Abstract
In this paper, we study the following nonlocal problem − a − b ∫ Ω ∇ u 2 d x Δ u = λ u + f x u p − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where a , b > 0 are constants, 1 < p < 2 , λ > 0 , f ∈ L ∞ Ω is a positive function, and Ω is a smooth bounded domain in ℝ N with N ≥ 3 . By variational methods, we obtain a pair of nontrivial solutions for the considered problem provided f ∞ is small enough.
Highlights
Introduction and Main ResultsThis paper is concerned with the existence and multiplicity of nontrivial solutions for the following nonlocal problem with Dirichlet boundary value conditions: 8 ð>< − a − b j∇uj2dx Δu = λu + f ðxÞjujp−2u,>: u = 0, Ω x ∈ Ω, x ∈ ∂Ω, ð1Þ where a, b > 0, 1 < p < 2, λ > 0, f ∈ L∞ðΩÞ is a positive function, and Ω is a smooth bounded domain in RN with N ≥ 3.In the past two decades, the following Kirchhoff type problems with Dirichlet boundary value conditions>< − a + b j∇uj2dx Δu = f ðx, uÞ, x ∈ Ω, ð2Þ x ∈ ∂Ω, have attracted great attention of many researchers
We study the following nonlocal problem
Kirchhoff has been some results on the existence and multiplicity of nontrivial solutions to this new nonlocal problem
Summary
This paper is concerned with the existence and multiplicity of nontrivial solutions for the following nonlocal problem with Dirichlet boundary value conditions:. Kirchhoff has been some results on the existence and multiplicity of nontrivial solutions to this new nonlocal problem (see [15,16,17,18,19,20,21,22]). >< − a − b j∇uj2dx Δu = jujp−2u, x ∈ Ω, ð4Þ x ∈ ∂Ω, where 2 < p < 2∗, and obtained the existence and multiplicity of solutions for the problem. B > 0, 1 < p < 2, λ > 0, f ∈ L∞ðΩÞ is a positive function, problem (1) has at least a pair of nontrivial solutions if j f j∞ is small enough.
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