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

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.

Problem (1.1) was proposed in [10] and generalizes some problems in [3,5,6,8,[17][18][19][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 -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: (1. 3) with the assumption that α, β ≥ 0 with 0 < α + β < κ -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 α ≥ 0, -1 < β < 0, with 0 < α < κ -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 β ∈ (-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 (ρ 3 ) , it is easily to see that the 2 condition is true for (t) (see [10]).
then w * (x) is called a subsolution of problem (1.1).
For more information on L ( ) and its norm, please refer to the literature [10]. Let In addition, and are N-functions satisfying the 2 condition, and they are also nondecreasing on [0, +∞). For an N-function , the corresponding Orlicz-Sobolev space is defined as the Banach space Specially, For their properties, one can refer to the literature [10].

Lemma 2.4 ([10])
Let λ > 0, let be given by (1.2), and suppose ⊂ R N is an admissible domain. Consider the problem where z λ is the unique solution. Define Here C * > 0 and C * > 0 depend on n, s, N , and .
For z λ which is defined in Lemma 2.4, it follows that z λ ∈ C 1 ( ) with z λ > 0 in .

Proofs of the main results
Proof of Theorem 2. 6 We consider We have the following claims: where ρ satisfies (ρ 1 ), (ρ 2 ), and (ρ 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 From the Lemma 2.3 and Lemma 2.1 in [12], we have It follows that Hence we can conclude that the operator B is coercive.
According to Lemma 2.2.2 in [21], there is a u ∈ W 1, 0 ( ) ∩ L ∞ ( ) such that for ∀w ∈ W 1, 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).
A similar argument shows that u ≥ 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 ≥ 1. We will use Theorem 2.6 to discuss problem (3.15).
In view of , choose a ϑ 0 > 0 large enough such that Thus, in the case < d(x) < 2τ for ϑ > 0 large enough.
(3) We consider the case d(x) > 2τ . Obviously, It is obvious that w * ≤ w * if M is large enough and μ is small enough. And (w * , 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 < μη ≤ u n ≤ z λ + M. Now we consider the set {u n }. From Lemma 2.2 in [12], one has that u 1, and |∇u | L defined on W 1, 0 are equivalent. And from the proof of the coercivity of the operator B, we know that if |∇u | L > 1, then |∇u| ≥ |∇u | L , that is, |∇u| ≥ u 1, , when u 1, > 1.
Denoting w = u nu, we have u n (u nu) = u n + 1 n β u n α L (u nu). Therefore, taking the limit as n → ∞ in (3.15), we have u = u β u α L .
The limit value u is just the solution which we are looking for, and it satisfies w * ≤ u ≤ w * , obviously. Therefore, the proof is finished.