Solvability of Strongly Nonlinear Degenerated Elliptic Problems

Abstract

Let us consider quasilinear elliptic boundary value problem (BVP)

$$ - div\left( {a\left( {x,u} \right){{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) = \lambda b\left( {x,u} \right){\left| u \right|^{p - 2}}u + f\left( {x,u} \right)in\Omega ,$$

(1.1a)

$$ u = 0on\partial \Omega ,$$

(1.1b)

where Ω is a bounded domain, p > 1 is a real number A is a real parameter and a(x, s)b(x, s)f (x, s) satisfy appropriate growth conditions. We will assume that the coefficient a(x, s) contains a degeneration or a singularity to a certain extent. The purpose of this note is to present results concerning the existence of the weak solution of the BVP (1.1). Under some additional assumptions on f (x, s) we prove the existence of nonnegative weak solution of the BVP (1.1). In order to deal with the degenerate (or singular) coefficient a(x, s) in the equation (1.la) we work in a suitable weighted Sobolev space with the weight which controls the degeneration (or singularity) of a(x, s) in a certain sense.