Non-Nehari manifold method for asymptotically periodic Schrödinger equations

Abstract

We consider the semilinear Schrödinger equationwhere f is a superlinear, subcritical nonlinearity. We mainly study the case whereN ) and lim |x|→∞ V 1 (x) = 0. Inspired by previous work of Li et al. [14], Pankov [20] and Szulkin and Weth [25], we develop a more direct approach to generalize the main result in [25] by removing the " strictly increasing" condition in the Nehari type assumption on f (x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N 0 by using the diagonal method.

Ground state solutions for asymptotically periodic Schrödinger–Poisson systems involving Hartree-type nonlinearities

“…Then, using the non-Nehari manifold approach developed by Tang [32,33] and the concentration compactness principle, we obtain a ground state solution for System (1.1) (see Sects. 2, 3).…”

Nehari-type ground state solutions for asymptotically periodic fractional Kirchhoff-type problems in RN$\mathbb{R}^{N}$

“…Proof We use the non-Nehari manifold approach developed in [42,43] to show (2.17). From (F1), (F2) and (2.1), we know that there exist δ 0 > 0 and ρ 0 > 0 such that…”

“…Inspired by the previously mentioned work, especially [21,42,43], we seek definite answers to overcome the above three difficulties. Firstly, we use a new trick to show the following set:…”

“…Secondly, for the asymptotically periodic case, we adopt a technique introducing in [21] by combining the quantitative deformation lemma and degree theory, to overcome the difficulties caused by the dropping of periodicity of V (x) and f (x, u) in x (see Section 4). Thirdly, because the argument based on Nehari manifold approach (see Szulkin and Weth [44]) becomes invalid under condition (F4) instead of (F4 ), we will use the non-Nehari manifold approach developed in [42,43]. It relies on finding a minimizing Cerami sequence for variational functional related to (1.1) outside the Nehari manifold by using the diagonal method (see Lemma 2.8).…”

Ground state solutions for a diffusion system

“…To our knowledge, there is no work devoted to this kind of problems under weaker assumptions, hence our result is new. As a motivation we recall that there are a large number of literatures devoted to the study of the existence of ground state solutions for Schrödinger equation or system, we refer readers to [25,24,26,27,23,22,[28][29][30][31][32][33][34][35][36][37][38] and the references therein.…”