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

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_λ$} \begin{cases} Δ^2u-μa(x)u=f(u)+λb(x)\quad\textrm{in $Ω$,}\\ u>0 \quad\textrm{in $Ω$,}\\ u=0=Δu \quad\textrm{on $\partialΩ$,} \end{cases} \end{equation} where $Ω$ 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. $μ,λ$ are positive constants. Under some suitable conditions on functions $a, b, f$ and the constant $μ$, we show that there exists $λ^*>0$ such that when $0<λ<λ^*$, ($P_λ$) admits a solution in $W^{2,2}(Ω)\cap W^{1,2}_0(Ω)$ and for $λ>λ^*$, it does not have any solution in $W^{2,2}(Ω)\cap W^{1,2}_0(Ω)$. Moreover as $λ\uparrowλ^*$, minimal positive solution of ($P_λ$) converges in $W^{2,2}(Ω)\cap W^{1,2}_0(Ω)$ to a solution of ($P_{λ^*}$). We also prove that there exists $\tildeλ^*<\infty$ such that $λ^*\leq\tildeλ^*$ and for $λ>\tildeλ^*$, the above problem ($P_λ$) 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 ($P_{λ^*}$) in $W^{2,2}(Ω)\cap W^{1,2}_0(Ω)$.

No reviews yet.

No comments yet.