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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