Fractional Choquard Research Articles

Symmetry and nonexistence results for a fractional Choquard equation with weights

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ u $\end{document}</tex-math></inline-formula> be a nonnegative solution to the equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * |x|^a u^p \right) |x|^a u^{p-1} \quad\text{ in } \mathbb{R}^n \setminus \{0\}, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ n \ge 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0 &lt; \alpha &lt; 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ 0 &lt; \beta &lt; n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ a&gt;\max\{ -\alpha, -\frac{\alpha+\beta}{2} \} $\end{document}</tex-math></inline-formula>. By exploiting the method of scaling spheres and moving planes in integral forms, we show that <inline-formula><tex-math id="M6">\begin{document}$ u $\end{document}</tex-math></inline-formula> must be zero if <inline-formula><tex-math id="M7">\begin{document}$ 1\le p&lt;\frac{n+\beta+2a}{n-\alpha} $\end{document}</tex-math></inline-formula> and must be radially symmetric about the origin if <inline-formula><tex-math id="M8">\begin{document}$ a&lt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \frac{n+\beta+2a}{n-\alpha} \le p \le \frac{n+\beta+a}{n-\alpha} $\end{document}</tex-math></inline-formula>.

Potential well theory for the focusing fractional Choquard equation

This note studies the non-linear fractional Schrödinger equation iu̇−(−Δ)su+(Iα*|u|p)|u|p−2u=0. In the mass super-critical and energy sub-critical regimes, the local solutions exist globally and scatter in the energy space or blow-up in finite time if the data belong to some stable sets.

Existence of nontrivial solutions for fractional Choquard equations with critical or supercritical growth

ABSTRACT In this paper, we study the following fractional Choquard equation with critical or supercritical growth where 0<s<1, denotes the fractional Laplacian of order s, N>2s, and . Under some suitable conditions, we prove that the equation has a nontrivial solution for small by the variational method.

Multiplicity and concentration results for fractional Choquard equations: Doubly critical case

In this paper, we consider the following fractional Choquard equation: ε2s(−Δ)su+V(x)u=ε−α(Iα∗F(u))F′(u),x∈RN,where N⩾3, s∈(0,1] and α∈(0,N), ε>0 is a parameter, Iα is the Riesz potential, and F(u)≔12α♯|u|2α♯+12α∗|u|2α∗, where 2α♯=N+αN and 2α∗=N+αN−2s are lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. Firstly, by the refined Sobolev inequality with Morrey norm, we show a generalization of Lions type theorem. Secondly, combining this theorem with variational methods, we show the multiplicity and concentration of positive solutions for above equation. Moreover, the multiplicity and concentration results are obtained in the case where F(u) has the lower critical exponent 2α♯, which remains unsolved in the existing literature.

Ground state solutions for fractional Choquard equations involving upper critical exponent

In this article, we study the following fractional Choquard equation involving upper critical exponent (−Δ)su+V(x)u=λf(x,u)+[|x|−μ∗|u|2μ,s∗]|u|2μ,s∗−2u,x∈RN, where λ>0, 0<s<1, (−Δ)s denotes the fractional Laplacian of order s, N>2s, 0<μ<2s and 2μ,s∗=2N−μN−2s. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by Nehari method.

Solutions to upper critical fractional Choquard equations with potential

In this paper, the upper critical fractional Choquard equation is considered. The interest in studying such equation comes from its strong connections with mathematical physics, for example, the fractional quantum mechanics, the Lévy random walk models, and the dynamics of pseudo-relativistic boson stars. Another strong motivation is from some mathematical difficulties such as the coexistence of two fractional diffusions, the lack of compactness, and the attendance of the upper critical exponent. To overcome these difficulties, some methods and techniques need to be combined and a radially symmetric solution is obtained.

Symmetry and nonexistence of positive solutions for fractional Choquard equations

This paper is devoted to study the following Choquard equation \begin{eqnarray*}\left\{ \begin{array}{lll} (-\triangle)^{\alpha/2}u=(|x|^{\beta-n}\ast u^p)u^{p-1},~~~&x\in R^n, u\geq0,\,\,&x\in R^n, \end{array} \right. \end{eqnarray*} where $0<\alpha,\beta<2$, $1\leq p<\infty$, and $n\geq2$. Using a direct method of moving planes, we prove the symmetry and nonexistence of positive solutions in the critical and subcritical case respectively.

Pseudoindex theory and Nehari method for a fractional Choquard equation

Pseudoindex theory and Nehari method for a fractional Choquard equation

Multiplicity and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent

In present paper, we study the fractional Choquard equation $$\varepsilon^{2s}(-\Delta)^s u+V(x)u=\varepsilon^{\mu-N}(\frac{1}{|x|^\mu}\ast F(u))f(u)+|u|^{2^\ast_s-2}u$$ where $\varepsilon>0$ is a parameter, $s\in(0,1),$ $N>2s,$ $2^*_s=\frac{2N}{N-2s}$ and $0<\mu<\min\{2s,N-2s\}$. Under suitable assumption on $V$ and $f$, we prove this problem has a nontrivial nonnegative ground state solution. Moreover, we relate the number of nontrivial nonnegative solutions with the topology of the set where the potential attains its minimum values and their's concentration behavior.

Classification of positive solutions to the critical fractional Choquard equation in

ABSTRACT The first aim of this paper is to classify the positive solutions of the fractional Choquard equation where is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Moreover, based on the uniqueness and non-degeneracy of the solution of the above equation, we then study the perturbed Choquard equation By using the finite-dimensional reduction, we obtain the existence of at least one positive solution if is suitable small.

Fractional Choquard Equations with Confining Potential With or Without Subcritical Perturbations

Abstract In this paper, we consider fractional Choquard equations with confining potentials. First, we show that they admit a positive ground state and infinitely many bound states. Then we prove the existence of two signed solutions when a superlinear and subcritical perturbation is added; in this case, the main feature is that such a perturbation does not satisfy the usual Ambrosetti–Rabinowitz condition.

Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth

<p style='text-indent:20px;'>We are concerned with nonlinear fractional Choquard equations involving critical growth in the sense of the Hardy-Littlewood-Sobolev inequality. Without the Ambrosetti-Rabinowitz condition or monotonicity condition on the nonlinearity, we establish the existence of radially symmetric ground state solutions.

A critical fractional Choquard–Kirchhoff problem with magnetic field

In this paper, we are interested in a fractional Choquard–Kirchhoff-type problem involving an external magnetic potential and a critical nonlinearity [Formula: see text] [Formula: see text] where [Formula: see text] with [Formula: see text], [Formula: see text] is the Kirchhoff function, [Formula: see text] is the magnetic potential, [Formula: see text] is the fractional magnetic operator, [Formula: see text] is a continuous function, [Formula: see text], [Formula: see text] is a parameter, [Formula: see text] and [Formula: see text] is the critical exponent of fractional Sobolev space. We first establish a fractional version of the concentration-compactness principle with magnetic field. Then, together with the mountain pass theorem, we obtain the existence of nontrivial radial solutions for the above problem in non-degenerate and degenerate cases.

On the multiplicity and concentration of positive solutions for a [formula omitted]-fractional Choquard equation in [formula omitted

In this paper we deal with the following fractional Choquard equation εsp(−Δ)psu+V(x)|u|p−2u=εμ−N1|x|μ∗F(u)f(u)inRN,u∈Ws,p(RN),u>0inRN,where ε>0 is a small parameter, s∈(0,1), p∈(1,∞), N>sp, (−Δ)ps is the fractional p-Laplacian, V is a positive continuous potential, 0<μ<sp, and f is a continuous superlinear function with subcritical growth. Using minimax arguments and the Ljusternik–Schnirelmann category theory, we obtain the existence, multiplicity and concentration of positive solutions for ε>0 small enough.

Liouville theorem and classification of positive solutions for a fractional Choquard type equation

Let n≥2, 0<α<2 and 0<β<n. We prove that the equation (−Δ)α2u=1|x|n−β∗upup−1inRn has no positive solution if 1≤p<n+βn−α. We also classify all positive solutions to the equation in the critical case p=n+βn−α.

Existence and multiplicity of normalized solutions for a class of fractional Choquard equations

In this paper, we study the existence and multiplicity of solutions with a prescribed ${L^2}$-norm for a class of nonlinear fractional Choquard equations in ${\R^N}$: where $N\geq3$, $s\in~(0,1)$, $\alpha~\in(0,N)$, $p\in(\max~\{~1~+~\frac{{\alpha~+~2s}}{N},2\},\frac{N+\alpha}{N-2s})$ and ${\kappa~_\alpha~}(x)~=~{~|~x~|^{\alpha~-~N}}$.To get such solutions, we look for critical points of the energy functional on the constraints 0. \end{equation}]]> For the value $p\in(\max~\{~1~+~\frac{{\alpha~+~2s}}{N},2\},\frac{N+\alpha}{N-2s})$ considered, the functional $I$ is unbounded from below on $S(c)$. By using the constrained minimization method on a suitable submanifold of $S(c)$, we prove that for any $c>0$, $I$ has a critical point on $S(c)$ with the least energy among all critical points of $I$ restricted on $S(c)$. After that, we describe a limiting behavior of the constrained critical point as $c$ vanishes and tends to infinity. Moreover, by using a minimax procedure, we prove that for any $c>0$, there are infinitely many radial critical points of $I$ restricted on $S(c)$.

Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity

In this paper, we study a class of nonlinear Choquard equation driven by the fractional Laplacian. When the potential function vanishes at infinity, we obtain the existence of a ground state solution for the fractional Choquard equation by using a non-Nehari manifold method. Moreover, in the zero mass case, we obtain a nontrivial solution by using a perturbation method. The results improve upon those in Alves, Figueiredo, and Yang (2015) and Shen, Gao, and Yang (2016).

Concentration phenomena for a fractional Choquard equation with magnetic field

We consider the following nonlinear fractional Choquard equation $$ \varepsilon^{2s}(-\Delta)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $0 0$ small enough.

Ground state solutions for asymptotically periodic fractional Choquard equations

Ground state solutions for asymptotically periodic fractional Choquard equations

Ground state solutions of fractional Choquard equations with general potentials and nonlinearities

In the present paper, we consider the following fractional Choquard equation with general potentials and nonlinearities of the form $$\begin{aligned} (-\Delta )^{s}u+V(x)u=\Big (I_{\alpha }*F(u)\Big )f(u),~~~\mathrm {in}~{\mathbb {R}}^{N}, \end{aligned}$$ where $$s\in (0,1)$$ , $$N>2s$$ , $$(-\Delta )^{s}$$ is the fractional Laplacian, $$\alpha \in (0,N)$$ , potential $$V\in C^{1}({\mathbb {R}}^{N},[0,\infty ))$$ , $$I_{\alpha }$$ is a Riesz potential, the nonlinearity F satisfies the general Berestycki–Lions-type assumptions. By introducing some new techniques, we establish the existence of ground state solution of Pohozaev-type to the above equation by variational methods.