Nontrivial Solution for the Fractional p-Laplacian Equations via Perturbation Methods

Huxiao Luo,Xianhua Tang,Shengjun Li

doi:10.1155/2017/5317213

Huxiao Luo, Xianhua Tang + Show 1 more

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https://doi.org/10.1155/2017/5317213

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Abstract

We study the existence of nontrivial solution of the following equation without compactness: (-Δ)pαu+up-2u=f(x,u), x∈RN, where N,p≥2, α∈(0,1), (-Δ)pα is the fractional p-Laplacian, and the subcritical p-superlinear term f∈C(RN×R) is 1-periodic in xi for i=1,2,…,N. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of fractional p-Laplacian type equation. To overcome this difficulty, by adding coercive potential term and using mountain pass theorem, we get the weak solution uλ of perturbation equations. And we prove that uλ→u as λ→0. Finally, by using vanishing lemma and periodic condition, we get that u is a nontrivial solution of fractional p-Laplacian equation.

Highlights

This article is concerned with the fractional p-Laplacian equations (−Δ)αp u + |u|p−2 u = f (x, u), x ∈ RN, (1)where N, p ≥ 2, α ∈ (0, 1), and f satisfies the following conditions.(f1) f ∈ C(RN×R), f is 1-periodic in xi for i = 1, 2, . . . , N, and f (x, t) lim |t|→∞ |t|q−1 = (2) f (x, t) t lim |t|p +∞
The fractional p-Laplacian is defined on smooth functions by
The main purpose of this paper is to consider the existence of nontrivial solutions for equation (1)

Summary

Introduction

The fractional p-Laplacian is defined on smooth functions by (−Δ)αp u (x) lim ε→0. The main purpose of this paper is to consider the existence of nontrivial solutions for equation (1). Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of (1) To overcome this problem, we apply the perturbation method [22, 24–26]. We prove that the energy functional of (5) has the geometry of the mountain pass theorem that it satisfies the Cerami condition and that the obtained solutions {uλ} have the uniform bounds. (iii) If {un} is a Palais Smale sequence of Φ (see Section 2) and un converges weakly to u0, one can not obtain that u0 is a weak solution of the fractional p-Laplacian type equation (1). (iv) o(1) denote being infinitely small (possibly different) when n → ∞

Variational Framework

Proofs of the Main Result