Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations onRN

Yu Duan,Chun-Lei Tang

doi:10.1155/2016/9503073

Abstract

The multiplicity of positive solutions for Kirchhoff type equations depending on a nonnegative parameterλonRNis proved by using variational method. We will show that if the nonlinearities are asymptotically linear at infinity andλ>0is sufficiently small, the Kirchhoff type equations have at least two positive solutions. For the perturbed problem, we give the result of existence of three positive solutions.

Highlights

Introduction and Main ResultsThe purpose of this article is to investigate the multiplicity of positive solutions to the following nonlocal Kirchhoff type equations:(a + λ (∫ (|∇u|2 + V (x) u2) dx)) (−Δu + V (x) u) RN (1)= q (x) f (u) + h (x) in RN, where N ≥ 3, a is a positive constant, λ > 0 is a parameter, and f : R → R is a continuous function.In recent years, the following Kirchhoff type equation− (a + b ∫ |∇u|2 dx) Δu + V (x) u = f (x, u) (2)in RN, has been studied by many researchers under variant assumptions on V and f
In all works for (6) mentioned above except for [39, 40], we found that the nonlinearities f are superlinear, sublinear, or local sublinear
By the assumptions imposed on f, g, and h, we know that Iλ(u) and Jλ(u) are well defined on H, and Iλ, Jλ ∈ C1(H, R) with the derivative given by

Summary

Introduction and Main Results

The purpose of this article is to investigate the multiplicity of positive solutions to the following nonlocal Kirchhoff type equations:. When f(x, u) is asymptotically linear with respect to u at infinity, Ye and Yin [39] studied (6) and proved the existence of positive solution for λ sufficiently small and the nonexistence result for λ sufficiently large. We will try to study multiplicity of positive solutions for problem (1) when f is asymptotically linear at infinity. It is well known that Sobolev embedding H 󳨅→ Lp(RN) is continuous but not compact for p ∈ [2, 2∗), and it is usually difficult to prove that a minimizing sequence or a Palais-Smale sequence is strongly convergent if we seek solutions of problem (1) by variational methods To overcome this difficulty, we make full use of integrability of potential function q(x) and perturbation h(x). Throughout this paper, C and Ci are used in various places to denote distinct constants

Proof of Theorem 1

Proof of Theorem 5