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A new existence result for some nonlocal problems involving Orlicz spaces and its applications

Abstract

This paper studies some quasilinear elliptic nonlocal equations involving Orlicz–Sobolev spaces. On the one hand, a new sub-supersolution theorem is proved via the pseudomonotone operator theory; on the other hand, using the obtained theorem, we present an existence result on the positive solutions of a singular elliptic nonlocal equation. Our work improves the results of some previous researches.

Introduction

This paper is concerned with the problem

$$ \textstyle\begin{cases} -\Delta _{\Phi}u=h_{1}(u) \Vert u \Vert _{L^{\Psi}}^{\alpha}+h_{2}(u) \Vert u \Vert _{L^{ \Lambda}}^{\gamma}, \quad x\in \Omega , \\ u=0, \quad x\in \partial \Omega , \end{cases} $$
(1.1)

where α, γ are positive constants, \(\|\cdot \|_{L^{\Psi}}\) (resp. \(\|\cdot \|_{L^{\Lambda}}\)) is a norm in \(L^{\Psi}(\Omega )\) (resp. \(L^{\Lambda}(\Omega )\)) and the nonlinearities \(h_{1}\), \(h_{2}\): \([0, +\infty )\to [0, +\infty )\) are continuous functions, Ω R N (\(N\ge 3\)) is bounded with \(\partial \Omega \in C^{2}\), \(\Delta _{\Phi}u= \mathrm{div}(\rho (|\nabla u|)\nabla u)\), where

$$ \begin{aligned} \Phi (t):= \int _{0}^{|t|}\rho (s)s\,ds. \end{aligned} $$
(1.2)

Here \(\rho \in C^{1}: [0, +\infty )\to [0, +\infty )\) and it satisfies (see [10])

\((\rho _{1})\):

\(t\rho(t)\) is differentiable for \(\forall t>0\),

\((\rho _{2})\):

\(\lim_{t\to 0^{+}}t\rho (t)=0\), \(\lim_{t\to +\infty}t \rho (t)=+\infty \),

and that there exist \(\kappa , s\in (1,N)\) such that

\((\rho _{3})\):

\(\kappa -1\le \frac{(\rho (t)t)'}{\rho (t)}\le s-1\), \(\forall t>0\).

Note that \((\rho _{3})\) implies that

\((\rho _{3})'\):

\(\kappa \le \frac{\rho (t)t^{2}}{\Phi (t)}\le s\), \(\forall t>0\).

Problem (1.1) was proposed in [10] and generalizes some problems in [3, 5, 6, 8, 1720]. As the authors of [10] pointed out, there are some difficulties to study problem (1.1): (1) variational methods cannot be used directly because of the nonlocal terms; (2) the presence of the concave–convex nonlinearities leads to invalidness of the Galerkin method; (3) there is no ready-made sub-supersolutions method as in [2] and [7] because of the Φ-Laplacian operator. In [10], for the first time, using monotone iterative technique, Figueiredo et al. obtained the sub-supersolution theorem for problem (1.1) in which they needed an important condition that \(h_{1}\), \(h_{2}\): [0,+)R are nondecreasing. As its application, the authors discussed the following problem:

$$ \textstyle\begin{cases} -\Delta _{\Phi}u=u^{\beta} \Vert u \Vert _{L^{\Psi}}^{\alpha}, \quad x\in \Omega , \\ u=0, \quad x\in \partial \Omega . \end{cases} $$
(1.3)

with the assumption that α, \(\beta \ge 0\) with \(0<\alpha +\beta <\kappa -1\), and got the existence of a positive solution.

Another interesting work appeared in [9], in which Dos Santos et al. studied the problem as follows:

$$ \textstyle\begin{cases} -A(x, \Vert u \Vert _{L^{r(x)}})\Delta _{p_{1}(x)}u=h_{1}(u,x) \Vert u \Vert _{L^{q(x)}}^{ \alpha _{1}(x)}+h_{2}(u,x) \Vert u \Vert _{L^{s(x)}}^{\gamma _{1}(x)}, \quad x \in \Omega , \\ u=0, \quad x\in \partial \Omega . \end{cases} $$

Note that \(h_{1}\) and \(h_{2}\) are not nondecreasing in this paper.

Motivated by [10] and [9], we try to present the sub-supersolution approach for problem (1.1) without the assumptions that \(h_{1}\) and \(h_{2}\) are nondecreasing.

Our paper is divided into four sections. In Sect. 2, some needed properties of Orlicz spaces and the main results are listed. In Sect. 3, we prove a new sub-supersolution theorem for problem (1.1) via the pseudomonotone operator theory and, using obtained theorem, we present a new existence result on positive solutions of problem (1.3) when \(\alpha \ge 0\), \(-1<\beta <0\), with \(0<\alpha <\kappa -1\). Our work complements the conclusions in [10] and [9]: (1) we obtain the existence of a nontrivial solution of problem (1.1) when \(h_{1}\) and \(h_{2}\) have no monotonicity; (2) problem (1.3) is studied when \(\beta \in (-1,0)\).

Preliminaries and main results

Now we shall list some main definitions, properties, and conclusions in the setting of Orlicz–Sobolev spaces. For more information, please refer to the literature [1, 4, 13, 15, 16, 22].

In (1.1), because of the existence of assumption \((\rho _{3})'\), it is easily to see that the \(\Delta _{2}\) condition is true for \(\Phi (t)\) (see [10]).

Lemma 2.1

The function Φ is nondecreasing on \([0, +\infty )\).

Proof

Obviously, it is enough to prove that for any \(0<\omega _{1}<\omega _{2}\), we always have the result \(\Phi (\omega _{1})\le \Phi (\omega _{2})\). Since Φ is convex from the definition of an N-function, we have

$$ \frac{\Phi (\omega _{1})-\Phi (0)}{\omega _{1}-0}\le \frac{\Phi (\omega _{2})-\Phi (\omega _{1})}{\omega _{2}-\omega _{1}}, $$

that is,

$$ \begin{aligned} \frac{\Phi (\omega _{1})-0}{\omega _{1}-0}\le \frac{\Phi (\omega _{2})-\Phi (\omega _{1})}{\omega _{2}-\omega _{1}}. \end{aligned} $$

Then we have \(\Phi (\omega _{2})-\Phi (\omega _{1})\ge 0\), that is, \(\Phi (\omega _{2})\ge \Phi (\omega _{1})\). Therefore, the function Φ is nondecreasing on \([0, +\infty )\). □

Definition 2.2

If a positive function \(\overline{w}^{*}\) with \(\overline{w}^{*}\in W^{1,\Phi}(\Omega )\cap L^{\infty}(\Omega )\) satisfies

$$ \textstyle\begin{cases} -\Delta _{\Phi}\overline{w}^{*} \ge h_{1}(\overline{w}^{*})J_{1}( \overline{w}^{*})+h_{2}(\overline{w}^{*})J_{2}(\overline{w}^{*}), \quad x\in \Omega , \\ \overline{w}^{*}\ge 0, \quad x\in \partial \Omega , \end{cases} $$

then \(\overline{w}^{*}(x)\) is called a supersolution of problem (1.1).

If a positive function \(\underline{w}_{*}\) with \(\underline{w}_{*}\in W^{1,\Phi}(\Omega )\cap L^{\infty}(\Omega )\) satisfies

$$ \textstyle\begin{cases} -\Delta _{\Phi}\underline{w}_{*}\le h_{1}(\underline{w}_{*})J_{1}( \underline{w}_{*})+h_{2}(\underline{w}_{*})J_{2}(\underline{w}_{*}), \quad x\in \Omega , \\ \underline{w}_{*}\leq 0, \quad x\in \partial \Omega , \end{cases} $$

then \(\underline{w}_{*}(x)\) is called a subsolution of problem (1.1).

For more information on \(L^{\Phi}(\Omega )\) and its norm, please refer to the literature [10]. Let

$$\begin{aligned}& \zeta (u,x):=\max \bigl\{ \underline{w}_{*}(x), \min \bigl\{ u, \overline{w}^{*}(x) \bigr\} \bigr\} , \\& \nu \in (0,1), \quad \gamma (t,x):=-(\underline{w}_{*}-t)_{+}^{\nu}+ \bigl(t- \overline{w}^{*}\bigr)_{+}^{\nu}, \\& J_{1}(u):= \bigl\Vert \zeta (u,x) \bigr\Vert _{L^{\Psi}}^{\alpha}={ \inf}^{\alpha} \biggl\{ \varsigma >0: \int _{\Omega}\Psi \biggl(\frac{ \vert \zeta (u,x) \vert }{\varsigma}\biggr) \le 1 \biggr\} , \\& J_{2}(u):= \bigl\Vert \zeta (u,x) \bigr\Vert _{L^{\Lambda}}^{\gamma}={ \inf}^{\gamma} \biggl\{ \varsigma >0: \int _{\Omega}\Lambda \biggl( \frac{ \vert \zeta (u,x) \vert }{\varsigma}\biggr)\le 1 \biggr\} . \end{aligned}$$

In addition, Ψ and Λ are N-functions satisfying the \(\Delta _{2}\) condition, and they are also nondecreasing on \([0, +\infty )\).

For an N-function Φ, the corresponding Orlicz–Sobolev space is defined as the Banach space

$$ W^{1, \Phi}(\Omega ):= \biggl\{ v\in L^{\Phi}(\Omega )\Bigm| \frac{\partial v}{\partial x_{i}}\in L^{\Phi}(\Omega )\text{ for } i=1, \dots , N \biggr\} $$

endowed with the norm

$$ \Vert v \Vert _{1, \Phi}= \Vert \nabla v \Vert _{L^{\Phi}}+ \Vert v \Vert _{L^{\Phi}}. $$

Specially,

$$ W_{0}^{1, \Phi}(\Omega ):= \biggl\{ v\in L^{\Phi}(\Omega )\Bigm| \frac{\partial v}{\partial x_{i}}\in L^{\Phi}(\Omega )\text{ for } i=1, \dots , N \text{ and } v=0, x\in \partial \Omega \biggr\} . $$

For their properties, one can refer to the literature [10].

Lemma 2.3

([10])

Let Φ be an N-function defined in (1.2) and satisfying \((\rho _{1})\), \((\rho _{2})\), and \((\rho _{3})\). Denote

$$ \xi _{0}(t)=\min \bigl\{ t^{\kappa}, t^{s}\bigr\} $$

and

$$ \xi _{1}(t)=\max \bigl\{ t^{\kappa}, t^{s}\bigr\} ,\quad t \ge 0, $$

then

$$\begin{aligned} &\xi _{0}(t)\Phi (\varrho )\le \Phi (\varrho t)\le \xi _{1}(t)\Phi ( \varrho ),\quad \varrho , t>0, \\ &\xi _{0}\bigl( \Vert u \Vert _{L^{\Phi}}\bigr)\le \int _{\Omega}\Phi (u)\le \xi _{1}\bigl( \Vert u \Vert _{L^{ \Phi}}\bigr),\quad u\in L^{\Phi}(\Omega ). \end{aligned}$$

Lemma 2.4

([10])

Let \(\lambda >0\), let Φ be given by (1.2), and suppose Ω R N is an admissible domain. Consider the problem

$$ \textstyle\begin{cases} -\Delta _{\Phi}z_{\lambda}=\lambda , \quad x\in \Omega , \\ z_{\lambda}=0, \quad x\in \partial \Omega . \end{cases} $$
(2.1)

where \(z_{\lambda}\) is the unique solution. Define

$$ \rho _{0}=\frac{1}{2 \vert \Omega \vert ^{\frac{1}{N}}C_{0}}.$$

If \(\lambda \ge \rho _{0}\), then

$$ \vert z_{\lambda} \vert _{L^{\infty}}\le C^{*}\lambda ^{\frac{1}{\kappa -1}},$$

and

$$ \vert z_{\lambda} \vert _{L^{\infty}}\le C_{*}\lambda ^{\frac{1}{s-1}}$$

if \(\lambda <\rho _{0}\). Here \(C^{*}>0\) and \(C_{*}>0\) depend on n, s, N, and Ω.

For \(z_{\lambda}\) which is defined in Lemma 2.4, it follows that \(z_{\lambda}\in C^{1}(\overline{\Omega})\) with \(z_{\lambda}>0\) in Ω.

Lemma 2.5

([11])

There is a \(k_{0}>0\) satisfying

$$ \bigl(\rho \bigl( \vert \zeta \vert \bigr)\zeta -\rho \bigl( \vert \epsilon \vert \bigr)\epsilon \bigr)\cdot (\zeta - \epsilon )\ge k_{0} \frac{\Phi ( \vert \zeta -\epsilon \vert )^{\frac{\kappa +1}{\kappa}}}{(\Phi ( \vert \zeta \vert )+\Phi ( \vert \epsilon \vert ))^{\frac{1}{\kappa}}} $$

for ζ,ϵ R N , \(\zeta \neq 0\).

Theorem 2.6

If the functions h 1 , h 2 :[0,+)R are continuous and nonnegative, \(\alpha , \gamma \ge 0\), \(\overline{w}^{*}\) is a supersolution and \(\underline{w}_{*}\) is a subsolution with \(0<\underline{w}_{*}\leq \overline{w}^{*}\), problem (1.1) possesses a nontrivial solution u with \(\underline{w}_{*}\le u\le \overline{w}^{*}\).

Theorem 2.7

Suppose that \(0<\alpha <\kappa -1\) and \(-1<\beta <0\), where κ is given in \((\rho _{3})\). Then equation (1.3) has a positive solution.

Proofs of the main results

Proof of Theorem 2.6

We consider

$$ \textstyle\begin{cases} -\Delta _{\Phi}u= H(u, x, h_{1}(\zeta (u,x)), h_{2}(\zeta (u,x))), \quad x\in \Omega , \\ u=0, \quad x\in \partial \Omega , \end{cases} $$
(3.1)

where

$$ H(u, x, s, t)= J_{1}(u)s+ J_{2}(u)t-\gamma (u,x). $$

We have the following claims:

Claim 1. Problem (3.1) has a solution in \(W^{1,\Phi}_{0}(\Omega )\cap L^{\infty}(\Omega )\).

Define \(B:W_{0}^{1, \Phi}(\Omega ): \to W^{-1, \Phi}(\Omega )\) as

$$ \begin{aligned} \bigl(B(u), w\bigr)&= \int _{\Omega}-\Delta _{\Phi}uw- \int _{\Omega}\bigl[h_{1}\bigl(\zeta (u,x) \bigr)J_{1}(u)+h_{2}\bigl(\zeta (u,x)\bigr)J_{2}(u) - \gamma (u,x)\bigr]w \\ &= \int _{\Omega}\rho \bigl( \vert \nabla u \vert \bigr) (\nabla u \cdot \nabla w)- \int _{\Omega}\bigl[h_{1}\bigl(\zeta (u,x) \bigr)J_{1}(u)+h_{2}\bigl(\zeta (u,x)\bigr)J_{2}(u) \\ &\quad {}-\gamma (u,x)\bigr]w,\quad \forall u, w\in W_{0}^{1, \Phi}( \Omega ), \end{aligned} $$

where ρ satisfies \((\rho _{1})\), \((\rho _{2})\), and \((\rho _{3})\).

First, we want to show that B is continuous, bounded, and coercive.

It is easy to see that the conditions on ρ and the continuity of \(h_{1}\) and \(h_{2}\) guarantees that B is bounded and continuous.

According to \((\rho _{3})'\), there exist \(\kappa , s\in (1,N)\) such that

$$ \kappa \le \frac{\rho (t)t^{2}}{\Phi (t)}\le s,\quad \forall t>0, $$

which implies that

$$ \begin{aligned} \frac{(B(u), u)}{ \Vert u \Vert _{1, \Phi}}&= \frac{\int _{\Omega}\rho ( \vert \nabla u \vert ) \vert \nabla u \vert ^{2}-\int _{\Omega}[h_{1}(\zeta (u,x))J_{1}(u)+h_{2}(\zeta (u,x))J_{2}(u)-\gamma (u,x)]u}{ \Vert u \Vert _{1, \Phi}} \\ &\ge \frac{\kappa \int _{\Omega}\Phi ( \vert \nabla u \vert )-\int _{\Omega}[h_{1}(\zeta (u,x))J_{1}(u)+h_{2}(\zeta (u,x))J_{2}(u)-\gamma (u,x)]u}{ \Vert u \Vert _{1, \Phi}}. \end{aligned} $$

From the Lemma 2.3 and Lemma 2.1 in [12], we have

$$ \min \bigl\{ \Vert \nabla u \Vert _{L^{\Phi}}^{\kappa}, \Vert \nabla u \Vert _{L^{\Phi}}^{s} \bigr\} =\xi _{0}\bigl( \Vert \nabla u \Vert _{L^{\Phi}}\bigr)\le \int _{\Omega}\Phi \bigl( \vert \nabla u \vert \bigr) $$

and

$$ \int _{\Omega}\Phi \bigl( \vert \nabla u \vert \bigr)\ge \int _{\Omega}\Phi \biggl(\frac{ \vert u \vert }{d}\biggr), $$

then we deduce

$$ \begin{aligned} \frac{(B(u), u)}{ \Vert u \Vert _{1, \Phi}}&\ge \frac{\frac{\kappa}{2}\min \{ \Vert \nabla u \Vert _{L^{\Phi}}^{\kappa}, \Vert \nabla u \Vert _{L^{\Phi}}^{s}\}+\frac{\kappa}{2}\min \{ \Vert \frac{u}{d} \Vert _{L^{\Phi}}^{\kappa}, \Vert \frac{u}{d} \Vert _{L^{\Phi}}^{s}\}}{ \Vert u \Vert _{1, \Phi}} \\ &\quad {}- \frac{\int _{\Omega}[h_{1}(\zeta (u,x))J_{1}(u)+h_{2}(\zeta (u,x))J_{2}(u)-\gamma (u,x)]u}{ \Vert u \Vert _{1, \Phi}}. \end{aligned} $$

It follows that

$$ \begin{aligned} \frac{\kappa \int _{\Omega}\Phi ( \vert \nabla u \vert )}{ \Vert u \Vert _{1, \Phi}} &= \frac{\kappa \int _{\Omega}\Phi ( \vert \nabla u \vert )}{ \vert \nabla u \vert _{L^{\Phi}}+ \vert u \vert _{L^{\Phi}}} \\ &\ge \frac{\frac{\kappa}{2}\min \{ \Vert \nabla u \Vert _{L^{\Phi}}^{\kappa}, \Vert \nabla u \Vert _{L^{\Phi}}^{s}\}+\frac{\kappa}{2}\min \{ \Vert \frac{u}{d} \Vert _{L^{\Phi}}^{\kappa}, \Vert \frac{u}{d} \Vert _{L^{\Phi}}^{s}\}}{ \vert \nabla u \vert _{L^{\Phi}}+ \vert u \vert _{L^{\Phi}}} \\ &=\frac{\kappa}{2} \frac{\min \{ \Vert \nabla u \Vert _{L^{\Phi}}^{\kappa}, \Vert \nabla u \Vert _{L^{\Phi}}^{s}\}+\min \{ \Vert \frac{u}{d} \Vert _{L^{\Phi}}^{\kappa}, \Vert \frac{u}{d} \Vert _{L^{\Phi}}^{s}\}}{ \vert \nabla u \vert _{L^{\Phi}}+ \vert u \vert _{L^{\Phi}}} \to \infty \end{aligned} $$

if \(\|u\|_{1, \Phi}\to \infty \). Then we have

$$ \begin{aligned} \frac{(B(u), u)}{ \Vert u \Vert _{1, \Phi}}\to \infty \quad \bigl( \Vert u \Vert _{1, \Phi}\to \infty \bigr). \end{aligned} $$

Hence we can conclude that the operator B is coercive.

In the end, we will prove that operator B is pseudomonotone, i.e., if

$$ u_{n}\rightharpoonup u \quad \text{in } W_{0}^{1,\Phi}( \Omega )\cap L^{ \infty}(\Omega )$$

and

$$ \lim_{n\to \infty}\sup \bigl(B(u_{n}), (u_{n}-u) \bigr)\le 0,$$

then

$$ \lim_{n\to \infty}\inf \bigl(B(u_{n}), (u_{n}-w)\bigr)\ge \bigl(B(u), (u-w)\bigr), \quad \forall w\text{ in } W_{0}^{1,\Phi}(\Omega )\cap L^{\infty}( \Omega ). $$
(3.2)

From

$$ \int _{\Omega}\bigl[h_{1}\bigl(\zeta (u_{n},x)\bigr)J_{1}(u_{n})+g\bigl(\zeta (u_{n},x)\bigr)J_{2}(u_{n})- \gamma (u_{n},x)\bigr](u_{n}-u)\to 0 $$

and

$$ \limsup_{n\to \infty}\bigl(B(u_{n}), (u_{n}-u) \bigr)\le 0, $$

we obtain

$$ \limsup_{n\to \infty} \int _{\Omega}\rho \bigl( \vert \nabla u_{n} \vert \bigr) \bigl( \nabla u_{n}\cdot \nabla (u_{n}-u)\bigr)\le 0. $$
(3.3)

From Lemma 3.1 in [12], we infer

$$ \begin{aligned} \bigl\Vert \nabla (u_{n}-u) \bigr\Vert _{L^{\Phi}}\le \int _{\Omega}\Phi \bigl( \bigl\vert \nabla (u_{n}-u) \bigr\vert \bigr). \end{aligned} $$
(3.4)

From Lemma 2.5, we can obtain a \(k_{0}>0\) such that

$$ \begin{aligned} &\Phi \bigl( \bigl\vert \nabla (u_{n}-u) \bigr\vert \bigr) \\ &\quad\le \frac{[\Phi ( \vert \nabla u_{n} \vert )+\Phi ( \vert \nabla u \vert )]^{\frac{1}{\kappa +1}}}{k_{0}^{\frac{\kappa}{\kappa +1}}} \\ &\quad\quad {}\times \bigl[\rho \bigl( \vert \nabla u_{n} \vert \bigr) \bigl(\nabla u_{n}\cdot \nabla (u_{n}-u)\bigr)- \rho \bigl( \vert \nabla u \vert \bigr) \bigl(\nabla u\cdot \nabla (u_{n}-u) \bigr)\bigr]^{ \frac{\kappa}{\kappa +1}}, \end{aligned} $$
(3.5)

that is,

$$ \begin{aligned} & \int _{\Omega}\Phi \bigl( \bigl\vert \nabla (u_{n}-u) \bigr\vert \bigr) \\ &\quad\le \int _{\Omega} \biggl\{ \frac{[\Phi ( \vert \nabla u_{n} \vert )+\Phi ( \vert \nabla u \vert )]^{\frac{1}{\kappa +1}}}{k_{0}^{\frac{\kappa}{\kappa +1}}} \\ &\quad \quad {}\times \bigl[\rho \bigl( \vert \nabla u_{n} \vert \bigr) \bigl( \nabla u_{n}\cdot \nabla (u_{n}-u)\bigr)- \rho \bigl( \vert \nabla u \vert \bigr) \bigl(\nabla u\cdot \nabla (u_{n}-u) \bigr)\bigr]^{ \frac{\kappa}{\kappa +1}} \biggr\} \\ &\quad\le \biggl\{ \int _{\Omega} \biggl[ \frac{[\Phi ( \vert \nabla u_{n} \vert )+\Phi ( \vert \nabla u \vert )]^{\frac{1}{\kappa +1}}}{k_{0}^{\frac{\kappa}{\kappa +1}}} \biggr]^{\kappa +1} \biggr\} ^{\frac{1}{\kappa +1}} \\ &\quad\quad {}\times \biggl\{ \int _{\Omega}\bigl[\rho \bigl( \vert \nabla u_{n} \vert \bigr) \bigl(\nabla u_{n} \cdot \nabla (u_{n}-u) \bigr)-\rho \bigl( \vert \nabla u \vert \bigr) \bigl(\nabla u\cdot \nabla (u_{n}-u)\bigr)\bigr] \biggr\} ^{\frac{\kappa}{\kappa +1}}. \end{aligned} $$
(3.6)

Since \(u_{n}\rightharpoonup u\), we have

$$ \int _{\Omega}\rho \bigl( \vert \nabla u \vert \bigr) \bigl( \nabla u\cdot \nabla (u_{n}-u)\bigr) \to 0, $$

which, together with (3.3), guarantees that

$$ \int _{\Omega}\rho \bigl( \vert \nabla u_{n} \vert \bigr) \bigl(\nabla u_{n}\cdot \nabla (u_{n}-u)\bigr)- \rho \bigl( \vert \nabla u \vert \bigr) \bigl(\nabla u\cdot \nabla (u_{n}-u)\bigr)\to 0\quad \text{as } n \to +\infty . $$
(3.7)

From (3.5), (3.6), and (3.7), we have

$$ \int _{\Omega}\Phi \bigl( \bigl\vert \nabla (u_{n}-u) \bigr\vert \bigr)\to 0, $$

that is,

$$ \bigl\Vert \nabla (u_{n}-u) \bigr\Vert _{L^{\Phi}}\to 0. $$

Therefore,

$$ \begin{aligned} \Vert u_{n}-u \Vert _{1, \Phi}= \Vert u_{n}-u \Vert _{L^{\Phi}}+ \bigl\Vert \nabla (u_{n}-u) \bigr\Vert _{L^{ \Phi}}\to 0, \end{aligned} $$

which implies that (3.2) is true.

According to Lemma 2.2.2 in [21], there is a \(u\in W_{0}^{1,\Phi}(\Omega )\cap L^{\infty}(\Omega )\) such that for \(\forall w\in W_{0}^{1,\Phi}(\Omega )\),

$$ \bigl(B(u), w\bigr)=0. $$

Therefore, we know that u is a (weak) solution of problem (3.1).

Claim 2. We show that the solution u of problem (3.1) obtained above is a solution of (1.1).

We shall prove that

$$ \begin{aligned} \underline{w}_{*}\le u\le \overline{w}^{*} \quad \text{in } \Omega . \end{aligned} $$
(3.8)

Choosing \(w=(u-\overline{w}^{*})_{+}\) as a test function, we have

$$ \begin{aligned} \int _{\Omega}-\Delta _{\Phi}u\bigl(u-\overline{w}^{*} \bigr)_{+} &= \int _{\Omega}\bigl[H\bigl(x, u, h_{1}\bigl(\zeta (u,x) \bigr), h_{2}\bigl(\zeta (u,x)\bigr)\bigr)- \gamma (u,x)\bigr]\bigl(u- \overline{w}^{*}\bigr)_{+} \\ &= \int _{\Omega}\bigl[h_{1}\bigl(\zeta (u,x) \bigr)J_{1}(u)+ h_{2}\bigl(\zeta (u,x)\bigr)J_{2}(u)- \gamma (u,x)\bigr]\bigl(u-\overline{w}^{*}\bigr)_{+}. \end{aligned} $$
(3.9)

Define

$$ \Omega _{1}:=\bigl\{ x\in \Omega \mid u>\overline{w}^{*} \bigr\} . $$

Then

$$ \begin{aligned} & \int _{\Omega}\bigl[h_{1}\bigl(\zeta (u,x) \bigr)J_{1}(u)+ h_{2}\bigl(\zeta (u,x)\bigr)J_{2}(u)- \gamma (u,x)\bigr]\bigl(u-\overline{w}^{*}\bigr)_{+} \\ &\quad= \int _{\Omega _{1}}+ \int _{\Omega -\Omega _{1}}\bigl[h_{1}\bigl(\zeta (u,x) \bigr)J_{1}(u)+ h_{2}\bigl(\zeta (u,x)\bigr)J_{2}(u)- \gamma (u,x)\bigr]\bigl(u-\overline{w}^{*}\bigr)_{+} \\ &\quad= \int _{\Omega _{1}}\bigl[h_{1}\bigl(\zeta (u,x) \bigr)J_{1}(u)+ h_{2}\bigl(\zeta (u,x)\bigr)J_{2}(u)- \gamma (u,x)\bigr]\bigl(u-\overline{w}^{*}\bigr)_{+}+0 \\ &\quad= \int _{\Omega _{1}}\bigl[h_{1}\bigl(\overline{w}^{*} \bigr)J_{1}(u)+ h_{2}\bigl( \overline{w}^{*} \bigr)J_{2}(u)-\bigl(u-\overline{w}^{*}\bigr)_{+}^{\nu} \bigr]\bigl(u- \overline{w}^{*}\bigr)_{+}. \end{aligned} $$
(3.10)

Since Ψ and Λ are increasing, from Lemma 2.1 and \(|\zeta (u,x)|\leq \overline{w}^{*}\), we have

$$ \biggl\{ \varsigma >0\Bigm| \int _{\Omega} \Psi \biggl( \frac{ \vert \zeta (u,x) \vert }{\varsigma} \biggr) \le 1 \biggr\} \supseteq \biggl\{ \varsigma >0\Bigm| \int _{\Omega} \Psi \biggl( \frac{\overline{w}^{*}}{\varsigma} \biggr) \le 1 \biggr\} $$

and

$$ \biggl\{ \varsigma >0\Bigm| \int _{\Omega} \Lambda \biggl( \frac{ \vert \zeta (u,x) \vert }{\varsigma} \biggr) \le 1 \biggr\} \supseteq \biggl\{ \varsigma >0\Bigm| \int _{\Omega} \Lambda \biggl( \frac{\overline{w}^{*}}{\varsigma} \biggr) \le 1 \biggr\} , $$

which implies that

$$ J_{1}\bigl(\zeta (u,x)\bigr)\leq J_{1} \bigl(\overline{w}^{*}\bigr),\qquad J_{2}\bigl(\zeta (u,x)\bigr) \leq J_{2}\bigl(\overline{w}^{*}\bigr). $$
(3.11)

From (3.9), (3.10), and (3.11), we have

$$ \int _{\Omega}-\Delta _{\Phi}u\bigl(u-\overline{w}^{*} \bigr)_{+}\le \int _{ \Omega}\bigl[h_{1}\bigl(\overline{w}^{*} \bigr)J_{1}\bigl(\overline{w}^{*}\bigr)+h_{2} \bigl( \overline{w}^{*}\bigr)J_{2}\bigl(\overline{w}^{*} \bigr)-\bigl(u-\overline{w}^{*}\bigr)_{+}^{ \nu} \bigr]\bigl(u-\overline{w}^{*}\bigr)_{+}. $$

By Definition 2.2, we have

$$ \int _{\Omega}-\Delta _{\Phi}u\bigl(u-\overline{w}^{*} \bigr)_{+}\le \int _{ \Omega}\bigl[-\Delta _{\Phi}\overline{w}^{*}- \bigl(u-\overline{w}^{*}\bigr)_{+}^{\nu}\bigr] \bigl(u- \overline{w}^{*}\bigr)_{+}. $$

Hence

$$ \int _{\Omega}-\Delta _{\Phi}u\bigl(u-\overline{w}^{*} \bigr)_{+}+ \int _{\Omega} \Delta _{\Phi}\overline{w}^{*} \bigl(u-\overline{w}^{*}\bigr)_{+}\le \int _{ \Omega}\bigl[-\bigl(u-\overline{w}^{*} \bigr)_{+}^{\nu +1}\bigr]\le 0, $$

i.e.,

$$ \int _{\Omega}\bigl(\rho \bigl( \vert \nabla u \vert \bigr) \nabla u-\rho \bigl( \bigl\vert \nabla \overline{w}^{*} \bigr\vert \bigr) \nabla \overline{w}^{*}\bigr)\cdot \nabla \bigl(u- \overline{w}^{*}\bigr)_{+}\le \int _{\Omega}\bigl[-\bigl(u-\overline{w}^{*} \bigr)_{+}^{\nu +1}\bigr]\le 0. $$
(3.12)

From Lemma 2.5, there exists a \(k_{0}>0\) such that

$$ \begin{aligned} & \int _{\Omega}\bigl(\rho \bigl( \vert \nabla u \vert \bigr) \nabla u-\rho \bigl( \bigl\vert \nabla \overline{w}^{*} \bigr\vert \bigr)\nabla \overline{w}^{*}\bigr)\cdot \nabla \bigl(u- \overline{w}^{*}\bigr)_{+} \\ &\quad \ge \int _{\Omega}k_{0} \frac{\Phi ( \vert \nabla u-\nabla \overline{w}^{*} \vert )^{\frac{\kappa +1}{\kappa}}}{(\Phi ( \vert \nabla u \vert )+\Phi ( \vert \nabla \overline{w}^{*} \vert ))^{\frac{1}{\kappa}}} \frac{\nabla (u-\overline{w}^{*})_{+}}{\nabla (u-\overline{w}^{*})}. \end{aligned} $$
(3.13)

Since

$$ \int _{\Omega}k_{0} \frac{\Phi ( \vert \nabla u-\nabla \overline{w}^{*} \vert )^{\frac{\kappa +1}{\kappa}}}{(\Phi ( \vert \nabla u \vert )+\Phi ( \vert \nabla \overline{w}^{*} \vert ))^{\frac{1}{\kappa}}} \frac{\nabla (u-\overline{w}^{*})_{+}}{\nabla (u-\overline{w}^{*})}= \int _{\Omega _{1}}k_{0} \frac{\Phi ( \vert \nabla u-\nabla \overline{w}^{*} \vert )^{\frac{\kappa +1}{\kappa}}}{(\Phi ( \vert \nabla u \vert )+\Phi ( \vert \nabla \overline{w}^{*} \vert ))^{\frac{1}{\kappa}}} $$

and Φ is continuous, we obtain that there is an \(M_{1}>0\) such that

$$ \int _{\Omega _{1}}k_{0} \frac{\Phi ( \vert \nabla u-\nabla \overline{w}^{*} \vert )^{\frac{\kappa +1}{\kappa}}}{(\Phi ( \vert \nabla u \vert )+\Phi ( \vert \nabla \overline{w}^{*} \vert ))^{\frac{1}{\kappa}}}= \frac{k_{0}}{M_{1}} \int _{\{u>\overline{w}^{*}\}}\Phi \bigl( \bigl\vert \nabla u- \nabla \overline{w}^{*} \bigr\vert \bigr)^{\frac{\kappa +1}{\kappa}}. $$
(3.14)

From (3.12), (3.13), and (3.14), we have

$$ \int _{\{u>\overline{w}^{*}\}}\Phi \bigl( \bigl\vert \nabla u-\nabla \overline{w}^{*} \bigr\vert \bigr)^{ \frac{\kappa +1}{\kappa}}\le 0. $$

From Lemma 2.2 in [11] and [14], we obtain

$$ \int _{\{u>\overline{w}^{*}\}}\Phi \biggl(\frac{ \vert u-\overline{w}^{*} \vert }{d} \biggr)^{\frac{\kappa +1}{\kappa}} \le \int _{\{u>\overline{w}^{*}\}} \Phi \bigl( \bigl\vert \nabla u-\nabla \overline{w}^{*} \bigr\vert \bigr)^{\frac{\kappa +1}{\kappa}} \le 0, $$

where \(d=\mathrm{diam}(\Omega )\). Therefore, we can conclude that

$$ \bigl\vert \bigl\{ u>\overline{w}^{*}\bigr\} \bigr\vert =0, $$

and then \(u\le \overline{w}^{*}\).

A similar argument shows that \(u\geq \underline{w}_{*}\).

Therefore, (3.8) is true and thus u is a solution of problem (1.1).

The proof is completed. □

Proof of Theorem 2.7

In order to get positive solutions of problem (1.3), we study the following problem:

$$ \textstyle\begin{cases} -\Delta _{\Phi}u=(u+\frac{1}{n})^{\beta} \Vert u \Vert _{L^{\Psi}}^{\alpha}, \quad x\in \Omega , \\ u=0, \quad x\in \partial \Omega , \end{cases} $$
(3.15)

for \(n\geq 1\). We will use Theorem 2.6 to discuss problem (3.15).

First, we will construct a supersolution of problem (3.15).

From Lemma 2.4, problem (2.1) has a unique positive \(z_{\lambda}\in W_{0}^{1, \Psi}(\Omega )\) which satisfies

$$ \begin{aligned} 0< z_{\lambda}(x)\le K\lambda ^{\frac{1}{\kappa -1}}, \quad x\in \Omega \end{aligned} $$
(3.16)

for \(\lambda >0\) big enough, where K is independent of λ.

Let \(M=K\lambda ^{\frac{1}{\kappa -1}}\). Then

$$ \begin{aligned} K\lambda ^{\frac{1}{\kappa -1}}< z_{\lambda}(x)+M\le 2K \lambda ^{ \frac{1}{\kappa -1}}, \quad x\in \Omega . \end{aligned} $$

The condition \(0<\alpha <\kappa -1\) implies that there is a \(\lambda >1\) big enough such that

$$ \lambda ^{\frac{\alpha}{\kappa -1}} \Vert 2K \Vert _{L^{\Psi}}^{\alpha}\le \lambda ,\qquad M=K\lambda ^{\frac{1}{\kappa -1}}>1$$

and (3.16) holds. Hence

$$ \biggl(z_{\lambda}+M+\frac{1}{n}\biggr)^{\beta} \Vert z_{\lambda}+M \Vert _{L^{\Psi}}^{ \alpha}\le \Vert z_{\lambda}+M \Vert _{L^{\Psi}}^{\alpha}\le \lambda ^{ \frac{\alpha}{\kappa -1}} \Vert 2K \Vert _{L^{\Psi}}^{\alpha}\le \lambda $$

and

$$ -\Delta _{\Phi}(z_{\lambda}+M)= -\Delta _{\Phi}z_{\lambda}= \lambda \geq \biggl(z_{\lambda}+M+\frac{1}{n} \biggr)^{\beta} \Vert z_{\lambda}+M \Vert _{L^{ \Psi}}^{\alpha}. $$

Therefore, \(z_{\lambda}+M\) is a supersolution of (3.15).

Second, we will construct a positive subsolution \(\underline{u}_{*}\) of problem (3.15).

Define \(d(x):=\mathrm{dist}(x,\partial \Omega )\), then by a direct calculation one can deduce that \(|\nabla d(x)|=1\). Because Ω is \(C^{2}\), we can get a constant \(\tau >0\) such that \(d\in C^{2}(\overline{\Omega _{3\tau}})\) with \(\overline{\Omega _{3\tau}}:=\{x\in \overline{\Omega}:d(x)\le 3\tau \} \) (see [9, 10]). Let \(\varpi \in (0, \tau )\). Define

$$ \eta (x):= \textstyle\begin{cases} e^{\vartheta d(x)}-1, &\text{for } d(x)< \varpi , \\ e^{\vartheta \varpi}-1+\int _{\varpi}^{d(x)}\vartheta e^{\vartheta d(x)}( \frac{2\tau -t}{2\tau -\varpi})^{\frac{s}{\kappa -1}}\,dt,&\text{for } \varpi \le d(x)\le 2\tau , \\ e^{\vartheta \varpi}-1+\int _{\varpi}^{2\tau}ke^{\vartheta d(x)}( \frac{2\tau -t}{2\tau -\varpi})^{\frac{s}{\kappa -1}}\,dt,&\text{for } 2\tau < d(x), \end{cases} $$

where \(\vartheta >0\) is an arbitrary number. Direct computations imply that

$$ -\Delta _{\Phi}(\mu \eta ) = \textstyle\begin{cases} -\vartheta \Theta (x)\frac{d}{dt}(\rho (t)t)|_{t=\Theta (x)} - \rho (\Theta (x))\Theta (x)\Delta d ,&\text{for } d(x)< \varpi , \\ \frac{\Theta _{0}(\frac{s}{\kappa -1})\chi (x)^{\frac{s}{\kappa -1}-1}}{2\tau -\varpi} \frac{d}{dt}(\rho (t)t)|_{t=\Theta _{0}\chi (x)^{ \frac{s}{\kappa -1}}} \\ \quad {}-\rho (\Theta _{0}\chi (x)^{\frac{s}{\kappa -1}})\Theta _{0}\chi (x)^{ \frac{s}{\kappa -1}}\Delta d,&\text{for } \varpi \le d(x) \le 2\tau , \\ 0, &\text{for } 2\tau < d(x), \end{cases} $$

with \(\Theta (x)=\mu \vartheta e^{\vartheta d(x)}\), \(\Theta _{0}=\mu \vartheta e^{\vartheta \varpi}\), and \(\chi (x)=\frac{2\tau -d(x)}{2\tau -\varpi}\) for all \(\mu >0\).

There are three cases: (1) \(d(x)<\varpi \); (2) \(\varpi < d(x)<2\tau \); and (3) \(d(x)>2\tau \).

(1) We consider the case \(d(x)<\varpi \).

Since Δd is a bounded function near Ω and \(\kappa >1\), there is a ϑ large enough such that

$$ \begin{aligned} -\Delta _{\Phi}(\mu \eta )&= - \mu \vartheta ^{2}e^{\vartheta d(x)} \frac{d}{dt}\bigl(\rho (t)t \bigr)\bigg|_{t=\mu \vartheta e^{\vartheta d(x)}}- \rho \bigl(\mu \vartheta e^{\vartheta d(x)}\bigr)\mu \vartheta e^{\vartheta d(x)} \Delta d \\ &\le -\vartheta ^{2}\mu e^{\vartheta d(x)}(\kappa -1)\rho \bigl(\mu \vartheta e^{\mu \vartheta e^{\vartheta d(x)}}\bigr)-\rho \bigl(\mu \vartheta e^{ \vartheta d(x)}\bigr) \mu \vartheta e^{\vartheta d(x)}\Delta d \\ &=\mu \vartheta e^{\vartheta d(x)}\rho \bigl(\mu \vartheta e^{\vartheta d(x)}\bigr) \bigl(- \vartheta (\kappa -1)-\Delta d\bigr) \\ &\le 0, \end{aligned} $$

which implies that

$$ \begin{aligned} -\Delta _{\Phi}(\mu \eta )\le 0\le (\mu \eta )^{\beta} \vert \mu \eta \vert _{L^{ \Psi}}^{\alpha}, \end{aligned} $$

when \(d(x)<\varpi \) and ϑ is large enough.

(2) We consider the case \(\varpi < d(x)<2\delta \).

From the condition \((\rho _{3})\) and Lemma 2.3, we have

$$ \begin{aligned} &\mu \vartheta e^{\vartheta \varpi} \biggl(\frac{s}{\kappa -1} \biggr) \biggl( \frac{2\tau -d(x)}{2\tau -\varpi} \biggr)^{\frac{s}{\kappa -1}-1} \biggl( \frac{1}{2\tau -\varpi} \biggr)\frac{d}{dt}\bigl(\rho (t)t \bigr)\bigg|_{t=\mu \vartheta e^{\vartheta \varpi} (\frac{2\tau -d(x)}{2\tau -\varpi} )^{\frac{s}{\kappa -1}}} \\ &\quad\le \mu \vartheta e^{\vartheta \varpi} \biggl(\frac{s}{\kappa -1} \biggr) \biggl( \frac{2\tau -d(x)}{2\tau -\varpi} \biggr)^{\frac{s}{\kappa -1}-1} \biggl(\frac{s-1}{2\tau -\varpi} \biggr)\rho \biggl(\mu \vartheta e^{ \vartheta \varpi} \biggl(\frac{2\tau -d(x)}{2\tau -\varpi} \biggr)^{ \frac{s}{\kappa -1}} \biggr) \\ &\quad\le \biggl(\frac{s}{\kappa -1} \biggr) \biggl(\frac{s-1}{2\tau -\varpi} \biggr) \frac{s\Phi (\mu \vartheta e^{\vartheta \varpi} (\frac{2\tau -d(x)}{2\tau -\varpi} )^{\frac{s}{\kappa -1}} )}{\mu \vartheta e^{\vartheta \varpi} (\frac{2\delta -d(x)}{2\tau -\varpi} )^{\frac{s}{\kappa -1}}} \frac{1}{\frac{2\tau -d(x)}{2\tau -\varpi}} \\ &\quad\le \biggl(\frac{s^{2}}{\kappa -1} \biggr) \biggl(\frac{s-1}{2\tau -\varpi} \biggr)\max \biggl\{ \bigl(\mu \vartheta e^{\vartheta \varpi}\bigr)^{s-1} \biggl( \frac{2\tau -d(x)}{2\tau -\varpi} \biggr)^{s(\frac{s}{\kappa -1})-( \frac{s}{\kappa -1}+1)}, \\ &\quad\quad \bigl(\mu \vartheta e^{ \vartheta \varpi}\bigr)^{\kappa -1} \biggl( \frac{2\tau -d(x)}{2\tau -\varpi} \biggr)^{\kappa (\frac{s}{\kappa -1})-(\frac{s}{\kappa -1}+1)} \biggr\} \Phi (1). \end{aligned} $$
(3.17)

Now \(s, \kappa >1\) implies \(\kappa (\frac{s}{\kappa -1} )-s (\frac{s}{\kappa -1}+1 ), s (\frac{s}{\kappa -1} )-s (\frac{s}{\kappa -1}+1 )>0\), which, together with \(0\le \frac{2\tau -d(x)}{2\tau -\varpi}\le 1\) and (3.17), guarantees that

$$\begin{aligned} &\mu \vartheta e^{\vartheta \varpi} \biggl(\frac{s}{\kappa -1} \biggr) \biggl( \frac{2\tau -d(x)}{2\tau -\varpi} \biggr)^{\frac{s}{\kappa -1}-1} \biggl( \frac{1}{2\tau -\varpi} \biggr)\frac{d}{dt}\bigl(\rho (t)t \bigr)\bigg|_{t=\mu \vartheta e^{\vartheta \varpi} ( \frac{2\delta -d(x)}{2\tau -\varpi} )^{\frac{s}{\kappa -1}}} \\ &\quad \le \biggl(\frac{s^{2}}{\kappa -1} \biggr) \biggl(\frac{s-1}{2\tau -\varpi} \biggr)\Phi (1) \max \bigl\{ \bigl(\mu \vartheta e^{\vartheta \varpi}\bigr)^{s-1}, \bigl(\mu \vartheta e^{\vartheta \varpi}\bigr)^{\kappa -1}\bigr\} \\ &\quad =C_{1} \biggl(\frac{1}{2\tau -\varpi} \biggr)\max \bigl\{ \bigl(\mu \vartheta e^{ \vartheta \varpi}\bigr)^{s-1}, \bigl(\mu \vartheta e^{\vartheta \varpi}\bigr)^{ \kappa -1}\bigr\} , \end{aligned}$$
(3.18)

where \(C_{1}=\frac{s^{2}(s-1)\Phi (1)}{\kappa -1}\) is a constant independent of μ and ϑ. Similarly, one has

$$ \begin{aligned} & \biggl\vert \rho \biggl(\mu \vartheta e^{\vartheta \varpi} \biggl( \frac{2\tau -d(x)}{2\tau -\varpi} \biggr)^{\frac{s}{\kappa -1}} \biggr)\mu \vartheta e^{\vartheta \varpi} \frac{(2\tau -d(x))^{\frac{s}{\kappa -1}}}{(2\tau -\varpi )^{\frac{s}{\kappa -1}}} \Delta d \biggr\vert \\ &\quad\le \rho \biggl(\mu \vartheta e^{\vartheta \varpi} \biggl( \frac{2\tau -d(x)}{2\tau -\varpi} \biggr)^{\frac{s}{r-1}}\biggr)\mu \vartheta e^{ \vartheta \varpi} \biggl( \frac{2\tau -d(x)}{2\tau -\varpi} \biggr)^{ \frac{s}{\kappa -1}}\sup_{\overline{\Omega _{3\tau}}} \vert \Delta d \vert \\ &\quad\le C \frac{\Phi (\mu \vartheta e^{\vartheta \varpi} (\frac{2\tau -d(x)}{2\tau -\varpi} )^{\frac{s}{\kappa -1}} )}{\mu \vartheta e^{\vartheta \varpi} (\frac{2\tau -d(x)}{2\tau -\varpi} )^{\frac{s}{\kappa -1}}} \\ &\quad\le C\max \biggl\{ \bigl(\mu \vartheta e^{\vartheta \varpi}\bigr)^{s-1} \biggl( \frac{2\tau -d(x)}{2\tau -\varpi} \biggr)^{s(\frac{s}{\kappa -1})-( \frac{s}{\kappa -1}+1)}, \\ &\quad\quad \bigl(\mu \vartheta e^{\vartheta \varpi}\bigr)^{\kappa -1} \biggl( \frac{2\tau -d(x)}{2\tau -\varpi} \biggr)^{\kappa ( \frac{s}{\kappa -1})-(\frac{s}{\kappa -1}+1)} \biggr\} \\ &\quad\le C_{2}\max \bigl\{ \bigl(\mu \vartheta e^{\vartheta \varpi} \bigr)^{s-1}, \bigl( \mu \vartheta e^{\vartheta \varpi}\bigr)^{\kappa -1} \bigr\} , \end{aligned} $$
(3.19)

where \(C_{2}\) is a constant independent of ϖ, ϑ, and μ. Thus from (3.18) and (3.19) we have

$$ \begin{aligned} -\Delta _{\Phi}u\le \max \biggl\{ \frac{C_{1}}{2\tau -\varpi}, C_{2} \biggr\} \max \bigl\{ \bigl(\mu \vartheta e^{\vartheta \varpi}\bigr)^{s-1}, \bigl(\mu \vartheta e^{\vartheta \varpi} \bigr)^{\kappa -1} \bigr\} , \end{aligned} $$

when \(\varpi < d(x)<2\tau \).

Let \(\varpi =\frac{ln2}{\vartheta }\) and \(\mu =e^{-\vartheta }\), then \(e^{\vartheta \varpi}=2\). Since

$$ \begin{aligned} \eta (x)&=e^{\vartheta \varpi}-1+ \int _{\varpi}^{d(x)}\vartheta e^{ \vartheta d(x)} \biggl( \frac{2\tau -t}{2\tau -\varpi} \biggr)^{ \frac{s}{\kappa -1}}\,dt \\ &>2-1+2\vartheta \int _{\varpi}^{d(x)} \biggl( \frac{2\tau -t}{2\tau -\varpi} \biggr)^{\frac{s}{\kappa -1}}\,dt \\ &=1+\vartheta C_{3} \\ &\ge 1, \end{aligned} $$

where \(C_{3}>0\) is a constant, we have that when μ is small enough and n is large enough,

$$\begin{aligned} \biggl(\mu \eta +\frac{1}{n} \biggr)^{\beta} \vert \mu \eta \vert _{L^{\Psi}}^{\alpha}& \ge \vert \mu \eta \vert _{L^{\Psi}}^{\alpha} \\ &={\inf}^{\alpha} \biggl\{ \varsigma >0: \int _{\Omega}\Psi \biggl( \frac{ \vert \mu \eta \vert }{\varsigma} \biggr)< 1 \biggr\} \\ &={\inf}^{\alpha} \biggl\{ \tau \mu >0: \int _{\Omega}\Psi \biggl( \frac{ \vert \mu \eta \vert }{\tau \mu} \biggr)< 1 \biggr\} \\ &=\mu ^{\alpha}{\inf}^{\alpha} \biggl\{ \tau >0: \int _{\Omega}\Psi \biggl( \frac{ \vert \eta \vert }{\tau} \biggr)< 1 \biggr\} \\ &\ge \mu ^{\alpha}C_{4}, \end{aligned}$$

where \(C_{4}>0\) is a constant independent of \(\vartheta >0\).

Since \(0<\alpha <\kappa -1\), we have the result

$$ \begin{aligned} \lim_{\vartheta \to +\infty} \frac{\vartheta ^{\kappa -1}}{e^{\vartheta (\kappa -1-\alpha )}}=0. \end{aligned} $$

In view of

$$ \begin{aligned} -\Delta _{\Phi}(\mu \eta )\le \max \biggl\{ \frac{C_{1}}{2\tau -\varpi}, C_{2} \biggr\} \max \bigl\{ 2^{s-1}, 2^{\kappa -1}\bigr\} \biggl( \frac{\vartheta }{e^{\vartheta }} \biggr)^{\kappa -1}, \end{aligned} $$

choose a \(\vartheta _{0}>0\) large enough such that

$$ \begin{aligned} C_{4}\ge \max \biggl\{ \frac{C_{1}}{2\tau -\frac{\ln 2}{\vartheta }}, C_{2} \biggr\} \max \bigl\{ 2^{s-1}, 2^{\kappa -1}\bigr\} \biggl( \frac{\vartheta ^{\kappa -1}}{e^{\vartheta (\kappa -1-\alpha )}} \biggr) \end{aligned} $$

for all \(\vartheta \ge \vartheta _{0}\).

Thus,

$$ \begin{aligned} -\Delta _{\Phi}(\mu \eta )\le \biggl(\mu \eta +\frac{1}{n} \biggr)^{\beta} \vert \mu \eta \vert _{L^{\Psi}}^{\alpha} \end{aligned} $$

in the case \(\varpi < d(x)<2\tau \) for \(\vartheta >0\) large enough.

(3) We consider the case \(d(x)>2\tau \).

Obviously,

$$ \begin{aligned} -\Delta _{\Phi}(\mu \eta )=0\le \biggl(\mu \eta +\frac{1}{n} \biggr)^{ \beta} \vert \mu \eta \vert _{L^{\Psi}}^{\alpha}. \end{aligned} $$

It is obvious that \(\underline{w}_{*}\leq \overline{w}^{*}\) if M is large enough and μ is small enough. And \((\underline{w}_{*},\overline{w}^{*})\) is a sub-supersolution pair of problem (3.15). Now Theorem 2.6 guarantees that problem (3.15) has a solution \(u_{n}\) which satisfies \(0<\mu \eta \le u_{n}\le z_{\lambda}+M\).

Now we consider the set \(\{u_{n}\}\).

From Lemma 2.2 in [12], one has that \(\|u\|_{1, \Phi}\) and \(|\!|\!|\nabla u|\!|\!|_{L^{\Phi}}\) defined on \(W_{0}^{1, \Phi}\) are equivalent. And from the proof of the coercivity of the operator B, we know that if \(|\!|\!|\nabla u|\!|\!|_{L^{\Phi}}>1\), then

$$ \int _{\Omega}\Phi \bigl( \vert \nabla u \vert \bigr)\ge |\!|\!|\nabla u|\!|\!|_{L^{\Phi}}, $$

that is,

$$ \int _{\Omega}\Phi \bigl( \vert \nabla u \vert \bigr)\ge \Vert u \Vert _{1, \Phi}, $$

when \(\|u\|_{1, \Phi}>1\).

If \(\|u_{n}\|_{1, \Phi}\le 1\), then \({u_{n}}\) is bounded in \(W_{0}^{1, \Phi}(\Omega )\) naturally.

If \(\|u_{n}\|_{1, \Phi}>1\), then

$$ \Vert u_{n} \Vert _{1, \Phi}\le \int _{\Omega}\Phi \bigl( \vert \nabla u_{n} \vert \bigr). $$

By the condition \((\rho _{3})'\) and due to

$$ \int _{\Omega}-\Delta _{\Phi}u_{n}u_{n}= \int _{\Omega}u_{n} \biggl(u_{n}+ \frac{1}{n} \biggr)^{\beta} \Vert u_{n} \Vert _{L^{\Psi}}^{\alpha}, $$

we have

$$ \kappa \int _{\Omega}\Phi \bigl( \vert \nabla u_{n} \vert \bigr)\le \int _{\Omega}\phi \bigl( \vert \nabla u_{n} \vert \bigr) \vert \nabla u_{n} \vert ^{2}= \int _{\Omega}u_{n} \biggl(u_{n}+ \frac{1}{n} \biggr)^{\beta} \Vert u_{n} \Vert _{L^{\Psi}}^{\alpha}, $$

which, together with \(\alpha \ge 0\), \(-1<\beta <0 \), gives

$$ \int _{\Omega}\Phi \bigl( \vert \nabla u_{n} \vert \bigr)\le \frac{1}{\kappa} \int _{\Omega} \overline{w}^{* \beta +1} \bigl\Vert \overline{w}^{*} \bigr\Vert _{L^{\Psi}}^{\alpha}, $$

that is,

$$ \Vert u_{n} \Vert _{1, \Phi}\le \frac{1}{\kappa} \int _{\Omega}\overline{w}^{* \beta +1} \bigl\Vert \overline{w}^{*} \bigr\Vert _{L^{\Psi}}^{\alpha}. $$

Therefore, \(\{u_{n}\}\) is bounded in \(W_{0}^{1, \Phi}(\Omega )\).

Since \(W_{0}^{1, \Phi}(\Omega )\) is reflexive, \(\{u_{n}\}\) has weakly convergent subsequences in \(W_{0}^{1,\Phi}(\Omega )\cap L^{\infty}(\Omega )\), and we still use \(u_{n}\) to denote its subsequence. From the analysis in [3], we have

$$ \begin{aligned} u_{n}\rightharpoonup u\quad \text{in } W_{0}^{1,\Phi}(\Omega )\cap L^{ \infty}(\Omega ) \end{aligned} $$

and

$$ \begin{aligned} u_{n}(x)\stackrel{\text{a.e.}}{\to} u(x), \quad x\in \Omega . \end{aligned} $$

Since

$$ \underline{w}_{*}\le u_{n}\le \overline{w}^{*}, \quad x\in \Omega ,$$

Lebesgue theorem implies

$$ u_{n} \to u \quad \text{in } L^{q}(\Omega ) \ \forall q\in [1, + \infty ).$$
(3.20)

Since \(u_{n}\) is a (weak) solution of (3.15) for all n N + , we have

$$ \begin{aligned} \int _{\Omega}-\Delta _{\Phi}u_{n}w= \int _{\Omega} \biggl(u_{n}+ \frac{1}{n} \biggr)^{\beta} \Vert u_{n} \Vert _{L^{\Psi}}^{\alpha}w, \end{aligned} $$

for all \(w\in W_{0}^{1,\Phi}(\Omega )\).

Denoting \(w=u_{n}-u\), we have

$$ \begin{aligned} \int _{\Omega}-\Delta _{\Phi}u_{n}(u_{n}-u)= \int _{\Omega} \biggl(u_{n}+ \frac{1}{n} \biggr)^{\beta} \Vert u_{n} \Vert _{L^{\Psi}}^{\alpha}(u_{n}-u). \end{aligned} $$

Since

$$ \biggl(u_{n}+\frac{1}{n} \biggr)^{\beta}\le \underline{w}_{*}^{\beta}, \quad x\in \Omega , $$

one has

$$ \begin{aligned} \int _{\Omega} \biggl(u_{n}+\frac{1}{n} \biggr)^{\beta} \Vert u_{n} \Vert _{L^{\Psi}}^{ \alpha} \vert u_{n}-u \vert &\le \int _{\Omega}\underline{w}_{*}^{\beta} \vert u_{n}-u \vert \Vert u_{n} \Vert _{L^{ \Psi}}^{\alpha} \\ &\le \biggl[ \int _{\Omega} \bigl(\underline{w}_{*}^{\beta} \Vert u_{n} \Vert _{L^{ \Psi}}^{\alpha} \bigr)^{p} \biggr]^{\frac{1}{p}} \biggl[ \int _{\Omega} \vert u_{n}-u \vert ^{q} \biggr]^{\frac{1}{q}}, \end{aligned} $$

where \(p, q>1\), \(\frac{1}{p}+\frac{1}{q}=1\), and \(\beta \in (-1,0)\). From (3.20), we have

$$ \biggl[ \int _{\Omega} \bigl(\underline{w}_{*}^{\beta} \Vert u_{n} \Vert _{L^{\Psi}}^{ \alpha} \bigr)^{p} \biggr]^{\frac{1}{p}} \biggl[ \int _{\Omega} \vert u_{n}-u \vert ^{q} \biggr]^{\frac{1}{q}}\to 0,$$

and so

$$ \begin{aligned} \int _{\Omega} \biggl(u_{n}+\frac{1}{n} \biggr)^{\beta} \Vert u_{n} \Vert _{L^{\Psi}}^{ \alpha} \vert u_{n}-u \vert \to 0\quad \text{as }n\to +\infty , \end{aligned} $$

which implies

$$ \begin{aligned} \int _{\Omega}-\Delta _{\Phi}u_{n}(u_{n}-u) \to 0. \end{aligned} $$

Obviously,

$$ \begin{aligned} \int _{\Omega}-\Delta _{\Phi}u (u_{n}-u)\to 0. \end{aligned} $$
(3.21)

Similar to the previous proof, from (3.4), (3.6), and (3.21), we have

$$ u_{n}\to u\quad \text{in } W_{0}^{1,\Phi}(\Omega ) \cap L^{\infty}( \Omega ),$$

and so

$$ \Vert u_{n} \Vert _{L^{\Psi}}^{\alpha}\to \Vert u \Vert _{L^{\Psi}}^{\alpha}.$$

Therefore, taking the limit as \(n\to \infty \) in (3.15), we have

$$ -\Delta _{\Phi}u=u^{\beta} \Vert u \Vert _{L^{\Psi}}^{\alpha}.$$

The limit value u is just the solution which we are looking for, and it satisfies \(\underline{w}_{*}\le u\le \overline{w}^{*}\), obviously. Therefore, the proof is finished. □

Availability of data and materials

Not applicable.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (62073203).

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This research was funded by the NSFC of China (62073203).

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Methodology and investigation, BY; Formal analysis, XQ. All authors have read and agreed to the published version of the manuscript.

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Correspondence to Baoqiang Yan.

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Qiu, X., Yan, B. A new existence result for some nonlocal problems involving Orlicz spaces and its applications. Bound Value Probl 2022, 62 (2022). https://doi.org/10.1186/s13661-022-01641-x

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Keywords

  • Orlicz–Sobolev space
  • Sub-supersolution
  • Pseudomonotone Operator Theorem
  • Luxemburg norm
  • Φ-Laplace operator