Biharmonic equation with singular nonlinearity

Abstract

We consider the following problem: \begin{eqnarray*} ( P)\qquad \displaystyle\left\{\begin{array} {ll} & \Delta^2 u = K(x)u^{-\alpha} \quad \mbox{ in }\,\Omega , \\ &u> 0\quad \mbox{ in }\,\Omega, \;\;u\vert_{\partial\Omega}=0, \,\Delta u\vert_{\partial\Omega} = 0. \end{array}\right. \end{eqnarray*} We prove the main existence result: Assume that $\alpha+\beta<2$. Then there exists a unique solution $u$ to $(P)$. Furthermore, there exist $c_1, c_2>0$ such that \begin{eqnarray}\label{behaviour-bound} c_1 \rho(x)\leq u(x)\leq c_2 \rho(x) \end{eqnarray} where $\rho(x)=d(x,\partial\Omega)$. This result is sharp: Assume that $\alpha+\beta\geq 2$. Then, there is no solution to $(P)$.