Biharmonic equation with singular nonlinearity

TL;DR

This paper establishes existence and uniqueness of solutions for a biharmonic equation with singular nonlinearity under certain conditions, and describes the solution's behavior near the boundary, with sharp nonexistence results when conditions are not met.

Contribution

It provides the first rigorous proof of existence, uniqueness, and boundary behavior for solutions to a biharmonic problem with singular nonlinearity, including sharp nonexistence criteria.

Findings

Existence and uniqueness of solutions when mp;eta<2.

Solutions behave proportionally to the distance to the boundary.

No solutions exist when mp;eta2;2.

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 $(P)$.