Solutions to semilinear elliptic PDE's with biharmonic operator and singular potential
Mousomi Bhakta · DOAJ (DOAJ: Directory of Open Access Journals) · 2016
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.