Nontrivial Solutions for Asymmetric Kirchhoff Type Problems

Ruichang Pei,Jihui Zhang

doi:10.1155/2014/163645

Abstract

We consider a class of particular Kirchhoff type problems with a right-hand side nonlinearity which exhibits an asymmetric growth at+∞and−∞inℝN(N=2,3). Namely, it is 4-linear at−∞and 4-superlinear at+∞. However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semiaxis. Some existence results for nontrivial solution are established by combining Mountain Pass Theorem and a variant version of Mountain Pass Theorem with Moser-Trudinger inequality.

Highlights

We consider the following nonlocal Kirchhoff type problem:− (1 + ∫ |∇u|2) Δu (x) = f (x, u), in Ω, Ω (1)u = 0, on ∂Ω, where Ω is a smooth bounded domain in RN (N = 2, 3) and f : Ω × R → R is continuous.It is pointed out in [1] that the problem (1) models several physical and biological systems where u describes a process which depends on the average of itself
U = 0, on ∂Ω, where Ω is a smooth bounded domain in RN (N = 2, 3) and f : Ω × R → R is continuous
This problem is related to the stationary analogue of the Kirchhoff equation utt − (1 + ∫ |∇u|2) Δu = g (x, t), (2)

Summary

Introduction

We consider the following nonlocal Kirchhoff type problem:. u = 0, on ∂Ω, where Ω is a smooth bounded domain in RN (N = 2, 3) and f : Ω × R → R is continuous. Let N = 3 and assume that f has the improved subcritical polynomial growth on Ω (condition (SCPI)) and satisfies (H1)–(H4). When N = 2 and f has the subcritical (exponential) growth (SCE), our work is again studying the asymmetric problem (1) without the (AR) condition in the positive semiaxis. Let N = 2 and assume that f has the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H1)–(H4).

Some Lemmas

Proofs of the Main Results