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

Abstract

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda }$} \begin{cases} Δ^2u-μa(x)u=f(u)+λb(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\ u=0=Δu \quad\textrm{on $\partial \Omega$,} \end{cases} \end{equation} where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$, $N\ge 5$, $a,b,f$ 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 ${\lambda }^{\ast }>0$ such that when $0<\lambda <{\lambda }^{\ast }$, ($P_{\lambda }$) admits a solution in $W^{2,2}(\Omega )\cap W_0^{1,2}(\Omega )$ and for $\lambda >{\lambda }^{\ast }$, it does not have any solution in $W^{2,2}(\Omega )\cap W_0^{1,2}(\Omega )$. Moreover as $\lambda \uparrow {\lambda }^{\ast }$, minimal positive solution of ($P_{\lambda }$) converges in $W^{2,2}(\Omega )\cap W_0^{1,2}(\Omega )$ to a solution of ($P_{{\lambda }^{\ast }}$). We also prove that there exists ${\tilde{\lambda }}^{\ast }<\infty$ such that ${\lambda }^{\ast }\le {\tilde{\lambda }}^{\ast }$ and for $\lambda >{\tilde{\lambda }}^{\ast }$, 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 ($P_{{\lambda }^{\ast }}$) in $W^{2,2}(\Omega )\cap W_0^{1,2}(\Omega )$.

arXiv comment21 pages