Existence of a bound state solution for quasilinear Schrödinger equations

Abstract

In this article, we establish the existence of bound state solutions for a class of quasilinear Schrödinger equations whose nonlinear term is asymptotically linear in {\mathbb{R}^{N}} . After changing the variables, the quasilinear equation becomes a semilinear equation, whose respective associated functional is well defined in {H^{1}(\mathbb{R}^{N})} . The proofs are based on the Pohozaev manifold and a linking theorem.

On a Schrödinger–Poisson system with singularity and critical nonlinearities

“…We can learn more details about physical background from [2,3] and the references therein. System (1.2) has been extensively studied, focusing on the existence of positive solutions, multiplicity of solutions, ground state solutions, sign-changing solutions, radial solutions, by using the variational methods and critical point theory under various assumptions of potential V and nonlocal term f , see for example [4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein. In addition, existence and multiplicity of the Schrödinger-Poisson problem in a bounded domain has been paid attention to by many authors, we can see [18][19][20][21][22][23][24].…”

Existence of standing waves for quasi-linear Schrödinger equations on T n

“…Zhang-Zhang-Xiang [27] obtained the existence of ground states for fractional Schrödinger equations involving a critical nonlinearity. Xue-Tang [25] showed that the existence of a bound state solution for quasilinear Schrödinger equations. One can see [8,9,12,21] for more results on the existence of solution for elliptic equations in R n .…”

Multiplicity and concentration results for magnetic relativistic Schrödinger equations

“…We also refer to d'Avenia and Squassina [26] for the existence of ground states and other useful estimates. Lastly, for the existence and multiplicity results of semilinear or quasilinear Schrödinger equations, we refer readers to [27][28][29] and references therein. Motivated by the about results, in this paper we deal with multiplicity and concentration results of the more general class of pseudo-relativistic magnetic Schrödinger equation (1.2) .…”