A sub-supersolution approach for some classes of nonlocal problems involving Orlicz spaces

Abstract

In the present paper we study the existence of solutions for some nonlocal problems involving Orlicz-Sobolev spaces. The approach is based on sub-supersolutions.

A Sub‐Supersolution Method for p‐Laplacian Equation with Non‐Local Term

“…For one thing, there are many ways to prove the sub-supersolution theorem. For example, Schaefer's fixed point theorem is used in [7,8,11,18,19], pseudomonotone operator theorem is used in [4,5,10], and the idea of monotone iteration is used in [20][21][22] to explore the sub-supersolution theorem of the equation. We must consider the applicability and feasibility of the above methods.…”

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

“…This paper is concerned with the problem Here ρ ∈ C 1 : [0, +∞) → [0, +∞) and it satisfies (see [10]) (ρ 1 ) tρ(t) is differentiable for ∀t > 0, (ρ 2 ) lim t→0 + tρ(t) = 0, lim t→+∞ tρ(t) = +∞, and that there exist κ, s ∈ (1, N) such that (ρ 3 ) κ -1 ≤ (ρ(t)t) ρ(t) ≤ s -1, ∀t > 0. Note that (ρ 3 ) implies that (ρ 3 ) κ ≤ ρ(t)t 2 (t) ≤ s, ∀t > 0.…”

“…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):…”

“…(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:…”

“…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.…”