## Solvability of quasilinear elliptic equations with strong dependence on the gradient.(English)Zbl 1005.35043

From the introduction: The aim of this paper is to study existence of weak and strong solutions of the following quasilinear elliptic problem: $-\Delta_pu= G\bigl(|x|,u,|\nabla u|\bigr) \text{ in }B\setminus\{0\},$
$u=0\text{ on }\partial B,\tag{1}$
$u(x)\text{ spherically symmetric and decreasing },$ where we assume strong dependence on both the unknown and the gradient. Here $$B=B_R(0)$$ is the ball of radius $$R$$ in $$\mathbb{R}^N$$, $$N\geq 1$$, $$1<p<\infty$$, $$\Delta_pv=\text{div} (|\nabla v|^{p-2}\nabla v)$$ is the $$p$$-Laplace operator. The elliptic problem (1) is studied by relating it to the corresponding singular ordinary integro-differential equation. Solvability range is obtained in the form of simple inequalities involving the coefficients describing the problem. We also study a posteriori regularity of solutions. An existence result is formulated for elliptic equations on arbitrary bounded domains in dependence of the outer radius of the domain.

### MSC:

 35J60 Nonlinear elliptic equations 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 45J05 Integro-ordinary differential equations
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