Multiple sign-changing solutions for nonlinear fractional Kirchhoff equations

Abstract

This paper is concerned with the following nonlinear fractional Kirchhoff equation: a + b R 3 (-) s 2 u 2 dx (-) s u + V(x)u = f (u), x ∈ R 3 , where s ∈ (3 4 , 1), a > 0, b ≥ 0, (-) s denotes the fractional Laplacian operator. Based on the methods of mountain pass theorem and invariant sets of descending flow, the existence of a positive solution, a negative solution, and multiple sign-changing solutions is obtained.

Existence and convergence results of meromorphic solutions to the equilibrium system with angular velocity

“…By using (18) and (19), it follows that θ 1 , θ 2 ∈ (0, 1). Hence, (30) implies that {( n , σ n )} is a Halton sequence in H 1 × H 2 .…”

“…As one of the universal equilibrium systems used in the description of pattern formation in spatially extended dissipative systems, the general equilibrium differential equation can also be found in the study of convective hydrodynamics, plasma confinement in toroidal devices, viscous film flow, and bifurcating solutions of the modified equilibrium differential equation [6,10,11]. In recent years, some references such as Sheng et al [12], Zhai et al [13], Zhang [14], Wu et al [15], Sun et al [16], Li et al [17], Bai and Sun [18], Wang et al [19], and so on, introduced many beautiful patterns to satisfy practical requirements of modern computing systems with multi-processors. There is the potential of considering the linearization characteristics to be further developed for the system of equilibrium boundary value problems.…”

“…Motivated and inspired by the references [18][19][20][21], in this paper we further consider the following universal equilibrium equations with nondifferentiable boundary conditions:…”

Explicit monotone iterative sequences for positive solutions of a fractional differential system with coupled integral boundary conditions on a half-line

“…On the other hand, integral boundary conditions are considered to be more reasonable than the local boundary conditions, which can depict phenomena of heat transmission, population dynamics, blood flow, etc. A large number of results about fractional differential equations with integral boundary condition have been obtained, see [9,10,[30][31][32][33][34][35][36][37][38][39][40][41][42][43] and the references cited therein. Meanwhile, we note that the coupled systems of fractionalorder differential equations have also attracted much attention due to their extensive applications, we refer to [3,9,10,14,22,29,[32][33][34][35][36][37][38][39][40][41][42][43].…”

Two nontrivial solutions for a nonhomogeneous fractional Schrödinger–Poisson equation in $\mathbb{R}^{3}$

“…where is Planck's constant, W : R 3 → R is an external potential, and m is a suitable nonlinearity. In the research of fractional quantum mechanics, this equation is an important model; therefore, it has been extensively studied, for example, see [4,5,13,20,22,25,26,29,32,33] and their references. When s = t = 1, system (E) reduces to the following Schrödinger-Poisson type equations:…”