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

Žubrinić, Darko

Abstr. Appl. Anal. 5, No. 3, 159-173 (2000).

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

Keywords:

\(p\)-Laplace operator; spherically symmetric; decreasing; weak and strong solutions; singular ordinary integro-differential equation; a posteriori regularity