Ground state solutions of scalar field fractional Schrödinger equations

Abstract

In this paper, we study the existence of multiple ground state solutions for a class of parametric fractional Schrödinger equations whose simplest prototype iswhere n > 2, (− ) s stands for the fractional Laplace operator of order s ∈ (0, 1), and λ is a positive real parameter. The nonlinear term f is assumed to have a superlinear behaviour at the origin and a sublinear decay at infinity. By using variational methods, we establish the existence of a suitable range of positive eigenvalues for which the problem admits at least two nontrivial solutions in a suitable weighted fractional Sobolev space. Mathematics Subject Classification

Multiplicity solutions of a class fractional Schrödinger equations

“…Moreover, .f 2 / is weaker than the condition .h/. So Theorem 1.1 extends Theorem 1 of [24] and Theorem 2 of [25].…”

“…The main purpose of this paper is to generalize the main results of [24,25]. Now we state our main results: This paper is organized as follows.…”

“…In [7], the authors used the concentration compactness principle to show that (2) (V .x/ D 1) has at least two nontrivial radial solutions without the .AR/ condition. In [25], Bisci and Rǎdulescu studied the following equation…”

Existence of three solutions for impulsive nonlinear fractional boundary value problems

“…In fact, they found sufficient conditions under which the problem has at least two or infinitely many nontrivial solutions. In [2,15,21,22,32,[40][41][42][43] by using variational methods and critical point theory the existence of multiple solutions for fractional boundary value problems was investigated, and in [16] the authors exploited a critical point result for differentiable functionals in order to prove that a suitable class of one-dimensional fractional problems admits at least one non-trivial solution under an asymptotical Existence of three solutions. .…”

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).…”