Multiplicity solutions of a class fractional Schrödinger equations

Abstract

In this paper, we study the existence of nontrivial solutions to a class fractional Schrödinger equations (−Δ)su+V(x)u=λf(x,u)inRN, $$ {( - \Delta )^s}u + V(x)u = \lambda f(x,u)\,\,{\rm in}\,\,{\mathbb{R}^N}, $$ where (−Δ)su(x)=2limε→0∫RN∖Bε(X)u(x)−u(y)|x−y|N+2sdy,x∈RN $ {( - \Delta )^s}u(x) = 2\lim\limits_{\varepsilon \to 0} \int_ {{\mathbb{R}^N}\backslash {B_\varepsilon }(X)} {{u(x) - u(y)} \over {|x - y{|^{N + 2s}}}}dy,\,\,x \in {\mathbb{R}^N} $ is a fractional operator and s ∈ (0, 1). By using variational methods, we prove this problem has at least two nontrivial solutions in a suitable weighted fractional Sobolev space.

Keywords

• fractional laplacian
• variational methods
• nontrivial solution
• 35p15
• 35p30
• 35r11