Quasilinear elliptic problems under asymptotically linear conditions at infinity and at the origin

Abstract

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.

Quasilinear Schrödinger equations with unbounded or decaying potentials

“…Recently, in the authors developed a perturbation method whose the main idea is adding a regularizing term to recover the smoothness of the energy functional, so the standard minimax theory can be applied. Along this line, there have been a large number of works about existence and multiplicity of solutions for the problem , we refer the reader for instance to and references therein. It is worth pointing out that the aforementioned authors always assumed that the potential V in satisfies the hypotheses …”

Standing waves for quasilinear Schrödinger equations with indefinite potentials

“…al. [14], the asymptotically linear case (the nonlinearity behaves like t at the origin and like t 3 at infinity) is investigated, where the potential V satisfies the same conditions as in [37]. For equations with concave and convex nonlinearities, one can consult do Ó and Severo [11].…”

Positive solutions for generalized quasilinear Schrödinger equations with potential vanishing at infinity

“…In [8], we also studied the existence of positive solutions for problem (1.1) with critical growth where the potential and the nonlinearity are both asymptotically periodic in x. For more results related to elliptic equations, we refer the readers to [4,5,9,10] and the references therein.…”