Existence result for nonuniformly degenerate semilinear elliptic systems in Rn

Abstract

We study the existence of solutions for a class of nonuniformly degenerate elliptic systems in RN, ?N 3, of the form -div(h 1(x)??u) + a(x)u = f(x,u,v) inRN -div(h 2(x)?? v) + b(x)v = g(x,u,v) inRN where hi ??L1 loc(RN), hi(x) ? ??0|x| ? with ? ??(0, 2) and ??0 > 0, i = 1, 2. The proofs rely essentially on a variant of the Mountain pass theorem (D. M. Duc, Nonlinear singular elliptic equations, J. Lond. Math. Soc. 40(2) (1989), 420-440) combined with the Caffarelli-Kohn-Nirenberg inequality (First order interpolation inequalities with weights, Composito Math. 53 (1984), 259275). ?? 2009 Glasgow Mathematical Journal Trust.