Nontrivial Solutions for Time Fractional Nonlinear Schrödinger-Kirchhoff Type Equations

Nontrivial solutions of Kirchhoff type problems

In this paper, we study Kirchhoff type problems on a bounded domain. We consider the cases that the nonlinearity is superlinear near zero but asymptotically 4-linear at infinity, and the nonlinearity is asymptotically linear near zero but 4-superlinear at infinity. By computing the relevant critical groups, we obtain nontrivial solutions via Morse theory.

Multiplicity of Nontrivial Solutions for Kirchhoff Type Problems

Copyright q 2010 Bitao Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using variational methods, we study the multiplicity of solutions for Kirchhoff type problems −a b Ω |∇u| 2 Δu f x, u, in Ω; u 0, on ∂Ω. Existe...

Numerical study of fractional nonlinear Schrödinger equations.

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the ...

Nontrivial Periodic Solutions for Nonlinear Second-Order Difference Equations

Positive solutions for asymptotically periodic Kirchhoff-type equations with critical growth

In this paper‎, ‎we consider the following Kirchhoff-type equations‎: ‎$-‎left(a+bint_{mathbb{R}^{3}}|nabla u|^{2}right)Delta u+V(x) u=lambda$ $f(x,u)+u^{5}‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$u(x)>0‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$uin H^{1}(mathbb{R}^{3})‎ ,‎$ ‎ ‎‎‎where $a,b>0$ are constants and $lambda$ is a positive parameter‎. ‎The aim of this paper is to study the existence of positive ...