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 $0<\lambda<\lambda^*$, there is a solution and when $\lambda>\lambda^*$ 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 $\tilde{\lambda}^*<\infty$ with $\lambda^*\leq\tilde{\lambda}^*$, and for $\lambda>\tilde{\lambda}^*$, 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. Submitted February 5, 2016. Published September 28, 2016. Math Subject Classifications: 35B09, 35B25, 35B35, 35G30, 35J91. Key Words: Semilinear biharmonic equation; singular potential; Navier boundary condition; existence; nonexistence; blow-up phenomenon; stability; uniqueness of extremal solution.