Positive weak solutions of elliptic Dirichlet problems with singularities in both the dependent and the independent variables

Abstract

We consider singular problems of the form $-\Delta u=k\left( \cdot,u\right) -h\left( \cdot,u\right) $ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u>0$ in $\Omega,$ where $\Omega$ is a bounded $C^{1,1}$ domain in $\mathbb{R}^{n},$ $n\geq2,$ $h:\Omega\times\left[ 0,\infty\right) \rightarrow\left[0,\infty\right) $ and $k:\Omega\times\left( 0,\infty\right) \rightarrow \left[ 0,\infty\right) $ are Carathéodory functions such that $h\left(x,\cdot\right) $ is nondecreasing, and $k\left( x,\cdot\right) $ is nonincreasing and singular at the origin a.e. $x\in\Omega$. Additionally, $k\left(\cdot,s\right) $ and $h\left( \cdot,s\right) $ are allowed to be singular on $\partial\Omega$ for $s>0$. Under suitable additional hypothesis on $h$ and $k,$ we prove that the stated problem has a unique weak solution $u\in H_{0}^{1}\left( \Omega\right) $, and that $u$ belongs to $C\left( \overline{\Omega}\right) $. The behavior of the solution near $\partial \Omega$ is also addressed.

Keywords

• singular elliptic problems
• variational problems
• sub-supersolutions method
• finite energy solutions
• positive solutions