Biharmonic equation with singular nonlinearity

Abstract

We consider the following problem: \begin{eqnarray*} ( P)\qquad \displaystyle\left\{\begin{array} {ll} & Δ^2 u = K(x)u^{-α} \quad \mbox{ in }\,Ω, \\ &u> 0\quad \mbox{ in }\,Ω, \;\;u\vert_{\partialΩ}=0, \,Δu\vert_{\partialΩ} = 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 ρ(x)\leq u(x)\leq c_2 ρ(x) \end{eqnarray} where $\rho (x)=d(x,\partial \Omega )$. This result is sharp: Assume that $\alpha +\beta \ge 2$. Then, there is no solution to $(P)$.

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