Nonlinear equations with gradient natural growth and distributional data, with applications to a Schrödinger type equation

Abstract

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