Semilinear elliptic PDE's with biharmonic operator and a singular potential

TL;DR

This paper investigates the existence, nonexistence, and behavior of positive solutions to a semilinear biharmonic PDE with a singular potential, establishing critical parameter thresholds and solution convergence properties.

Contribution

It introduces new existence and nonexistence results for positive solutions of a biharmonic PDE with singular potential, including solution behavior at critical parameters.

Findings

Existence of solutions for 0<λ<λ*

Nonexistence for λ>λ*

Solution convergence as λ approaches λ*

Abstract

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda}$} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\ u=0=\Delta u \quad\textrm{on $\partial\Omega$,} \end{cases} \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 5$, $a, b, f$ are nonnegaive functions satisfying certain hypothesis which we will specify later. $\mu,\lambda$ are positive constants. Under some suitable conditions on functions $a, b, f$ and the constant $\mu$, we show that there exists $\lambda^*>0$ such that when $0<\lambda<\lambda^*$, ($P_{\lambda}$) admits a solution in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$ and for $\lambda>\lambda^*$, it does not have any solution in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$. Moreover as $\lambda\uparrow\lambda^*$, minimal positive solution of ($P_{\lambda}$) converges in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$ to a solution of ($P_{\lambda^*}$). We also prove that there exists $\tilde{\lambda}^*<\infty$ such that $\lambda^*\leq\tilde{\lambda}^*$ and for $\lambda>\tilde{\lambda}^*$, the above problem ($P_{\lambda}$) does not have any solution even in the distributional sense/very weak sense and there is complete {\it blow-up}. Under an additional integrability condition on $b$, we establish the uniqueness of positive solution of ($P_{\lambda^*}$) in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$.