Multiple positive bound states for critical Schrödinger-Poisson systems

Abstract

Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schrödinger-Poisson system:We remark that (SP ) exhibits a "double" lack of compactness because of the unboundedness of R 3 and the critical growth of the nonlinear term and that in our assumptions ground state solutions of (SP ) do not exist. MSC: 35J20, 35J60

On a class of nonlinear Schrödinger–Poisson systems involving a nonradial charge density

“…Lions [36] and Benci-Cerami [9], which include more recent contributions in the context of Schrödinger-Poisson systems, see e.g. [21], [50], [19]. We point out that these recent results are mostly in the range p > 3, for Palais-Smale sequences constrained on Nehari manifolds, and for functionals without positive parts, unlike our result.…”

“…[22]). In the presence of potentials, however, existence may occur when p = 5, as it has been recently shown in [19]. Ambrosetti and Ruiz [5] improved upon these early results by using the so-called 'monotonicity trick' introduced by Struwe [47] and formulated in the context of the nonlinear Schrödinger equations by Jeanjean [30] and Jeanjean-Tanaka [31], in order to show the existence of multiple bound state solutions to (1.3).…”

On the Schrödinger-Poisson system with indefinite potential and 3-sublinear nonlinearity

“…It is also possible to obtain solutions when no ground state exists, see e.g. [18] for p ∈ ]3, 5[ and [19] in the critical case p = 5 with Q ≡ 1. Sun et al [20] found at least k positive solutions for (1.6) for sufficiently large λ assuming Q has k strict positive maxima.…”

Multiple positive bound state solutions for a critical Choquard equation

“…Lions [30]. After the publication of [9], many authors studied problems related to (5), see for example [3], [7], [8], [10], [12], [13], [24], [32], [33], [34], [40] and references therein.…”

“…In particular, the case λ > 0 presents some more difficulties, because in such a case the natural space to work in is H 1 (R N ) while typically the splitting theorem works in D 1,2 (R N ). To overcome these difficulties we follow some ideas from [12] and [25] together with a nonexistence result contained in [37]. Let us remark that in the critical case no extra regularity assumptions are needed for the nonexistence result, see Theorem 2.2.…”