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 (-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 (uin 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 (fin 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 (umapsto 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}), (vge 0) in (Omega ), and (v=1) on (partial Omega ), which is interesting in its own right.