VARIATIONAL METHODS FOR A RESONANT PROBLEM WITH THE p-LAPLACIAN IN RN

The solvability of the resonant Cauchy problem -Delta(p)u = lambda(1)m(vertical bar x vertical bar)vertical bar u vertical bar(p-2) u + f(x) in R-N ; u is an element of D-1,D-p (R-N), in the entire Euclidean space R-N (N >= 1) is investigated as a part of the Fredholm alternative at the first (smallest) eigenvalue lambda(1) of the positive p-Laplacian -Delta(p) on D-1,D- p (R-N) relative to the weight m(vertical bar x vertical bar). Here, Delta(p) stands for the p-Laplacian, m: R+ -> R+ is a weight function assumed to be radially symmetric, m not equivalent to 0 in R+, and f : R-N -> R is a given function satisfying a suitable integrability condition. The weight m(r) is assumed to be bounded and to decay fast enough as r -> +infinity. Let phi(1) denote the (positive) eigenfunction associated with the (simple) eigenvalue lambda(1) of -Delta(p). If integral(RN) f phi(1) dx = 0, we show that problem has at least one solution u in the completion D-1,D-p (R-N) of C-c(1)(R-N) endowed with the norm (integral(RN) vertical bar del u vertical bar(p) dx)(1/ p). To establish this existence result, we employ a saddle point method if 1 < p < 2, and an improved Poincare inequality if 2 <= p < N. We use weighted Lebesgue and Sobolev spaces with weights depending on phi(1). The asymptotic behavior of phi(1)(x) = phi(1)(vertical bar x vertical bar) as vertical bar x vertical bar -> infinity plays a crucial role.