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 $(P)$.
Categories: math.AP