Mountain pass and linking type sign-changing solutions for nonlinear problems involving the fractional Laplacian

Boundary Value Problems

, 2017:108

First Online: 17 July 2017Received: 09 December 2016Accepted: 01 July 2017

Abstract

In this paper we study the existence of sign-changing solutions for nonlinear problems involving the fractional Laplacian $$\left \{ \textstyle\begin{array}{l@{\quad}l} -\Delta^{s}u-\lambda u=fx,u,andx\in\Omega, \\ u=0,andx\in\mathbb{R}^{n}\setminus\Omega, \end{array}\displaystyle ight .$$ 0.1 where \\Omega\subset\mathbb{R}^{n}\ \geq2\ is a bounded smooth domain, \s\in0,1\, \-\Delta^{s}\ denotes the fractional Laplacian, λ is a real parameter, the nonlinear term f satisfies superlinear and subcritical growth conditions at zero and at infinity. When \\lambda\leq0\, we prove the existence of a positive solution, a negative solution and a sign-changing solution by combing minimax method with invariant sets of descending flow. When \\lambda\geq \lambda {1}^{s}\ where \\lambda {1}^{s}\ denotes the first eigenvalue of the operator \-\Delta^{s}\ in Ω with homogeneous Dirichlet boundary data, we prove the existence of a sign-changing solution by using a variation of linking type theorems.Keywordsfractional Laplacian sign-changing solutions mountain pass and linking type invariant sets of descending flow MSC35R11 58E30  Download fulltext PDF

Author: Huxiao Luo - Xianhua Tang - Shengjun Li