Nonlinear equations with gradient natural growth and distributional data, with applications to a Schr\"odinger type equation

We obtain necessary and sufficient conditions with sharp constants on the distribution $\sigma$ for the existence of a globally finite energy solution to the quasilinear equation with a gradient source term of natural growth of the form $-\Delta_p u = |\nabla u|^p + \sigma$ in a bounded open set $\Omega\subset \mathbb{R}^n$. Here $\Delta_p$, $p>1$, is the standard $p$-Laplacian operator defined by $\Delta_p u={\rm div}\, (|\nabla u|^{p-2}\nabla u)$. The class of solutions that we are interested in consists of functions $u\in W^{1,p}_0(\Omega)$ such that $e^{{\mu} u}\in W^{1,p}_0(\Omega)$ for some ${\mu}>0$ and the inequality \begin{equation*} \int_{\Omega} |\varphi|^p |\nabla u|^p dx \leq A \int_\Omega |\nabla \varphi|^p dx \end{equation*} holds for all $\varphi\in C_c^\infty(\Omega)$ with some constant $A>0$. This is a natural class of solutions at least when the distribution $\sigma$ is nonnegative. The study of $-\Delta_p u = |\nabla u|^p + \sigma$ is applied to show the existence of globally finite energy solutions to the quasilinear equation of Schr\"odinger type $-\Delta_p v = \sigma\, v^{p-1}$, $v\geq 0$ in $\Omega$, and $v=1$ on $\partial\Omega$, via the exponential transformation $u\mapsto v=e^{\frac{u}{p-1}}$.