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
$u_{tt}+u_{xxxx}=av+\frac{2\alpha }{\alpha +beta}{u}_{+}^{\alpha -1}{v}^{\beta }+s{\varphi }_{00}$ in $(-\frac{\pi }{2},\frac{\pi }{2})\times R$,

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

where α, β > 1 are real constants, u+ = max{u, 0}, ${\varphi }_{00}$ 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
$A=\left(\begin{array}{cc}0& a\\ b& 0\end{array}\right)$
, 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.