Sign-changing solutions for non-local elliptic equations with asymptotically linear term

Huxiao Luo,Zu Gao,Xianhua Tang

doi:10.3934/cpaa.2018055

Abstract

In this article, we study the existence of sign-changing solutions for a problem driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary condition \begin{document}$\left\{ \begin{array}{ll}-\mathcal{L}_Ku = f(x,u) &\text{in}~Ω, \\u = 0 &\text{in}~\mathbb{R}^n\setminusΩ, \end{array} \right. \left( 1 \right)$ \end{document} where $Ω\subset\mathbb{R}^n(n≥2)$ is a bounded, smooth domain and $f(x, u)$ is asymptotically linear at infinity with respect to $u$. By introducing some new ideas and combining constraint variational method with the quantitative deformation lemma, we prove that there exists a sign-changing solution of problem (1).

Highlights

In recent years, the fractional and non-local operators of elliptic type have been widely investigated
The fractional and non-local operators of elliptic type have been widely investigated. This type of operators arise in several areas such as anomalous diffusion, the thin obstacle problem, optimization, finance, phase transitions, stratified materials, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, quasi-geostrophic flows, multiple scattering and materials science
Many papers [3, 5, 6, 7, 8, 13, 15, 18, 21, 22, 23] are devoted to the study of the existence of sign-changing solutions of the nonlinear problem

Summary

Introduction

The fractional and non-local operators of elliptic type have been widely investigated. Variational method, sign-changing solutions, non-local elliptic equations, asymptotically linear, deformation lemma. Note that even in the model case, where K(x) = |x|−(n+2s), the norms in (5) and (6) are not the same, because Ω × Ω is strictly contained in Q This leads that the classical fractional Sobolev space approach is not sufficient for studying problem (3). I (u), u+ = I (u+), u+ − (u−(x)u+(y) + u−(y)u+(x))K(x − y)dxdy, R2n where I is the energy functional of problem (4) given by. In present paper, motivated by [1, 3, 5, 6, 21, 28], we try to get the sign-changing solution for problem (4) by seeking the minimizer of the energy functional I over the following constraint:.

We choose ε

Let ε

It follows from