On the minimal solution of bi-Laplacian equation

Abstract

We study the existence of positive solution to the problem
\begin{gather*}
\Delta^2u-q\Delta u-\mu \alpha(x)u=h(u)+\lambda \beta(x)\quad\text{in }\Omega,\\
u>0 \quad\text{in }\Omega,\\
u=0=\Delta u \quad\text{on }\partial\Omega,
\end{gather*}
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$.
We verify the existence of a value $\lambda_0>0$ such that when
$0<\lambda<\lambda_0$, then one can find a positive solution
in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$.
As $\lambda\uparrow\lambda_0$, then $u_\lambda$ of minimal positive solutions
converge to a solution of the main problem but for $\lambda_0$.