPhi} (\varOmega)\) such that \(\Vert u_{\lambda} \Vert >\rho\) and \(\mathcal{J}_{\lambda} ( u_{\lambda} ) <0 \).

Proof

Take a compact set \(S \subset \varOmega\) with positive measure, we can define \(u_{0} \in C_{c}^{\infty} (\varOmega)\) such that \(u_{0} ( x ) =1\) for \(x \in S\), \(0\leq u_{0} ( x ) \leq1\) for \(x \in \varOmega \) (please see [30]). Then \(u_{0} \in W_{0}^{1, \varPhi} (\varOmega)\).

By condition \(( f_{3} )\), we deduce that for \(M_{0} := \frac{2 \Vert u_{0} \Vert ^{\varphi^{0}}}{\mu S} >0\) there exists \(C_{ 1} >0\) such that

$$F ( x,t ) \geq M_{0} \vert t \vert ^{\varphi^{0}}- C_{ 1}, \quad \forall x \in \varOmega, t\in \mathbb{R}. $$

Let \(t >1 \) large enough such that \(\Vert t u_{0} \Vert >1\), by Lemma 2.1,

$$\begin{aligned} \mathcal{J}_{\lambda} ( t u_{0} ) =& \int_{\varOmega} \varPhi \bigl( \vert \nabla tu_{0} \vert \bigr)\,dx- \lambda \int_{\varOmega} G ( tu_{0} )\,dx- \int_{\varOmega} F ( x, tu_{0} )\,dx \\ \leq& t^{\varphi^{0}} \Vert u_{0} \Vert ^{\varphi^{0}}- M_{0} t^{\varphi^{0}} \int_{\varOmega} \vert u_{0} \vert ^{\varphi^{0}} \,dx +C_{ 1} \mu\varOmega \\ \leq& t^{\varphi^{0}} \biggl( \Vert u_{0} \Vert ^{\varphi^{0}}- \frac{2 \Vert u_{0} \Vert ^{\varphi^{0}}}{\mu S} \int_{S} \vert u_{0} \vert ^{\varphi^{0}} \,dx \biggr) +C_{ 1} \mu\varOmega \\ =&- t^{\varphi^{0}} \Vert u_{0} \Vert ^{\varphi^{0}} +C_{ 1} \mu\varOmega. \end{aligned}$$

Due to \(\Vert u_{0} \Vert >0\), we see \(\mathcal{J}_{\lambda} ( t u_{0} )\rightarrow-\infty \) as \(t\rightarrow+\infty \).

Taking t large enough such that \(t> \max \{1, \frac{\rho+1}{ \Vert u_{0} \Vert }\}\), set \(u_{\lambda} =t u_{0}\), which completes the proof. □

Lemma 3.4

Given that \(( \varPhi_{1} )\), \(( g_{1} )\), and \(( f_{2} )\) hold. Then, for any \(\lambda >0\), \(\rho >0\), there exists a function \(\tilde{u}_{\lambda} \in W_{0}^{1, \varPhi} (\varOmega)\) such that \(\Vert \tilde{u}_{\lambda} \Vert <\rho\) and \(\mathcal{J}_{\lambda} ( \tilde{u}_{\lambda} )<0\).

Proof

Take a compact set \(\tilde{S}\subset \varOmega\) with positive measure, we can define \(\tilde{u}_{0} \in C_{c}^{\infty} (\varOmega)\) such that \(\tilde{u}_{0} ( x ) =1\) for \(x \in \tilde{S}\), \(0\leq \tilde{u}_{0} ( x ) \leq1\) for \(x \in \varOmega\) (please see [30]). Then \(\tilde{u}_{0} \in W_{0}^{1, \varPhi} (\varOmega)\).

We take \(t\in(0,\delta)\) (where δ is defined in (3.2)) such that \(\Vert t \tilde{u}_{0} \Vert <1\) and \(\Vert t \tilde{u}_{0} \Vert _{( G )} <1\). By (3.2), we have \(\vert F ( x,t \tilde{u}_{0} (x) ) \vert \leq (\lambda _{1} - \varepsilon_{0} ) \varPhi ( t \tilde{u}_{0} (x) )\) for all \(x \in \varOmega\). From Lemma 2.1 and (2.1), it follows

$$\begin{aligned} \mathcal{J}_{\lambda} ( t \tilde{u}_{0} ) =& \int_{\varOmega} \varPhi \bigl( \vert \nabla t \tilde{u}_{0} \vert \bigr)\,dx- \lambda \int_{\varOmega} G ( t \tilde{u}_{0} )\,dx- \int_{\varOmega} F ( x,t \tilde{u}_{0} )\,dx \\ \leq& \int_{\varOmega} \varPhi \bigl( \vert \nabla t \tilde{u}_{0} \vert \bigr)\,dx- \lambda \int_{\varOmega} G ( t \tilde{u}_{0} )\,dx- (\lambda_{ 1} - \varepsilon_{0} ) \int_{\varOmega} \varPhi ( t \tilde{u}_{0} )\,dx \\ \leq& \biggl(1+ \frac{ (\lambda_{1} - \varepsilon_{0} )}{ \lambda_{ 1}} \biggr) \int_{\varOmega} \varPhi \bigl( \vert \nabla t \tilde{u}_{0} \vert \bigr)\,dx- \lambda \int_{\varOmega} G ( t \tilde{u}_{0} )\,dx \\ \leq& \biggl(2- \frac{\varepsilon_{0}}{\lambda_{ 1}} \biggr) \int_{\varOmega} \varPhi (C _{1} t )\,dx- \lambda \int_{\tilde{S}} G ( t )\,dx \\ \leq& C _{2} \varPhi ( t )- \lambda G ( t ) \mu \tilde{S} \\ =& G ( t ) \biggl[C _{2} \frac{\varPhi ( t )}{G ( t )}- \lambda\mu \tilde{S} \biggr]. \end{aligned}$$

Due to \(( g_{1} )\), we can find \(t>0\) small enough such that for \(\Vert \tilde{u}_{\lambda} \Vert = \Vert t \tilde{u}_{0} \Vert <\rho\) and \(\mathcal{J}_{\lambda} ( t \tilde{u}_{0} )<0 \). □

Lemma 3.5

Given that \(( \varPhi_{1} )\), \(( \varPhi_{2} )\), \(( g_{1} )\), and \(( f_{1} )\)–\(( f_{4} )\) hold. Then, for each \(\lambda >0 \), the functional \(\mathcal{J}_{\lambda}\) satisfies \(C_{c}\) condition for any \(c >0\).

Proof

Given \(\lambda >0\), \(c >0\). Let \(\{ u_{n} \}\subset W_{0}^{1, \varPhi} (\varOmega)\) be a Cerami sequence at the level c of \(\mathcal{J}_{\lambda}\), i.e.,

$$ \mathcal{ J}_{\lambda} ( u_{n} ) \rightarrow c\quad \mbox{and}\quad \bigl\Vert \mathcal{J}_{\lambda} ' ( u_{n} ) \bigr\Vert _{*} \bigl( 1+ \Vert u_{n} \Vert \bigr) \rightarrow0, \quad n \rightarrow \infty. $$
(3.5)

First, we shall show that \(\{ u_{n} \}\) is bounded.

Otherwise, there is a subsequence, still denoted by \(\{ u_{n} \}\), such that \(\lim_{n\rightarrow\infty} \Vert u_{n} \Vert =\infty\) and \(\Vert u_{n} \Vert >1\) (\(\forall n\in \mathbb{N}\)).

We denote \(w_{n} (x):= \frac{u_{n} (x)}{ \Vert u_{n} \Vert }\), \(x \in \varOmega\), \(n =1,2,\ldots \) . Then \(\{ w_{n} \}\subset W_{0}^{1, \varPhi} (\varOmega)\) and \(\Vert w_{n} \Vert =1 \) for every \(n\in \mathbb{N}\). Applying the Eberlein–Smulian theorem, we may assume that there exists \(w \in W_{0}^{1, \varPhi} (\varOmega)\) such that \(w_{n}\) converges weakly to w. From Remark 3.1, it follows that

$$\begin{aligned}& \Vert w_{n} - w \Vert _{L^{1} (\varOmega)} \rightarrow0,\qquad \Vert w_{n} - w \Vert _{( G )} \rightarrow0,\qquad \Vert w_{n} - w \Vert _{( H )} \rightarrow0,\quad n \rightarrow\infty, \\ \end{aligned}$$
(3.6)
$$\begin{aligned}& w_{n} (x)\rightarrow w ( x )\quad \mbox{a.e. }x \in \varOmega, n \rightarrow\infty. \end{aligned}$$
(3.7)

Claim: \(w ( x ) =0\) a.e. \(x \in \varOmega\).

We suppose \(\mu \varOmega_{0} := \mu \{ x \in \varOmega:w ( x ) \neq0 \}>0\). Given \(x \in \varOmega_{0} \), (3.7) implies that \(| u_{n} ( x ) |= | w_{n} ( x ) |\cdot \Vert u_{n} \Vert \rightarrow\infty\) as \(n\rightarrow\infty\). Furthermore, by \(( f_{3} )\) we obtain that, for given \(x \in \varOmega_{0} \),

$$ \frac{F ( x, u_{n} ( x ) )}{ \Vert u_{n} \Vert ^{\varphi^{0}}} = \frac{F ( x, u_{n} ( x ) )}{ \vert u_{n} ( x ) \vert ^{\varphi^{0}}} \bigl\vert w_{n} ( x ) \bigr\vert ^{\varphi^{0}} \rightarrow\infty,\quad n\rightarrow \infty. $$
(3.8)

From \(( f_{3} )\) and the continuity of F on \(\overline{\varOmega} \times \mathbb{R}\), there exists a constant \(C_{ 1} \) such that

$$F ( x,t ) \geq C_{ 1}, \quad \forall({x,t)} \in \varOmega\times \mathbb{R}, $$

which implies that

$$ \frac{F ( x, u_{n} ( x ) ) -C_{ 1}}{ \Vert u_{n} \Vert ^{\varphi^{0}}} = \frac{F ( x, u_{n} ( x ) ) -C_{ 1}}{ \vert u_{n} ( x ) \vert ^{\varphi^{0}}} \bigl\vert w_{n} ( x ) \bigr\vert ^{\varphi^{0}} \geq0, \quad \forall x \in \varOmega, \forall t\in \mathbb{R}. $$
(3.9)

From (3.5), it follows that

$$c+o ( 1 ) = \mathcal{J}_{\lambda} ( u_{n} ) = \int_{\varOmega} \varPhi \bigl( \vert \nabla u_{n} \vert \bigr)\,dx- \lambda \int_{\varOmega} G ( u_{n} )\,dx- \int_{\varOmega} F ( x, u_{n} )\,dx. $$

Dividing the above equality by \(\Vert u_{n} \Vert ^{\varphi^{0}} \), by Lemma 2.1 and \(\Vert u_{n} \Vert >1\),

$$\begin{aligned}& \mathop{\lim \inf}_{n\rightarrow\infty} \int_{\varOmega} \frac{F ( x, u_{n} ( x ) )}{ \Vert u_{n} \Vert ^{\varphi^{0}}}\,dx \\& \quad = \frac{\mathop{\lim \inf}_{n\rightarrow\infty} \int_{\varOmega} F ( x, u_{n} ( x ) )\,dx}{ \Vert u_{n} \Vert ^{\varphi^{0}}} \\& \quad =\mathop{\lim \inf}_{n\rightarrow\infty} \biggl( \frac{\int_{\varOmega} \varPhi ( \vert \nabla u_{n} \vert )\,dx}{ \Vert u_{n} \Vert ^{\varphi^{0}}} - \frac{\lambda \int_{\varOmega} G ( u_{n} )\,dx}{ \Vert u_{n} \Vert ^{\varphi^{0}}} - \frac{c+o ( 1 )}{ \Vert u_{n} \Vert ^{\varphi^{0}}} \biggr) \\& \quad \leq \mathop{\lim \inf}_{n\rightarrow\infty} \biggl( \frac{\int_{\varOmega} \varPhi ( \vert \nabla u_{n} \vert )\,dx}{ \Vert u_{n} \Vert ^{\varphi^{0}}} - \frac{c+o ( 1 )}{ \Vert u_{n} \Vert ^{\varphi^{0}}} \biggr) \leq1. \end{aligned}$$
(3.10)

By Fatou’s lemma and (3.7)–(3.10),

$$\begin{aligned} \infty =& \int_{\varOmega_{0}} \lim_{n\rightarrow\infty} \frac{F ( x, u_{n} ( x ) ) -C_{ 1}}{ \Vert u_{n} \Vert ^{\varphi^{0}}}\,dx \\ \leq& \mathop{\lim \inf}_{n\rightarrow\infty} \int_{\varOmega_{0}} \frac{F ( x, u_{n} ( x ) ) -C_{ 1}}{ \Vert u_{n} \Vert ^{\varphi^{0}}}\,dx \\ \leq& \mathop{\lim \inf}_{n\rightarrow\infty} \int_{\varOmega} \frac{F ( x, u_{n} ( x ) ) -C_{ 1}}{ \Vert u_{n} \Vert ^{\varphi^{0}}}\,dx \\ =& \mathop{\lim \inf}_{n\rightarrow\infty} \frac{\int_{\varOmega} F ( x, u_{n} ( x ) )\,dx}{ \Vert u_{n} \Vert ^{\varphi^{0}}} - \mathop{\lim \sup}_{n\rightarrow\infty} \frac{\int_{\varOmega}C_{ 1}\,dx}{ \Vert u_{n} \Vert ^{\varphi^{0}}} \\ =& \mathop{\lim \inf}_{n\rightarrow\infty} \frac{\int_{\varOmega} F ( x, u_{n} ( x ) )\,dx}{ \Vert u_{n} \Vert ^{\varphi^{0}}} \leq1. \end{aligned}$$

Consequently, we get a contradiction, which implies that \(w ( x ) =0\) a.e. \(x \in \varOmega\).

Since \(\mathcal{J}_{\lambda} ( t u_{n} )\) is continuous on \([ 0,1 ]\) for each \(n\in \mathbb{N}\), there exists \(t_{n} \in [ 0,1 ]\) such that \(\mathcal{J}_{\lambda} ( t_{n} u_{n} ) = \max_{t\in[0,1]} \mathcal{J}_{\lambda} ( t u_{n} )\). Due to \(\Vert \mathcal{J}_{\lambda} ' ( u_{n} ) \Vert _{*} (1+ \Vert u_{n} \Vert )\rightarrow0\), we deduce

$$ \bigl\langle \mathcal{J}_{\lambda} ' ( t_{n} u_{n} ), t_{n} u_{n} \bigr\rangle \rightarrow0,\quad n\rightarrow\infty. $$
(3.11)

Take \(\{ s_{k} \}_{k=1}^{\infty}\subset(1,\infty)\) with \(s_{k} \rightarrow+\infty\) as \(k\rightarrow\infty\). Then, for each \(n,k \in \mathbb{N}\), one has \(\Vert s_{k} w_{n} \Vert = s_{k} >1\). From (3.6) and the claim, combining conditions \(( g_{1} ) \) and \(( f_{1} ) \), we deduce

$$\begin{aligned} \int_{\varOmega} F \bigl( x, s_{k} w_{n} ( x ) \bigr)\,dx \leq& C \int_{\varOmega} \bigl[ \vert s_{k} w_{n} \vert +H ( s_{k} w_{n} ) \bigr] \,dx \\ \leq& C\big( \Vert s_{k} w_{n} \Vert _{L^{1} ( \varOmega )} + \Vert s_{k} w_{n} \Vert _{(H)}\big) \rightarrow0,\quad n\rightarrow\infty, \end{aligned}$$
(3.12)

and

$$ \int_{\varOmega} G \bigl( s_{k} w_{n} ( x ) \bigr)\,dx\leq \Vert s_{k} w_{n} \Vert _{(G)} \rightarrow0,\quad n\rightarrow\infty. $$
(3.13)

Due to \(\lim_{n\rightarrow\infty} \Vert u_{n} \Vert =\infty\), given \(k\in \mathbb{N}\), there exists \(n_{k} \geq k\). For all \(n\geq n_{k} \geq k\), one has \(\Vert u_{n} \Vert > s_{k}\), i.e., \(0< \frac{s_{k}}{ \Vert u_{n} \Vert } <1\).

From \(\Vert s_{k} w_{n} \Vert >1\), Lemma 2.1 and (3.12), (3.13), for large \(n\in \mathbb{N}\),

$$\begin{aligned} \mathcal{J}_{\lambda} ( t_{n} u_{n} ) =& \max _{t\in[0,1]} \mathcal{J}_{\lambda} ( t u_{n} ) \geq \mathcal{J}_{\lambda} \biggl( \frac{s_{k}}{ \Vert u_{n} \Vert } u_{n} \biggr) = \mathcal{J}_{\lambda} ( s_{k} w_{n} ) \\ =& \int_{\varOmega} \varPhi \bigl( \vert \nabla s_{k} w_{n} \vert \bigr)\,dx- \lambda \int_{\varOmega} G ( s_{k} w_{n} )\,dx- \int_{\varOmega} F ( x, s_{k} w_{n} )\,dx \\ \geq& \Vert s_{k} w_{n} \Vert ^{\varphi_{0}}- \lambda \int_{\varOmega} G ( s_{k} w_{n} )\,dx - \int_{\varOmega} F ( x, s_{k} w_{n} )\,dx \\ \geq& \frac{1}{2} \Vert s_{k} w_{n} \Vert ^{\varphi_{0}}= \frac{1}{2} s_{k}^{\varphi_{0}}. \end{aligned}$$

Let \(s_{k} = \Vert u_{k} \Vert ^{\gamma}>1\), where \(\gamma\in( \frac{\varphi^{0}}{\varphi_{0}},+\infty)\) is a constant. For all \(n\geq n_{k} \geq k\), one has

$$ \mathcal{ J}_{\lambda} ( t_{n} u_{n} ) \geq \frac{1}{2} \Vert u_{k} \Vert ^{{\gamma} \varphi_{0}}. $$
(3.14)

Applying (3.11), \(( f_{3} )\), \(( f_{4} )\), and \(( \varPhi_{2} )\), for large \(n \mathbb{\in \mathrm{N}}\),

$$\begin{aligned} \mathcal{ J}_{\lambda} ( t_{n} u_{n} ) =& \mathcal{J}_{\lambda} ( t_{n} u_{n} ) - \frac{1}{\varphi^{0}} \bigl\langle \mathcal{J}_{\lambda} ' ( t_{n} u_{n} ), t_{n} u_{n} \bigr\rangle +o(1) \\ =& \int_{\varOmega} \varPhi \bigl( \vert \nabla t_{n} u_{n} \vert \bigr)\,dx- \lambda \int_{\varOmega} G ( t_{n} u_{n} )\,dx- \int_{\varOmega} F ( x, t_{n} u_{n} )\,dx \\ &{}- \frac{1}{\varphi^{0}} \int_{\varOmega} \varphi \bigl( \vert \nabla t_{n} u_{n} \vert \bigr) \vert \nabla t_{n} u_{n} \vert \,dx+ \frac{\lambda}{\varphi^{0}} \int_{\varOmega} t_{n} u_{n} g ( t_{n} u_{n} )\,dx \\ &{}+ \frac{1}{\varphi^{0}} \int_{\varOmega} t_{n} u_{n} f ( x, t_{n} u_{n} )\,dx+o(1) \\ =& \frac{1}{\varphi^{0}} \int_{\varOmega} \overline{\varPhi} \bigl( t_{n} \vert \nabla u_{n} \vert \bigr)\,dx+ \frac{1}{\varphi^{0}} \int_{\varOmega} \overline{F} ( x, t_{n} u_{n} )\,dx \\ &{}+ \frac{\lambda}{\varphi^{0}} \int_{\varOmega} \bigl[t_{n} u_{n} g ( t_{n} u_{n} ) - \varphi^{0} G ( t_{n} u_{n} ) \bigr]\,dx+o(1) \\ \leq& \frac{1}{\varphi^{0}} \int_{\varOmega} \bigl[D _{1} \overline{\varPhi} \bigl( \vert \nabla u_{n} \vert \bigr) +\beta ( x ) \bigr]\,dx+ \frac{1}{\varphi^{0}} \int_{\varOmega} \bigl[D _{1} \overline{F} ( x, u_{n} ) +\alpha ( x ) \bigr]\,dx+o(1) \\ =& \frac{D_{ 1}}{\varphi^{0}} \int_{\varOmega} \bigl[ \overline{\varPhi} \bigl( \vert \nabla u_{n} \vert \bigr) + \overline{F} ( x, u_{n} ) \bigr] \,dx+ C _{2 }+o(1) \\ =& D_{ 1} \mathcal{J}_{\lambda} ( u_{n} ) - \frac{D _{1}}{ \varphi^{0}} \bigl\langle \mathcal{J}_{\lambda} ' ( u_{n} ), u_{n} \bigr\rangle \\ &{}+ D_{ 1} \lambda \int_{\varOmega} \biggl[ G ( u_{n} ) - \frac{1}{\varphi^{0}} u_{n} g ( u_{n} ) \biggr]\,dx+ C_{ 2 }+o(1) \\ \leq& D_{ 1} c+ D _{ 1} \lambda \biggl( 1 - \frac{1}{\varphi^{0}} \biggr) \int_{\varOmega} G ( 2 u_{n} )\,dx+ C_{ 2} +o(1) \\ \leq& C_{ 3} + C_{ 3} \int_{\varOmega} \varPhi ( 2 u_{n} )\,dx\leq C_{ 3} + C_{ 3} \Vert u_{n} \Vert ^{\varphi^{0}}. \end{aligned}$$

Combined with (3.14), we have \(\frac{1}{2} \Vert u_{k} \Vert ^{\gamma \varphi_{0}}- C_{ 3} \Vert u_{n} \Vert ^{\varphi^{0}}\leq C_{ 3} \). Letting \(k\rightarrow\infty\), then \(n\geq n_{k} \geq k\rightarrow\infty\). From \(\gamma \varphi_{0} > \varphi^{0}\), we get \(\infty\leq C_{ 3}\). This contradiction shows that \(\{ \Vert u_{n} \Vert \}\) is bounded, that is, \(\sup_{n\in \mathbb{N}} \Vert u_{n} \Vert := K_{ 0} <\infty \).

Taking into account the reflexivity of \(W_{0}^{1, \varPhi} (\varOmega)\) and the Eberlein–Smulian theorem, we may assume that \(u_{n} \) converges weakly to \(u \in W_{0}^{1, \varPhi} (\varOmega)\). By using Remark 3.1, we obtain

$$ \Vert u_{n} - u \Vert _{L^{1} (\varOmega)} \rightarrow0,\qquad \Vert u_{n} - u \Vert _{( G )} \rightarrow0,\qquad \Vert u_{n} - u \Vert _{( H )} \rightarrow0,\quad n \rightarrow\infty. $$
(3.15)

Using \(( f_{1} )\) and Hölder’s inequality, we have

$$\begin{aligned}& \biggl\vert \lambda \int_{\varOmega} g ( u_{n} ) ( u_{n} -u ) \,dx+ \int_{\varOmega} f ( x, u_{n} ) ( u_{n} -u )\,dx \biggr\vert \\& \quad \leq\lambda \int_{\varOmega} \bigl\vert g ( u_{n} ) ( u_{n} -u ) \bigr\vert \,dx+ \int_{\varOmega} \bigl[C \vert u_{n} -u \vert +C \bigl\vert h ( x, u_{n} ) ( u_{n} -u ) \bigr\vert \bigr]\,dx \\& \quad \leq2\lambda \bigl\Vert g ( u_{n} ) \bigr\Vert _{ ( \tilde{G} )} \Vert u_{n} - u \Vert _{ ( G )} +C \Vert u_{n} - u \Vert _{L^{1} ( \varOmega )} \\& \qquad {}+2C \bigl\Vert h ( u_{n} ) \bigr\Vert _{ ( \tilde{H} )} \Vert u_{n} - u \Vert _{ ( H )}. \end{aligned}$$
(3.16)

Now, we will show that both \(\Vert g ( u_{n} ) \Vert _{ ( \tilde{G} )}\) and \(\Vert h ( u_{n} ) \Vert _{ ( \tilde{H} )} \) are bounded.

Applying Lemma 2.1,

$$\begin{aligned} \int_{\varOmega} \tilde{G} \bigl( g( u_{n} ) \bigr) \,dx \leq& \int_{\varOmega} u_{n} g( u_{n} )\,dx\leq \int_{\varOmega} G ( 2 u_{n} )\,dx\leq C _{4 }+ \int_{\varOmega} \varPhi ( 2 u_{n} )\,dx \\ \leq& C_{ 4} + C_{ 4} \Vert u_{n} \Vert ^{\varphi^{0}}< \infty. \end{aligned}$$

The definition of \(\Vert \cdot \Vert _{ ( \tilde{G} )}\) yields that \(\Vert g ( u_{n} ) \Vert _{ ( \tilde{G} )} \leq C _{4} + C_{4} K_{ 0}^{\varphi^{0}}\), \(n =1,2,\ldots \) . On the other hand, due to \(\lim_{t\rightarrow\infty} \frac{ H(2t)}{ \varPhi_{*} (t)} =0\), there exists \(t_{0} >0\) such that \(H(2t)\leq \varPhi_{*} (t)\) for all \(t \geq t_{0}\). By Lemma 2.4 in [23], we have \(d_{0} := \sup_{t>0} \frac{t \varPhi_{*} ' ( t )}{\varPhi_{*} (t)} \leq \frac{N \varphi^{0}}{N- \varphi^{0}} <\infty\). Since \(W_{0}^{1, \varPhi} (\varOmega) \hookrightarrow L_{ \varPhi_{*}} ( \varOmega )\),

$$\begin{aligned} \int_{\varOmega} \tilde{H} \bigl( h( u_{n} ) \bigr) \,dx \leq& \int_{\varOmega} H ( 2 u_{n} )\,dx\leq H_{ 4} (2 t_{0} )\mu \varOmega+ \int_{\varOmega} \varPhi_{*} ( u_{n })\,dx \\ \leq& C_{5} + C_{5} \Vert u_{n} \Vert _{ ( \varPhi_{*} )}^{d_{0}}\leq C_{ 6} + C_{6} K _{0}^{d_{0}}< \infty, \quad n =1,2, \ldots. \end{aligned}$$

Hence, \(\Vert h ( u_{n} ) \Vert _{ ( \tilde{H} )} \leq C_{ 6 }+ C_{6} K_{0}^{d_{0}}<\infty\), \(n =1,2,\ldots \) .

Combining (3.15) and (3.16), we have

$$ \int_{\varOmega} g ( u_{n} ) ( u_{n} -u ) \,dx+ \int_{\varOmega} f ( x, u_{n} ) ( u_{n} -u )\,dx \rightarrow0 ,\quad n\rightarrow\infty. $$
(3.17)

From (3.5), it follows that \(\int_{\varOmega} a ( \vert \nabla u_{n} \vert ) \nabla u_{n} \cdot \nabla( u_{n} -u)\,dx \rightarrow0\) as \(n\rightarrow\infty\). Since \(u_{n}\) converges weakly to u, Theorem 4 in [36] implies that \(\lim_{n\rightarrow\infty} \Vert u_{n} -u \Vert =0\). Therefore, \(\mathcal{J}_{\lambda}\) satisfies \(C_{c}\) condition. □

Next, we give the proof of our main result Theorem 3.1.

Proof

\(\lambda_{ *} >0\), \(\eta>0\), \(\rho >0 \) are constants defined in Lemma 3.2. For all \(\lambda\in (0, \lambda_{ *} )\), Lemma 3.2, Lemma 3.3, and Lemma 3.5 show that the functional \(\mathcal{J}_{\lambda}\) satisfies all the assumptions of Lemma 2.3. Then \(\mathcal{J}_{\lambda}\) has a critical value \(c \geq \eta>0\). This shows that problem (1.1) has a nontrivial weak solution u with \(\mathcal{J}_{\lambda} ({u} )=c\).

In the following, we prove there exists a second weak solution \(\tilde{u} \neq u\).

Let \(B_{\rho} := \{ u \in W_{0}^{1, \varPhi} ( \varOmega ): \Vert u \Vert \leq\rho \}\), \(U_{\rho} := \{ u \in W_{0}^{1, \varPhi} ( \varOmega ): \Vert u \Vert <\rho \}\). Applying Lemma 3.4, we deduce that

$$-\infty< \tilde{c} := \inf_{B_{\rho}} \mathcal{J}_{\lambda} ({u} ) < 0. $$

For each \(\sigma \in(0, \inf_{S_{\rho}} \mathcal{J}_{\lambda} ({u} ) - \inf_{U_{\rho}} \mathcal{J}_{\lambda} ({u} ) )\), by the Ekeland variational principle [11], there exists \(u_{\sigma} \in B_{\rho}\) such that

$$\mathcal{J}_{\lambda} ( u_{\sigma} )\leq \inf _{B_{\rho}} \mathcal{J}_{\lambda} ({u} ) +\sigma $$

and

$$ \mathcal{ J}_{\lambda} ( u_{\sigma} )< \mathcal{J}_{\lambda} ({u} )+\sigma \Vert u_{\sigma} -u \Vert , \quad \forall u\neq u_{\sigma}. $$
(3.18)

Therefore,

$$\mathcal{J}_{\lambda} ( u_{\sigma} )\leq \inf _{B_{\rho}} \mathcal{J}_{\lambda} ({u} ) +\sigma< \inf _{U_{\rho}} \mathcal{J}_{\lambda} ({u} ) + \inf _{S_{\rho}} \mathcal{J}_{\lambda} ({u} ) - \inf _{U_{\rho}} \mathcal{J}_{\lambda} ({u} ) = \inf _{S_{\rho}} \mathcal{J}_{\lambda} ({u} ), $$

which implies \(u_{\sigma} \in U_{\rho}\).

\(\forall v \in B_{1}\), take \(h \in(0, \rho- \Vert u_{\sigma} \Vert )\), then \(u_{\sigma} +hv\in B_{\rho}\). By (3.18), we have

$$\mathcal{J}_{\lambda} ( u_{\sigma} )- \mathcal{J}_{\lambda} ( u_{\sigma} +hv )\leq\sigma h \Vert v \Vert . $$

Dividing the above inequality by h and letting \(h\rightarrow 0^{+}\), one has

$$\bigl\langle \mathcal{J}_{\lambda} ' ( u_{\sigma} ), v \bigr\rangle \geq-\sigma \Vert v \Vert . $$

Replacing v with −v in the above inequality, we deduce \(\langle \mathcal{J}_{\lambda} ' ( u_{\sigma} ), v \rangle \leq\sigma \Vert v \Vert \). Therefore, \(\Vert \mathcal{J}_{\lambda} ' ( u_{\sigma} ) \Vert \leq\sigma\).

Summarily, there exist \(\{ \tilde{u}_{n} \}_{n=1}^{\infty} \subset U_{\rho}\) such that \(\mathcal{J}_{\lambda} ( \tilde{u}_{n} ) \rightarrow \tilde{c}\) and \(\Vert \mathcal{J}_{\lambda} ' ( \tilde{u}_{n} ) \Vert \leq \frac{1}{n} \rightarrow0\) as \(n\rightarrow\infty\). From the Eberlein–Smulian theorem, we may assume \(\tilde{u}_{n}\) converges to \(\tilde{u} \in B_{\rho}\). (3.17) and Theorem 4 in [36] imply that \(\lim_{n\rightarrow\infty} \Vert \tilde{u}_{n} - \tilde{u} \Vert =0\). Since \(\mathcal{J}_{\lambda} \in C^{1} ( W_{0}^{1, \varPhi} ( \varOmega ), \mathbb{R} )\) and \(\Vert \mathcal{J}_{\lambda} ' ( \tilde{u}_{n} ) \Vert \rightarrow0\), one has \(\mathcal{J}_{\lambda} ' ( \tilde{u} )= \lim_{n\rightarrow\infty} \mathcal{J}_{\lambda} ' ( \tilde{u}_{n} ) =\theta\) and \(\mathcal{J}_{\lambda} ' ( \tilde{u} )= \tilde{c}\), so \(\tilde{u} \neq\theta\) and \(\tilde{u} \neq u\), which completes the proof. □

By Lemma 3.2, Lemma 3.3, and Lemma 3.5, we can get the following corollary.

Corollary 3.6

([27])

Given that Φ satisfies \(( \varPhi_{1} )\) and \(( \varPhi_{2} )\), f satisfies \(( f_{1} )\)–\(( f_{4} )\). Then

$$\textstyle\begin{cases} -\operatorname{div}( a ( \vert \nabla u \vert ) \nabla u ) =f ( x,u ),& \textit{in } \varOmega,\\ u=0, &\textit{on }\partial\varOmega, \end{cases} $$

has a nontrivial weak solution.

4 Conclusions

Using variational arguments, we establish the existence of two nontrivial solutions for quasilinear elliptic problems in Orlicz–Sobolev spaces, where the nonlinear terms exhibit the combined effects of concave and convex without the Ambrosetti–Rabinowitz type condition.