Electronic Journal of Differential Equations (Sep 2016)
Solutions to semilinear elliptic PDE's with biharmonic operator and singular potential
Abstract
We study the existence and nonexistence of positive solution to the problem $$\displaylines{ \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\text{in }\Omega,\cr u>0 \quad\text{in }\Omega,\cr u=0=\Delta u \quad\text{on }\partial\Omega, }$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$. We show the existence of a value $\lambda^*>0$ such that when $0\lambda^*$ there is no solution in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$. Moreover as $\lambda\uparrow\lambda^*$, the minimal positive solution converges to a solution. We also prove that there exists $\tilde{\lambda}^*\tilde{\lambda}^*$, such that the above problem does not have solution even in the distributional sense/very weak sense, and there is a complete blow-up. Under an additional integrability condition on b, we establish the uniqueness of positive solution.
