Abstract
Abstract In this paper we investigate the existence, multiplicity and asymptotic behavior of positive solution for the nonlocal nonlinear Schrödinger equations. We exploiting the relationship between the Nehari manifold and eigenvalue problems to discuss how the Nehari manifold changes as parameters μ, λ changes and show how existence, multiplicity and asymptotic results for positive solutions of the equation are linked to properties of the manifold.
Highlights
IntroductionOn the with Γ being the Euler gamma function
In this paper we are concerned with the existence and multiplicity of positive solutions of the nonlocal nonlinear Schrödinger equation−∆u + Vμ,λ (x) u + Iα * up |u|p− u = f (x) |u| p− u in RN, u ∈ H RN, (Pμ,λ) where N ≥ N+α N ≤ p < N N−and Iα is the Riesz potential of orderEuclidean space RN, de ned for each point x ∈ RN\{ } byΓ N−α Iα (x) = Γ α πN/ α |x|N−α
In this paper we investigate the existence, multiplicity and asymptotic behavior of positive solution for the nonlocal nonlinear Schrödinger equations
Summary
On the with Γ being the Euler gamma function. Throughout this paper, we assume that the parameters μ, λ > and the functions Vμ,λ := μg − λa and f satisfy the following conditions: (V )g is a nonnegative continuous function on RN;. (V )there exists c > such that the set {g < c} := x ∈ RN | g (x) < c is nonempty and has nite measure;. (V )Ω = int x ∈ RN | g (x) = is nonempty bounded domain and has a smooth boundary with Ω =. By condition (V ) , the set x ∈ Ω : a (x) > has positive Lebesgue measure, we can assume that λ (aΩ) denote the positive principal eigenvalue of the problem
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