Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth

Abstract

In this paper we study the existence, multiplicity and concentration behaviour of ground states for a class of quasilinear Schrödinger equations with critical growth. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. We relate the number of positive solutions with the topology of the set where the potential attains its minimum value. The proofs are based on the Ljusternik-Schnirelmann theory and variational methods.

Study of a nonlinear Kirchhoff equation with non-homogeneous material

“…In [3] it is shown existence of solution for f subcritical and critical. Multiplicity of solutions were showed in [11], [14], [19], [27], [28] and [29] using genus or category theory. The case in which the Laplace operator is replaced by the p-Laplacian or the p(x)-Laplacian has been considered in [6] and [5] respectively.…”

High energy solutions of modified quasilinear fourth-order elliptic equation

“…It was pointed out in [1][2][3][4] that (1.1) models several physical and biological systems where u describes a process which relies on the mean of itself such as the population density. For more mathematical and physical background on Kirchhoff-type problems, we refer the reader to [1,[5][6][7][8] and the references therein.…”

Decay Rate and Energy Gap for the Singularity Solution of the Inhomogeneous Landau-Lifshitz Equation on S2

“…Hence, (5) can be regarded as a simple case of (2). Similar to the inhomogeneous Schrödinger equation, (2) (or (1)) can be extended to the inhomogeneous expression.…”

“…SMF can be regarded as a nonlinear Schrödinger equation that contains a derivative term. Although the existence, uniqueness, and the blowup problem of some nonlinear Schrödinger equations [2][3][4] are clear, the theorem of SMF becomes more complicated and some further work still needs to be done. Similarly, comparing HMF and some general harmonic system (or even biharmonic equation) [5][6][7][8], the mapping system is more complicated than the nonmapping system due to the curvature flow of the Riemannian manifolds.…”