In this paper, we are concerned with the Schrödinger-Poisson systemDue to its relevance in physics, the system has been extensively studied and is quite well understood in the case d ≥ 3. In contrast, much less information is available in the planar case d = 2 which is the focus of the present paper. It has been observed by Cingolani and the second author [6] that the variational structure of (0.1) differs substantially in the case d = 2 and leads to a richer structure of the set of solutions. However, the variational approach of [6] is restricted to the case p ≥ 4 which excludes some physically relevant exponents. In the present paper, we remove this unpleasant restriction and explore the more complicated underlying functional geometry in the case 2 < p < 4 with a different variational approach. MSC: 35J50; 35Q40

In this paper, we study the existence, nonexistence and mass concentration of L2-normalized solutions for nonlinear fractional Schrödinger equations. Comparingwith the Schrödinger equation, we encounter some new challenges due to the nonlocal nature of the fractional Laplacian. We first prove that the optimal embedding constant for the fractional Gagliardo–Nirenberg–Sobolev inequality can be expressed by exact form, which improves the results of [17, 18]. By doing this, we then establish the existence and nonexistence of L2-normalized solutions for this equation. Finally, under a certain type of trapping potentials, by using some delicate energy estimates we present a detailed analysis of the concentration behavior of L2-normalized solutions in the mass critical case.

In this paper, we are concerned with a class of Schrödinger-Poisson systems with the asymptotically linear or asymptotically 3-linear nonlinearity. Under some suitable assumptions on V , K, a, and f , we prove the existence, nonexistence, and asymptotic behavior of solutions via variational methods. In particular, the potential V is allowed to be sign-changing for the asymptotically linear case. C 2016 AIP Publishing LLC.

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For a class of asymptotically periodic Schrödinger-Poisson systems with critical growth, the existence of ground states is established. The proof is based on the method of Nehari manifold and concentration compactness principle. MSC:35J05, 35J50, 35J60

In this paper, we consider the following Schrödinger–Poisson system [Formula: see text] where [Formula: see text] are real parameters and [Formula: see text]. Suppose that [Formula: see text] represents a potential well with the bottom [Formula: see text], the system has been widely studied in the case [Formula: see text]. In contrast, no existence result of solutions is available for the case [Formula: see text] due to the presence of the nonlocal term [Formula: see text]. With the aid of the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for [Formula: see text] large and [Formula: see text] small in the case [Formula: see text]. Then we obtain the nonexistence of nontrivial solutions for [Formula: see text] large and [Formula: see text] large in the case [Formula: see text]. Finally, we explore the decay rate of the positive solutions as [Formula: see text] as well as their asymptotic behavior as [Formula: see text] and [Formula: see text].