The nonlinear (p,q)-Schrödinger equation with a general nonlinearity: Existence and concentration.

We investigate the following class of (p , q) -Laplacian problems: { − ε p Δ p v − ε q Δ q v + V (x) (| v | p − 2 v + | v | q − 2 v) = f (v) in R N , v ∈ W 1 , p (R N) ∩ W 1 , q (R N) , v > 0 in R N , where ε > 0 is a small parameter, N ≥ 3 , 1 < p < q < N , Δ s v : = div (| ∇ v | s − 2 ∇ v) , with s ∈ { p , q } , is the s -Laplacian operator, V : R N → R is a continuous potential such that inf R N ⁡ V > 0 and V 0 : = inf Λ ⁡ V < min ∂ Λ ⁡ V for some bounded open set Λ ⊂ R N , and f : R → R is a subcritical Berestycki-Lions type nonlinearity. Using variational arguments, we show the existence of a family of solutions (v ε) which concentrates around M : = { x ∈ Λ : V (x) = V 0 } as ε → 0. [ABSTRACT FROM AUTHOR]