Existence results for quasilinear Schrödinger equations with a general nonlinearity

Abstract

Consider the quasilinear Schrödinger equation −∆u + V (x)u − ∆(u 2)u = h(u) in R N , (A) where N ≥ 3, V : R N → R and h : R → R are functions. Under some general assumptions on V and h, we establish two existence results for problem (A) by using variational methods. The main novelty is that, unlike most other papers on this problem, we do not assume the nonlinear term to be 4-superlinear at infinity.

Quasilinear Schrödinger equations involving singular potentials

“…Using a constrained minimisation argument, Ruiz and Siciliano proved in [46] that (1.1) has a positive ground state for h(u) = |u| p−2 u and 2 < p < 4N N−2 . A complement result can be found in [30] where existence of a positive solution is established for (1.1) with a general nonlinearity. In [3,23], boundary value problems of parameter-dependent quasilinear elliptic equations on a bounded domain are studied and results on the existence of multiple solutions depending on the parameter are given in the case where the growth rate of nonlinearity is between 2 and 4.…”

Ground state solutions for quasilinear Schr\”{o}dinger equations with critical Berestycki-Lions nonlinearities

“…Up to our knowledge, there are few results on problem (1.2) under the Berestycki-Lions conditions (see [1,2]) which are almost optimal for the existence of solutions. Here we mention [4,11,26,27]. Colin and Jeanjean [4] proved the existence of a solution radially symmetric for problem (1.2) with V (x) ≡ 0 and the nonlinearity h(u) satisfies the Berestycki-Lions conditions.…”

“…The method is to analyze the behavior of solutions for subcritical problems and to take the limit as the exponent approaches to the critical exponent, which was also used in [3,18]. The rest results about Berestycki-Lions conditions [11,26] are in the subcritical cases.…”

“…The main purpose of this paper is to extend the result in [11] to the case of critical exponents. We borrow an idea from [29] to obtain the ground state solution for problem (1.2).…”

“…Remark 1.3. (1) The above two theorems can be regarded as a form of the subcritical BerestyckiLions theorem in [11] to the critical case. Due to the lack of compact embedding of H 1 (R N ) → L 2 * (R N ), for critical nonlinearity h, the existence of ground states of problem (1.2) becomes rather complicated.…”