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

We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form $-{\mathrm{\Delta }}_{p}u=\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p+\sigma$ in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$. Here ${\mathrm{\Delta }}_p$, $p>1$, is the standard $p$-Laplacian operator defined by $\Delta_p u={\rm div}, (|\nabla u|^{p-2}\nabla u)$,andthedatum$\sigma$isasigneddistributionin$\Omega$. The class of solutions that we are interested in consists of functions $u\in W_0^{1,p}(\mathrm{\Omega })$ such that $|\nabla u|\in M(W^{1,p}(\Omega)\rightarrow L^p(\Omega))$, a space pointwise Sobolev multipliers consisting of functions $f\in L^p(\mathrm{\Omega })$ 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 $\mathrm{\Omega }$. 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 $\sigma =\mathrm{d}\mathrm{i}\mathrm{v}F$ for a vector field $F$ such that $\mathrm{\mid }F{\mathrm{\mid }}^{\frac{1}{p-1}}\in M(W^{1,p}(\mathrm{\Omega })\to L^p(\mathrm{\Omega }))$. 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 $-{\mathrm{\Delta }}_{p}v=\sigma v^{p-1}$, $v\ge 0$ in $\mathrm{\Omega }$, and $v=1$ on $\mathrm{\partial }\mathrm{\Omega }$, which is interesting in its own right.