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.


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