Existence of a nontrivial solution for the (p, q)-Laplacian in RN without the Ambrosetti–Rabinowitz condition. (February 2015) - Chaves, Marcio Fialho; Ercole, Grey; Miyagaki, Olimpio Hiroshi
Abstract: In this paper we prove the existence of at least one nonnegative nontrivial weak solution in D 1, p ( R N ) ∩ D 1, q ( R N ) for the equation − Δ p u − Δ q u + a ( x ) | u | p − 2 u + b ( x ) | u | q − 2 u = f ( x, u ), x ∈ R N, where 1 < q < p < q ⋆ : = N q N − q, p < N, Δ m u : = div ( | ∇ u | m − 2 ∇ u ) is the m -Laplacian operator, the coefficients a and b are continuous, coercive and positive functions, and the nonlinearity f is a Carathéodory function satisfying some hypotheses which do not include the Ambrosetti–Rabinowitz condition.