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 $u\in W^{1,p}_0(\Omega )$ 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

$$\begin{aligned} \int _{\Omega } |f|^{p} |\varphi |^p dx \le C \int _{\Omega } (|\nabla \varphi |^p + |\varphi |^p) dx \quad \forall \varphi \in C^\infty (\Omega ), \end{aligned}$$

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 $|F|^{\frac{1}{p-1}}\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))$. 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 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.