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A new existence result for some nonlocal problems involving Orlicz spaces and its applications
Boundary Value Problems volume 2022, Article number: 62 (2022)
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.
1 Introduction
This paper is concerned with the problem
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, (\(N\ge 3\)) is bounded with \(\partial \Omega \in C^{2}\), \(\Delta _{\Phi}u= \mathrm{div}(\rho (|\nabla u|)\nabla u)\), where
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, 17–20]. 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 \(h_{1}\), \(h_{2}\): are nondecreasing. As its application, the authors discussed the following problem:
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:
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 \(\beta \in (-1,0)\).
2 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
that is,
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
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
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
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
endowed with the norm
Specially,
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
and
then
Lemma 2.4
([10])
Let \(\lambda >0\), let Φ be given by (1.2), and suppose is an admissible domain. Consider the problem
where \(z_{\lambda}\) is the unique solution. Define
If \(\lambda \ge \rho _{0}\), then
and
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
for , \(\zeta \neq 0\).
Theorem 2.6
If the functions 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.
3 Proofs of the main results
Proof of Theorem 2.6
We consider
where
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
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
which implies that
From the Lemma 2.3 and Lemma 2.1 in [12], we have
and
then we deduce
It follows that
if \(\|u\|_{1, \Phi}\to \infty \). Then we have
Hence we can conclude that the operator B is coercive.
In the end, we will prove that operator B is pseudomonotone, i.e., if
and
then
From
and
we obtain
From Lemma 3.1 in [12], we infer
From Lemma 2.5, we can obtain a \(k_{0}>0\) such that
that is,
Since \(u_{n}\rightharpoonup u\), we have
which, together with (3.3), guarantees that
From (3.5), (3.6), and (3.7), we have
that is,
Therefore,
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 \(\forall w\in W_{0}^{1,\Phi}(\Omega )\),
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
Choosing \(w=(u-\overline{w}^{*})_{+}\) as a test function, we have
Define
Then
Since Ψ and Λ are increasing, from Lemma 2.1 and \(|\zeta (u,x)|\leq \overline{w}^{*}\), we have
and
which implies that
From (3.9), (3.10), and (3.11), we have
By Definition 2.2, we have
Hence
i.e.,
From Lemma 2.5, there exists a \(k_{0}>0\) such that
Since
and Φ is continuous, we obtain that there is an \(M_{1}>0\) such that
From (3.12), (3.13), and (3.14), we have
From Lemma 2.2 in [11] and [14], we obtain
where \(d=\mathrm{diam}(\Omega )\). Therefore, we can conclude that
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:
for \(n\geq 1\). We will use Theorem 2.6 to discuss problem (3.15).
First, we will construct a supersolution u̅ 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
for \(\lambda >0\) big enough, where K is independent of λ.
Let \(M=K\lambda ^{\frac{1}{\kappa -1}}\). Then
The condition \(0<\alpha <\kappa -1\) implies that there is a \(\lambda >1\) big enough such that
and (3.16) holds. Hence
and
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
where \(\vartheta >0\) is an arbitrary number. Direct computations imply that
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
which implies that
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
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
where \(C_{1}=\frac{s^{2}(s-1)\Phi (1)}{\kappa -1}\) is a constant independent of μ and ϑ. Similarly, one has
where \(C_{2}\) is a constant independent of ϖ, ϑ, and μ. Thus from (3.18) and (3.19) we have
when \(\varpi < d(x)<2\tau \).
Let \(\varpi =\frac{ln2}{\vartheta }\) and \(\mu =e^{-\vartheta }\), then \(e^{\vartheta \varpi}=2\). Since
where \(C_{3}>0\) is a constant, we have that when μ is small enough and n is large enough,
where \(C_{4}>0\) is a constant independent of \(\vartheta >0\).
Since \(0<\alpha <\kappa -1\), we have the result
In view of
choose a \(\vartheta _{0}>0\) large enough such that
for all \(\vartheta \ge \vartheta _{0}\).
Thus,
in the case \(\varpi < d(x)<2\tau \) for \(\vartheta >0\) large enough.
(3) We consider the case \(d(x)>2\tau \).
Obviously,
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
that is,
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
By the condition \((\rho _{3})'\) and due to
we have
which, together with \(\alpha \ge 0\), \(-1<\beta <0 \), gives
that is,
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
and
Since
Lebesgue theorem implies
Since \(u_{n}\) is a (weak) solution of (3.15) for all , we have
for all \(w\in W_{0}^{1,\Phi}(\Omega )\).
Denoting \(w=u_{n}-u\), we have
Since
one has
where \(p, q>1\), \(\frac{1}{p}+\frac{1}{q}=1\), and \(\beta \in (-1,0)\). From (3.20), we have
and so
which implies
Obviously,
Similar to the previous proof, from (3.4), (3.6), and (3.21), we have
and so
Therefore, taking the limit as \(n\to \infty \) in (3.15), we have
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. □
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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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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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DOI: https://doi.org/10.1186/s13661-022-01641-x