Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers

Abstract

We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form $-{\Delta }_{p}u=|\nabla u{|}^p+\sigma$ in a bounded domain $\Omega \subset {\mathbb{R}}^n$. Here ${\Delta }_p$, $p>1$, is the standard $p$-Laplacian operator defined by ${\Delta }_{p}u=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$, and the datum $\sigma$ is a signed distribution in $\Omega$. The class of solutions that we are interested in consists of functions $u\in W_0^{1,p}(\Omega )$ such that $|\nabla u|\in M(W^{1,p}(\Omega )\to L^p(\Omega ))$, a space pointwise Sobolev multipliers consisting of functions $f\in L^p(\Omega )$ such that \begin{equation*} \int_Ω |f|^{p} |φ|^p dx \leq C \int_Ω (|\nabla φ|^p + |φ|^p) dx \quad \forall φ\in C^\infty(Ω), \end{equation*} for some $C>0$. This is a natural class of solutions at least when the distribution $\sigma$ is nonnegative and compactly supported in $\Omega$. We show essentially that, with only a gap in the smallness constants, the above equation has a solution in this class if and only if one can write $\sigma =\mathrm{div}F$ for a vector field $F$ such that $|F{|}^{\frac{1}{p-1}}\in M(W^{1,p}(\Omega )\to L^p(\Omega ))$. As an important application, via the exponential transformation $u\mapsto v=e^{\frac{u}{p-1}}$, we obtain an existence result for the quasilinear equation of Schrödinger type $-{\Delta }_{p}v=\sigma v^{p-1}$, $v\ge 0$ in $\Omega$, and $v=1$ on $\partial \Omega$, which is interesting in its own right.