A stability result for elliptic equations with singular nonlinearity and its applications to homogenization problems

Abstract

We consider model semilinear elliptic equations of the type \[ \begin{cases} - \mathrm{div} (A(x) \nabla u) = f u^{- λ}, \quad u > 0 \quad \text{in} \ Ω, \\ u \in H_{0}^{1}(Ω), \end{cases} \] where $\Omega$ is a bounded domain in ${\mathbf{R}}^N$, $N\ge 1$, $A\in L^{\infty }(\Omega {)}^{N\times N}$ is a coercive matrix, $0<\lambda \le 1$ and $f$ is a nonnegative function in $L_{loc}^1(\Omega )$, or more generally, nonnegative Radon measure on $\Omega$. We discuss $H^1$-stability of $u$ under a minimal assumption on $f$. Additionally, we apply the result to homogenization problems.

References in corpus (2)

• Spreading equilibria under mildly singular potentials: pancakes versus droplets
• Trace inequalities of the Sobolev type and nonlinear Dirichlet problems