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 $\Delta_p u={\rm div}\, (|\nabla u|^{p-2}\nabla u)$, and the datum $\sigma$ is a signed distribution in $\Omega$. The class of solutions that we are interested in consists of functions $u\in W^{1,p}_0(\Omega)$ such that $|\nabla u|\in M(W^{1,p}(\Omega)\rightarrow L^p(\Omega))$, a space pointwise Sobolev multipliers consisting of functions $f\in L^{p}(\Omega)$ such that \begin{equation*} \int_{\Omega} |f|^{p} |\varphi|^p dx \leq C \int_{\Omega} (|\nabla \varphi|^p + |\varphi|^p) dx \quad \forall \varphi\in C^\infty(\Omega), \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 $\sigma={\rm div}\, F$ for a vector field $F$ such that $|F|^{\frac{1}{p-1}}\in M(W^{1,p}(\Omega)\rightarrow L^p(\Omega))$. 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 Schr\"odinger type $-\Delta_p v = \sigma\, v^{p-1}$, $v\geq 0$ in $\Omega$, and $v=1$ on $\partial\Omega$, which is interesting in its own right.