Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in ℝ N

Abstract

In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrodinger equation $$- {\Delta _N}u + b{\left| u \right|^{N - 2}}u - {\Delta _N}\left( {{u^2}} \right)u = h\left( u \right)$$ , x ∈ RN where Δ N is the N-Laplacian operator, h(u) is continuous and behaves as exp(α|u| N/(N-1)) when |u| → ∞. Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution u(x) ∈ W 1,N (R N ) with u(x) → 0 as |x| → ∞ is established.