Multiplicity and concentration of solutions for fractional Schrödinger equation with sublinear perturbation and steep potential well

Ground state of solutions for a class of fractional Schrödinger equations with critical Sobolev exponent and steep potential well

“…4,5 Also, a detailed mathematical descriptions of Caffarell and Silvestre 6 can be found in the Appendix of Dávila et al 7 In recent years, with the aid of variational methods, the existence, nonexistence, and multiplicity results of various solutions for (1) have been extensively investigated, see, for instance, previous studies [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the references therein. Furthermore, there have been enormous results considered the concentration property of solutions, see, for instance, previous studies 3,7,24,25 and the references therein. In fact, most of the above results focused on the nonlinear model |u| p−1 u with 2 ≤ p < 2 * .…”

“…In fact, most of the above results focused on the nonlinear model |u| p−1 u with 2 ≤ p < 2 * . For example, in Yang and Liu, 24 the authors proved that the multiplicity and concentration of solutions for fractional Schrödinger Equation 2, when f(x, u) is of subcritical growth. In, 25 Shang and Zhang proved that the existence and multiplicity of solutions which concentrate near some critical points of (x) by a perturbative variational method.…”

Infinitely many small energy solutions for fractional coupled Schrodinger system with critical growth

“…It was discovered by Laskin [20,21] as a result of extending the Feynman path integral. In recent few years, may researchers have investigated the existence and multiplicity of (critical) fractional Schrödinger equations, see for instance [6,9,13,15,24,27,29,30,32,33,36,38]. In some work, the nonlinearity satisfies the AmbrosettiRabinowitz (A-R) condition, i.e., there exists θ > 2 such that 0 < θF(x, t) < tf(x, t).…”

Energy solutions and concentration problem of fractional Schrödinger equation

“…Obviously, the form of (1.1), whose nonlinear term combines asymptotic linearity and superlinear term with sublinear term h(x)|u| p-2 u, is more general than that of (1.2). In [38], the existence of at least two nontrivial solutions of (1.1) is showed. In the present paper, by variational methods, we have established existence criteria of infinitely many nontrivial high or small energy solutions without A-R condition.…”

“…As is well known, the compactness of the embedding fails when ρ is large enough. By using a new function space introduced in [38,39] and constructing some inequalities, we can obtain the concentration of the solutions of (1.1) under different conditions. Hence our results can be viewed as an extension to the main results in [16][17][18][19][20][21][22][23]38].…”