Quasilinear Schrödinger Equations with Asymptotically Linear Nonlinearities

Abstract

We deal with the existence of nonzero solution for the quasilinear Schrödinger equation−Δu + V(x)u − Δ(uwhere V is a positive potential and the nonlinearity g(x, s) behaves like K

Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure

“…Since f 2 is convex function we mention that f 2 (t − s) ≤ 4 f 2 (t) + f 2 (s) holds true for all s, t ∈ R, for instance see [14,13]. Using the item ii) we know that 0…”

“…The linking structure. In this section we shall prove the linking structure for the energy functional J θ given in (13). The basic idea is to guarantee that J θ has the linking geometry described in Theorem 2.1.…”

“…the function u = 0 is a local minimum for the energy functional. In this case, the Mountain Pass Theorem can be applied proving several results concerning existence and multiplicity of solutions for quasilinear Schrödinger equations, see [1,9,11,13,14,27,28,29,25]. In the present work we shall consider the following case…”

“…The proof of items (f 1 ) − (f 12 ) can be viewed in [9,13,14,44]. In what follows taking into account the change of variables u = f (v) in the functional given in (12) we obtain the following auxiliary functional…”