The Dirichlet problem for singular elliptic equations with general nonlinearities

Abstract

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\{\begin{array}{ll}-{\Delta }_{1}u=h(u)f& \text{in}\Omega ,\\ u\ge 0& \text{in}\Omega ,\\ u=0& \text{on}\partial \Omega ,\end{array}$$ where, ${\Delta }_1$ is the $1$-laplace operator, $\Omega$ is a bounded open subset of ${\mathbb{R}}^N$ with Lipschitz boundary, $h(s)$ is a continuous function which may become singular at $s=0^{+}$, and $f$ is a nonnegative datum in $L^{N,\infty }(\Omega )$ with suitable small norm. Uniqueness of solutions is also shown provided $h$ is decreasing and $f>0$. As a by-product of our method a general theory for the same problem involving the $p$-laplacian as principal part, which is missed in the literature, is established. The main assumptions we use are also further discussed in order to show their optimality.

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• 1-Laplacian type problems with strongly singular nonlinearities and gradient terms
• Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms
• Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula
• A double-phase Neumann problem with $p=1$
• The Sattinger iteration method for 1-Laplace type problems and its application to concave-convex nonlinearities