Abstract
In this paper we study the existence of sign-changing solutions for nonlinear problems involving the fractional Laplacian \t\t\t0.1{(−Δ)su−λu=f(x,u),x∈Ω,u=0,x∈Rn∖Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\left \\{ \\textstyle\\begin{array}{l@{\\quad}l} (-\\Delta)^{s}u-\\lambda u=f(x,u),&x\\in\\Omega, \\\\ u=0,&x\\in\\mathbb{R}^{n}\\setminus\\Omega, \\end{array}\\displaystyle \\right . $$\\end{document} where Omegasubsetmathbb{R}^{n} (ngeq2) is a bounded smooth domain, sin(0,1), (-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 lambdaleq0, 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 lambdageq 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.
Highlights
This paper is concerned with the existence of sign-changing solutions for nonlinear problems involving the fractional Laplacian (– )su – λu = f (x, u), x ∈, u =, x ∈ Rn \, where ⊂ Rn (n ≥ ) is a bounded domain with smooth boundary ∂, f is a Carathéodory function, < s
1 Introduction This paper is concerned with the existence of sign-changing solutions for nonlinear problems involving the fractional Laplacian
The fractional Laplacian operator (– )s arises in many different applications, such as continuum mechanics, phase transition phenomena, population dynamics and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes; see [ ] and the references therein
Summary
In the past few years, some existence results on sign-changing solutions of nonlinear elliptic equations have been obtained by combing minimax method with invariant sets of descending flow (see [ , ]). In Section , we discuss the case λ ≤ , a positive and a negative and a sign-changing solution have been found by constructing different invariant sets on which the functional is bounded below; In Section , we discuss the case λ > λs (where λs denotes the first eigenvalue of the operator (– )s in with homogeneous Dirichlet boundary data), by using a variation of the linking theorem. (see [ ]) Assume that M is a closed invariant set of descending flow and f satisfies the (PS) condition on M.
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