Fractional Laplacian System Research Articles

Kirchhoff‐type critical fractional Laplacian system with convolution and magnetic field

AbstractIn this paper, we consider a class of upper critical Kirchhoff‐type fractional Laplacian system with Choquard nonlinearities and magnetic fields. With the help of the limit index theory and the concentration–compactness principles for fractional Sobolev spaces, we establish the existence of infinitely many nontrivial radial solutions for the above system. A distinguished feature of this paper is that the above Kirchhoff‐type system is degenerate, that is, the Kirchhoff term is zero at zero.

Sign‐changing solutions of fractional Laplacian system with critical exponent

This paper deals with the following fractional Laplacian system with critical exponent: where is a bounded smooth open connected set in is the fractional critical Sobolev exponent, and is the first eigenvalue of fractional Laplacian under the condition in . We prove that, for each fixed and slightly smaller than , the above system with admits a sign‐changing solution in the following sense: one component changes sign, while the other one is positive. Our result includes the lower dimensional case . Compared with the classical Laplacian case, our problem is nonlocal and the first component of the solution is sign‐changing, some new difficulties arise and new arguments and estimates should be introduced.

Nonzero positive solutions of fractional Laplacian systems with functional terms

AbstractWe study the existence of nonzero positive solutions of a class of systems of differential equations driven by fractional powers of the Laplacian. Our approach is based on the notion of fixed point index, and allows us to deal with nonlocal functional weights and functional boundary conditions. We present two examples to shed light on the type of functionals and growth conditions that can be considered with our approach.

Existence of positive solutions for fractional Laplacian systems with critical growth

Existence of positive solutions for fractional Laplacian systems with critical growth

Multiplicity of Solutions to a p-q Fractional Laplacian System with Concave Singular Nonlinearities

Multiplicity of Solutions to a p-q Fractional Laplacian System with Concave Singular Nonlinearities

The sliding method for fractional Laplacian systems

In this paper, we generalize the sliding method to the fractional Laplacian system in the whole space and obtain a monotonicity result for bounded solutions of the system.

Existence and multiplicity of positive solutions for a class of critical fractional Laplacian systems

Existence and multiplicity of positive solutions for a class of critical fractional Laplacian systems

Radially Symmetric Solution for Fractional Laplacian Systems with Different Negative Powers

By developing the direct method of moving planes, we study the radial symmetry of nonnegative solutions for a fractional Laplacian system with different negative powers: (−Δ)α2u(x)+u−γ(x)+v−q(x)=0,x∈RN, (−Δ)β2v(x)+v−σ(x)+u−p(x)=0,x∈RN, u(x)≳|x|a,v(x)≳|x|bas|x|→∞, where α,β∈(0,2), and a,b>0 are constants. We study the decay at infinity and narrow region principle for the fractional Laplacian system with different negative powers. The same results hold for nonlinear Hénon-type fractional Laplacian systems with different negative powers.

Existence and concentration of positive solutions to a fractional system with saturable term

In this paper, we consider a fractional Laplacian system with saturable nonlinearity. Under some assumptions on the parameters and potential functions, we obtain the existence and concentration behavior of the positive ground state solution by variational methods.

Existence and multiplicity of positive solutions to fractional Laplacian systems with combined critical Sobolev terms

We deal with the existence and multiplicity of solutions for a non-local fractional system involving combined critical non-linearities. Some existence results are obtained by splitting the Nehari manifold and applying the Mountain–Pass theorem of Ambrosetti and Rabinowitz.

Positive solutions for fractional Laplacian system involving concave-convex nonlinearities and sign-changing weight functions

Positive solutions for fractional Laplacian system involving concave-convex nonlinearities and sign-changing weight functions

Combined effects of logarithmic and superlinear nonlinearities in fractional Laplacian systems

In this paper, we consider the existence and multiplicity of solutions for the following fractional Laplacian system with logarithmic nonlinearity $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}u=\lambda h_1(x)u\ln |u|+\frac{p}{p+q}b(x)|v|^{q}|u|^{p-2}u\ \ &{}x\in \Omega , \\ (-\Delta )^t v=\mu h_2(x)v\ln |v|+\frac{q}{p+q}b(x)|u|^p|v|^{q-2}v\ \ &{}x\in \Omega , \\ u=v=0\ \ &{}x\in \mathbb {R}^N{\setminus }\Omega , \end{array}\right. } \end{aligned}$$ where $$s,t\in (0,1),\ N>\max \{2s,2t\}$$ , $$\lambda ,\mu >0$$ , $$2<p+q<\min \{\frac{2N}{N-2s},\frac{2N}{N-2t}\}$$ , $$\Omega \subset \mathbb {R}^N$$ is a bounded domain with Lipschitz boundary, $$h_1,h_2,b\in C(\overline{\Omega })$$ and $$(-\Delta )^{s}$$ is the fractional Laplacian. When $$h_1,h_2,b$$ are positive functions, the existence of ground state solutions for the problem is obtained. When $$h_1,h_2$$ are sign-changing functions and b is a positive function, two nontrivial and nonnegative solutions are obtained. Our results are new even in the case of a single equation.

Complete classification of ground state solutions with different Morse index for critical fractional Laplacian system

In this paper, we first obtain the existence of positive ground state solutions for the following critical fractional Laplacian system: then we give a complete classification of positive ground state solutions with different Morse index. More precisely, we show that if (u, v) be any positive ground state solution of system (1.1), then (u, v) must be (C1Uϵ, y, C2Uϵ, y) type with Morse index 1 and Morse index 2, where Uϵ, y is a positive ground state solution for a given equation.

Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains

<p style='text-indent:20px;'>In this paper, we first establish a narrow region principle for systems involving the fractional Laplacian in unbounded domains, which plays an important role in carrying on the direct method of moving planes. Then combining this direct method with the sliding method, we derive the monotonicity of bounded positive solutions to the following fractional Laplacian systems in unbounded Lipschitz domains <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} \left(-\Delta\right)^{s}u&amp; = &amp;f(u,v), &amp; \mbox{in}\ \ \Omega\,, \\[0.05cm] \left(-\Delta\right)^{t}v&amp; = &amp;g(u,v),&amp; \mbox{in}\ \ \Omega\,, \\[0.05cm] u,\,v&amp;\equiv&amp;0, &amp; \mbox{on}\ \ \mathbb{R}^{n}\setminus\Omega\,, \end{array}\right. \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>without any decay assumptions on the solution pair <inline-formula><tex-math id="M2">\begin{document}$ (u,\,v) $\end{document}</tex-math></inline-formula> at infinity.

MULTIPLE SOLUTIONS FOR A FRACTIONAL LAPLACIAN SYSTEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND HOMOGENEOUS TERM

In this paper, we deal with a class of fractional Laplacian system with critical Sobolev-Hardy exponents and sign-changing weight functions in a bounded domain. By exploiting the Nehari manifold and variational methods, some new existence and multiplicity results are obtain.

Existence results for a fractional elliptic system with critical Sobolev‐Hardy exponents and concave‐convex nonlinearities

In this paper, we study the following nonlinear fractional Laplacian system with critical Sobolev‐Hardy exponent where is a smooth bounded domain in , , , , , , satisfy , is the critical Sobolev‐Hardy exponent, , are parameters, , and are nonnegative functions on . Using the variational methods and analytic techniques, we prove that the critical fractional Laplacian system admits at least two positive solutions when the pair of parameters belongs to a suitable subset of .

Liouville type theorem for Fractional Laplacian system

In this paper, using the method of moving planes combined with integral inequality to handle the fractional Laplacian system, we prove Liouville type theorems of nonnegative solution for the nonlinear system.

Liouville theorem and isolated singularity of fractional Laplacian system with critical exponents

This paper is devoted to the fractional Laplacian system with critical exponents. We use the method of moving spheres to derive a Liouville Theorem with at most three radial solutions, and then prove the solutions in Rn∖{0} are radially symmetric and monotonically decreasing. Together with blow up analysis, we get the upper bound of the local solutions in B1∖{0}. Our results is an extension of the classical works by Caffarelli et al. (1989), Caffarelli et al. (2014), Chen and Lin (2015) and Guo and Liu (2008).

Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling

It is well known that a single nonlinear fractional Schrödinger equation with a potential V (x) may have a positive solution that is concentrated at the nondegenerate minimum point of V (x) when the positive parameter ε is sufficiently small [see G. Chen and Y. Zheng, Commun. Pure Appl. Anal. 13, 2359 (2014); J. Dávila et al., J. Differential Equations 256, 858 (2014); and M. M. Fall et al., Nonlinearity 28, 1937 (2015)]. In this work, we find multiple different positive solutions for a class of weakly coupled fractional Schrödinger systems with two potentials V1(x) and V2(x) that have some of the same minimum points; the positive solutions are concentrated at these minimum points. By using energy estimates, the Nehari manifold technique, and the Lusternik–Schnirelmann theory of critical points, we obtain some multiplicity results for a class of weakly coupled fractional Schrödinger systems. The existence and nonexistence of least energy positive solutions are also explored.

Existence and multiplicity of solutions for fractional Laplacian system

ABSTRACT This paper is devoted to study the existence and multiplicity of solutions to fractional -Laplacian system by analyzing the relationship between the Nehari manifold and fibering maps.