Existence of solutions for fractional p-Laplacian problems via Leray-Schauder’s nonlinear alternative

Hong Qiu,Mingqi Xiang

doi:10.1186/s13661-016-0593-8

Abstract

In this paper, we are concerned with the existence of solutions for a class of quasilinear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary data. As a particular case, we study the following problem: $$\begin{aligned}& (-\Delta)_{p}^{s} u =f (x,u )\quad \mbox{in } \Omega, \\& u=0 \quad \mbox{in } \mathbb{R}^{N}\setminus\Omega, \end{aligned}$$ where $(-\Delta)_{p}^{s}$ is the fractional p-Laplace operator, Ω is an open bounded subset of $\mathbb{R}^{N}$ with Lipschitz boundary, and $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function. The existence of nonnegative solutions is obtained by using Leray-Schauder’s nonlinear alternative.

Highlights

1 Introduction and main results Recently, a great deal of attention has been paid to the study of problems involving fractional and nonlocal operators, both in pure mathematical research and in realworld applications, such as optimization, finance, continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces, and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes; see [ – ] and the references therein
The fractional Laplacian operators of the form (– )s can be viewed as the infinitesimal generators of stable Lévy processes; see for instance [ ]
There is no doubt that the literature on fractional and nonlocal operators is quite large; see for example [ – ]

Summary

Introduction

Introduction and main resultsRecently, a great deal of attention has been paid to the study of problems involving fractional and nonlocal operators, both in pure mathematical research and in realworld applications, such as optimization, finance, continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces, and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes; see [ – ] and the references therein. In this paper we are interested in the existence of solutions for the following problem: Where ⊂ RN is an open bounded set with Lipschitz boundary ∂ , f : × R → R is a Carathéodory function, and LK is a nonlocal operator defined as lim ε→ +

Results

Conclusion