Fractional Laplacian System Research Articles

Existence of ground state solutions to a class of fractional Schr\xf6dinger system with linear and nonlinear couplings

In this paper, we study the existence of ground state solutions to the following fractional Schrödinger system with linear and nonlinear couplings: \t\t\t{(−△)su+(λ1+V(x))u+kv=μ1u3+βuv2,in R3,(−△)sv+(λ2+V(x))v+ku=μ2v3+βu2v,in R3,u,v∈Hs(R3),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (-\\triangle )^{s}u+(\\lambda _{1}+V(x))u+kv=\\mu _{1}u^{3}+\\beta uv^{2}, \\quad \\text{in } R^{3},\\\\ (-\\triangle )^{s}v+(\\lambda _{2}+V(x))v+ku=\\mu _{2}v^{3}+ \\beta u^{2}v, \\quad \\text{in } R^{3},\\\\ u, v\\in H^{s}(R^{3}), \\end{cases} $$\\end{document} where (-triangle )^{s} denotes the fractional Laplacian of order sin (frac{3}{4},1). Under some assumptions of the potential V(x) and the linear and nonlinear coupling constants k, β, we prove some results for the existence of ground state solutions for the fractional Laplacian systems by using variational methods.

Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent

In this paper, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent \begin{document}$ \begin{cases} (-\Delta)^{s}u+\mu u = |u|^{p-1}u+\lambda v,& x\in\mathbb{R}^{N},\\ (-\Delta)^{s}v+\nu v = |v|^{2^{\ast}-2}v+\lambda u,& x\in\mathbb{R}^{N},\\ \end{cases} $\end{document} where $ (-\Delta)^{s} $ is the fractional Laplacian, $ 0<s<1,\ N>2s, \ \lambda <\sqrt{\mu\nu },\ 1<p<2^{\ast}-1\; \text{and}\; \ 2^{\ast} = \frac{2N}{N-2s} $ is the Sobolev critical exponent. By using the Nehari\ manifold, we show that there exists a $ \mu_{0}\in(0,1) $, such that when $ 0<\mu\leq\mu_{0} $, the system has a positive ground state solution. When $ \mu>\mu_{0} $, there exists a $ \lambda_{\mu,\nu}\in[\sqrt{(\mu-\mu_{0})\nu},\sqrt{\mu\nu}) $ such that if $ \lambda>\lambda_{\mu,\nu} $, the system has a positive ground state solution, if $ \lambda<\lambda_{\mu,\nu} $, the system has no ground state solution.

A priori bounds and existence result of positive solutions for fractional Laplacian systems

In this paper, we consider the fractional Laplacian system \begin{document}$\left\{\begin{array}{ll}(-\triangle)^{\frac{\alpha}{2}}u+\sum^{N}_{i = 1}b_{i}(x)\frac{\partial u}{\partial x_{i}}+C(x)u = f(x,v), \;\;x\in \Omega,\\(-\triangle)^{\frac{\beta}{2}}v+\sum^{N}_{i = 1}c_{i}(x)\frac{\partial v}{\partial x_{i}}+D(x)v = g(x,u),\;\; x\in \Omega,\\u>0, v>0, \;\; x\in \Omega,\\u = 0, v = 0, \;\; x\in \mathbb R^{N}\setminus \Omega,\end{array}\right.$ \end{document} where $Ω$ is a smooth bounded domain in $\mathbb R^{N}$, $α ∈ (1,2)$, $β ∈ (1,2)$, $N>\max\{α, β\}$. Under some suitable conditions on potential functions and nonlinear terms, we use scaling method to obtain a priori bounds of positive solutions for the fractional Laplacian system with distinct fractional Laplacians.

Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent

In this paper, we study the following critical system with fractional Laplacian: \t\t\t{(−Δ)su+λ1u=μ1|u|2∗−2u+αγ2∗|u|α−2u|v|βin Ω,(−Δ)sv+λ2v=μ2|v|2∗−2v+βγ2∗|u|α|v|β−2vin Ω,u=v=0in RN∖Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} (-\\Delta)^{s}u+\\lambda_{1}u=\\mu_{1}|u|^{2^{\\ast}-2}u+\\frac{\\alpha \\gamma}{2^{\\ast}}|u|^{\\alpha-2}u|v|^{\\beta} & \\text{in } \\Omega, \\\\ (-\\Delta)^{s}v+\\lambda_{2}v= \\mu_{2}|v|^{2^{\\ast}-2}v+\\frac{\\beta \\gamma}{2^{\\ast}}|u|^{\\alpha}|v|^{\\beta-2}v & \\text{in } \\Omega, \\\\ u=v=0 & \\text{in } \\mathbb{R}^{N}\\setminus\\Omega, \\end{cases} $$\\end{document} where (-Delta)^{s} is the fractional Laplacian, 0< s<1, mu_{1},mu_{2}>0, 2^{ast}=frac{2N}{N-2s} is a fractional critical Sobolev exponent, N>2s, 1<alpha, beta<2, alpha+beta=2^{ast}, Ω is an open bounded set of mathbb{R}^{N} with Lipschitz boundary and lambda_{1},lambda_{2}>-lambda_{1,s}(Omega), lambda_{1,s}(Omega) is the first eigenvalue of the non-local operator (-Delta)^{s} with homogeneous Dirichlet boundary datum. By using the Nehari manifold, we prove the existence of a positive ground state solution of the system for all gamma>0. Via a perturbation argument and using the topological degree and a pseudo-gradient vector field, we show that this system has a positive higher energy solution. Then the asymptotic behaviors of the positive ground state solutions are analyzed when gammarightarrow0.

On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling

In this paper, a class of systems of two coupled nonlinear fractional Laplacian equations are investigated. Under very weak assumptions on the nonlinear terms $f$ and $g$, we establish some results about the existence of positive vector solutions and vector ground state solutions for the fractional Laplacian systems by using variational methods. In addition, we also study the asymptotic behavior of these solutions as the coupling parameter $β$ tends to zero.

Fractional Laplacian system involving doubly critical nonlinearities in ℝ N

Fractional Laplacian system involving doubly critical nonlinearities in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> </mml:math>

Multiple Positive Solutions for a Fractional Laplacian System with Critical Nonlinearities

In this paper, we study the following nonlinear fractional Laplacian system with critical exponent $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u=\lambda |u|^{p-2}u+\frac{2\alpha }{\alpha +\beta }|u|^{\alpha -2}u|v|^{\beta }, &{}\quad \hbox {in} \;\ \Omega ,\\ (-\Delta )^{s}v=\mu |v|^{p-2}v+\frac{2\beta }{\alpha +\beta }|u|^{\alpha }|v|^{\beta -2}v, &{} \quad \hbox {in} \;\ \Omega ,\\ u=v=0, &{} \quad \hbox {in} \;\ {\mathbb {R}}^{N}\backslash \Omega , \end{array}\right. \end{aligned}$$ where \(\Omega \subset {\mathbb {R}}^{N}\) is a bounded domain with smooth boundary, \(0 1\) satisfy \(\alpha +\beta =2_{s}^{*}, 2_{s}^{*}=\frac{2N}{N-2s}\) is the critical Sobolev exponent, and \(N>4s, \lambda , \mu >0\) are parameters. Using the \({\mathcal {N}}\) ehari manifold, fibering maps and the Lusternik–Schnirelmann category, we prove that the problem has at least \(\hbox {cat}(\Omega )+1\) distinct positive solutions, where \(\hbox {cat} (\Omega )\) denotes the Lusternik–Schnirelmann category of \(\Omega \) in itself.

Radial symmetry results for fractional Laplacian systems

In this paper, we generalize the direct method of moving planes for the fractional Laplacian to the system case. Considering a coupled nonlinear system with fractional Laplacian, we first establish a decay at infinity principle and a narrow region principle. Using these principles, we then obtain two radial symmetry results for the decaying solutions of the fractional Laplacian systems. Finally, we apply our method to fractional Schrödinger systems and fractional Hénon systems.