We obtain existence and multiplicity of solutions for the quasilinear Schrödinger equationwhere V is a positive potential and the nonlinearity g(x, t) behaves like t at the origin and like t 3 at infinity. In the proof, we apply a changing of variables besides variational methods. The obtained solutions belong to W 1,2 (R N ). Mathematics Subject Classification (1991).Primary 35J20 · Secondary 35J60.

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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

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It establishes existence and multiplicity of solutions to the elliptic quasilinear Schrödinger equation −div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=h(x,u),x∈ℝN,where g, h, V are suitable smooth functions. The function g is asymptotically linear at infinity and, for each fixed x∈ℝN, the function h(x, s) behaves like s at the origin and s3 at infinity. In the proofs, we apply variational methods.