AT LEAST TWO SOLUTIONS FOR THE ASYMMETRIC BEAM SYSTEM WITH CRITICAL GROWTH

Abstract

We consider the multiplicity of the solutions for a class
of a system of critical growth beam equations with periodic condition on t and Dirichlet boundary condition
$u_{tt}+u_{xxxx}=av+\frac{2\alpha}{\alpha+beta}u_+ ^{\alpha-1}v^\beta +s\phi_{00}$ in $(-\frac{\pi}{2},\frac{\pi}{2})\times R$,

$v_{tt}+v_{xxxx}=bu+\frac{2\alpha}{\alpha+beta}u_+ ^{\alpha}v^{\beta-1} +t\phi_{00}$ in $(-\frac{\pi}{2},\frac{\pi}{2})\times R$,

where α, β > 1 are real constants, u+ = max{u, 0}, $\phi_{00}$ is the eigen-function corresponding to the positive eigenvalue $\lambda_{00}=1$ of the eigenvalue problem $u_{tt} + u_{xxxx} =\lambda_{mn} u$. We show that the system

has a positive solution under suitable conditions on the matrix
$A =\begin{pmatrix} 0&a\\ b&0\end{pmatrix}$
, s > 0, t > 0, and next show that the system has
b 0
another solution for the same conditions on A by the linking argu-
ments.