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

Abstract

In this paper, we studied the following fractional Kirchhoff-type equation:where a, b are positive constants, α ∈ (0, 1), N ∈ (2α, 4α), (-) α is the fractional Laplacian operator, V(x) and f (x, u) are periodic or asymptotically periodic in x. Under some weaker conditions on the nonlinearity, we obtain the existence of ground state solutions for the above problem in periodic case and asymptotically periodic case, respectively. In particular, our results unify both asymptotically cubic and super-cubic nonlinearities, which are new even for α = 1. MSC: 35J20; 35J25; 35J60

Positive and negative solutions for the nonlinear fractional Kirchhoff equation in $${\mathbb {R}}^{N}$$

“…Then, there has been increasing interest in studying fractional Kirchhoff equations. For instance, Peng, Tang, and Chen [22] investigated the existence of ground state solutions for Eq. (1) with k ¼ 0 by using the method of Nehari manifold.…”

Existence of Infinitely Many High Energy Solutions for a Fourth-Order Kirchhoff Type Elliptic Equation in R<sup>3</sup>

“…Meanwhile, many researchers pay attention to the Kirchhoff type problems when the domain is unbounded or is the whole space R N . For the existence and infinitely many solutions for Kirchhoff type problems in R N , please see the interesting results in [13][14][15] and the references therein. Now, we mention some recent works related to the Kirchhoff-type problem.…”

Multiple sign-changing solutions for nonlinear fractional Kirchhoff equations

“…Then, many papers have been devoted to studying the existence of solutions for fractional Kirchhoff equation like (1.1). For instance, Peng, Tang, and Chen [23] investigated the existence of ground state solutions for Eq. (1.1) by using the method of Nehari manifold.…”