Fourth order elliptic system with dirichlet boundary condition

Tacksun Jung,Q-Heung Choi

doi:10.1186/1029-242x-2011-60

Abstract

We investigate the multiplicity of the solutions of the fourth order elliptic system with Dirichlet boundary condition. We get two theorems. One theorem is that the fourth order elliptic system has at least two nontrivial solutions when λ k < c < λk+1and λk+n(λk+n- c) < a + b < λk+n+1(λk+n+1- c). We prove this result by the critical point theory and the variation of linking method. The other theorem is that the system has a unique nontrivial solution when λ k < c < λk+1and λ k (λ k - c) < 0, a+b < λk+1(λk+1- c). We prove this result by the contraction mapping principle on the Banach space.AMS Mathematics Subject Classification: 35J30, 35J48, 35J50

Highlights

Let Ω be a smooth bounded region in Rn with smooth boundary ∂Ω
In this paper we investigate the multiplicity of the solutions of the following fourth order elliptic system with Dirichlet boundary condition
Tarantello [10] studied problem (1.2) when c < l1 and b ≥ l1(l1 - c). She show that (1.2) has at least two solutions, one of which is a negative solution. She obtained this result by degree theory

Summary

Introduction

Let Ω be a smooth bounded region in Rn with smooth boundary ∂Ω. Let l1 < l2 ≤ ... ≤ lk ≤ ... be the eigenvalues of -Δ with Dirichlet boundary condition in Ω. In this paper we investigate the multiplicity of the solutions of the following fourth order elliptic system with Dirichlet boundary condition. Lazer and McKenna [6] studied the single fourth order elliptic equation with Dirichlet boundary condition. They show that (1.3) has at least two nontrivial solutions when c < l1, l1(l1 - c) < b < l2(l2 - c) and s 0 They obtained these results by using the variational reduction method. They [3] proved that when c < l1, l1(l1 - c) < b < l2(l2 - c) and s

We also have

We can write

Thus there exists