Fractional Choquard Research Articles

Sign-changing solutions for a fractional Choquard equation with power nonlinearity

In this paper, we study the following fractional Choquard equation (FC)(−Δ)su+V(x)u=(Iα∗|u|p)|u|p−2u,inRN,where s∈(0,1), N≥3, V(x) is continuous potential function, Iα:RN→R is the Riesz potential of order α defined by Iα(x)=|x|α−N for every x∈RN∖{0}, α∈(0,N), ∗ denotes the convolution operator, 2<p<N+αN−2s=2α,s∗, 2α,s∗ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and fractional Laplace operator and the operator (−Δ)s stands for the fractional Laplacian of order s. Combining constraint variational method, quantitative deformation lemma and the Brouwer degree theory, we prove that (FC) possesses one least energy sign-changing solution u0. Moreover, we show that the energy of u0 is strictly larger than 2p−2p−1 times and strictly less than four times the ground state energy.

Fractional Choquard Equations with an Inhomogeneous Combined Non-linearity

Fractional Choquard Equations with an Inhomogeneous Combined Non-linearity

Saddle solutions for the fractional Choquard equation

We study the saddle solutions for the fractional Choquard equation \begin{align*} (-\Delta)^{s}u+ u=(K_{\alpha}\ast|u|^{p})|u|^{p-2}u, \quad x\in \mathbb{R}^N \end{align*} where $s\in(0,1)$, $N\geq 3$ and $K_\alpha$ is the Riesz potential with order $\alpha\in (0,N)$. For every Coxeter group $G$ with rank $1\leq k\leq N$ and $p\in[2,\frac{N+\alpha}{N-2s})$, we construct a $G$-saddle solution with prescribed symmetric nodal configurations. This is a counterpart for the fractional Choquard equation of saddle solutions to the Choquard equation and further completes the existence of non-radial sign-changing solutions for this doubly nonlocal equation.

The Benci–Cerami problem for the fractional Choquard equation with critical exponent

The Benci–Cerami problem for the fractional Choquard equation with critical exponent

Existence of positive solutions for a class of fractional Choquard equation in exterior domain

&lt;p style='text-indent:20px;'&gt;In this paper we show existence of positive solutions for a class of problems involving the fractional Laplacian in exterior domain and Choquard type nonlinearity. We prove the main results using variational method combined with Brouwer theory of degree and Deformation Lemma..&lt;/p&gt;

On fractional Choquard–Kirchhoff equations with subcritical or critical nonlinearities

In this article, we study the fractional Choquard–Kirchhoff equation with subcritical or critical nonlinearities. We obtain some existence results of solutions using variational methods and the Nehari manifold under two different cases.

Existence Results for p1(x,·) and p2(x,·) Fractional Choquard–Kirchhoff Type Equations with Variable s(x,·)-Order

In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable s(x,·)-order fractional p1(x,·) and p2(x,·)-Laplacian. Assuming some reasonable conditions and with the help of variational methods, we reach a positive energy solution and a negative energy solution in an appropriate space of functions. The main difficulties and innovations are the Choquard nonlinearities and Kirchhoff functions with the presence of double Laplace operators involving two variable parameters.

Bound state solutions of fractional Choquard equation with Hardy–Littlewood–Sobolev critical exponent

Bound state solutions of fractional Choquard equation with Hardy–Littlewood–Sobolev critical exponent

On Fractional Choquard Equation with Subcritical or Critical Nonlinearities

On Fractional Choquard Equation with Subcritical or Critical Nonlinearities

On nodal solutions of the fractional Choquard equation

In this paper, we study the following fractional Choquard equation(−Δ)su+u=(Kα⁎|u|p)|u|p−2u,x∈RN. Using variational methods on Nehari manifold, we prove that there is an odd solution for this equation under an energy estimate and we also obtain a least action nodal solution for this equation. Finally, we give a nonexistence result.

Small linear perturbations of fractional Choquard equations with critical exponent

We are concerned with the qualitative analysis of positive solutions to the fractional Choquard equation{(−Δ)su+a(x)u=(Iα⁎|u|2α,s⁎)|u|2α,s⁎−2u,x∈RN,u∈Ds,2(RN),u(x)>0,x∈RN, where Iα(x) is the Riesz potential, s∈(0,1), N>2s, 0<α<min⁡{N,4s}, and 2α,s⁎=2N−αN−2s is the fractional critical Hardy-Littlewood-Sobolev exponent. We first establish a nonlocal global compactness property in the framework of fractional Choquard equations. In the second part of this paper, we prove that the equation has at least one positive solution in the case of small perturbations of the potential that describes the linear term.

A critical fractional choquard problem involving a singular nonlinearity and a radon measure

This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential. 0.1 $$\begin{aligned} \begin{aligned} (-\Delta )^su-\alpha \frac{u}{|x|^{2s}}&=\lambda u+ u^{-\gamma }+\beta \left( \int _{\Omega }\frac{u^{2_b^*}(y)}{|x-y|^b}dy\right) u^{2_b^*-1}+\mu ~\text {in}~\Omega ,\\ u&>0~\text {in}~\Omega ,\\ u&= 0~\text {in}~\mathbb {R}^N{\setminus }\Omega . \end{aligned} \end{aligned}$$ Here, $$\Omega $$ is a bounded domain of $$\mathbb {R}^N$$ , $$s\in (0,1)$$ , $$\alpha ,\lambda $$ and $$\beta $$ are positive real parameters, $$N>2s$$ , $$\gamma \in (0,1)$$ , $$0<b<\min \{N,4s\}$$ , $$2_b^*=\frac{2N-b}{N-2s}$$ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and $$\mu $$ is a bounded Radon measure in $$\Omega $$ .

Asymptotic behavior of ground states for a fractional Choquard equation with critical growth

<abstract> In this paper, we are concerned with the following fractional Choquard equation with critical growth: <p class="disp_formula">$ (-\Delta)^s u+\lambda V(x)u = (|x|^{-\mu} \ast F(u))f(u)+|u|^{2^*_s-2}u \; \hbox{in}\; \mathbb{R}^N, $ where $ s\in (0, 1) $, $ N &gt; 2s $, $ \mu\in (0, N) $, $ 2^*_s = \frac{2N}{N-2s} $ is the fractional critical exponent, $ V $ is a steep well potential, $ F(t) = \int_0^tf(s)ds $. Under some assumptions on $ f $, the existence and asymptotic behavior of the positive ground states are established. In particular, if $ f(u) = |u|^{p-2}u $, we obtain the range of $ p $ when the equation has the positive ground states for three cases $ 2s &lt; N &lt; 4s $ or $ N = 4s $ or $ N &gt; 4s $. </abstract>

Soliton Solutions for Fractional Choquard Equations

Soliton Solutions for Fractional Choquard Equations

Soliton Solutions for Fractional Choquard Equations

Soliton Solutions for Fractional Choquard Equations

Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

AbstractIn this paper, we study the singularly perturbed fractional Choquard equationε2s(−Δ)su+V(x)u=εμ−3(∫R3|u(y)|2μ,s∗+F(u(y))|x−y|μdy)(|u|2μ,s∗−2u+12μ,s∗f(u))inR3,$$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$whereε&gt; 0 is a small parameter, (−△)sdenotes the fractional Laplacian of orders ∈(0, 1), 0 &lt;μ&lt; 3,2μ,s⋆=6−μ3−2s$2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator.Fis the primitive offwhich is a continuous subcritical term. Under a local condition imposed on the potentialV, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

Multiple bound state solutions for fractional Choquard equation with Hardy–Littlewood–Sobolev critical exponent

In this paper, we study the nonlinear Choquard equation ε2s(−Δ)su+V(x)u=Iα*|u|2α,s*|u|2α,s*−2u,u∈Ds,2(RN), where s ∈ (0, 1), N ≥ 3, ɛ is the positive parameter, and 2α,s*=N+αN−2s is the critical exponent with respect to the Hardy–Littlewood–Sobolev inequality. V(x)∈LN2s(RN), where V(x) is assumed to be zero in some region of RN, which means that it is of the critical frequency case. In virtue of a global compactness result in fractional Sobolev space and Lusternik–Schnirelmann theory of critical points, we succeed in proving the multiplicity of bound state solutions.

Existence and multiplicity results for p(⋅)&q(⋅) fractional Choquard problems with variable order

This paper is concerned with the existence and multiplicity of solutions for the fractional variable order Choquard type problem where and are two fractional Laplace operators with variable order and with different variable exponents and . Here is a bounded smooth domain with at least , λ is a real parameter, β, μ and k are continuous variable parameters, while F is the primitive function of a suitable f. Under some appropriate conditions on β and k, through variational methods, we prove existence and multiplicity of solutions for the above problem.

Well-posedness and blow-up of Virial type for some fractional inhomogeneous Choquard equations

ABSTRACT In the subcritical energy case, local well-posedness is established in the radial energy space for a class of fractional inhomogeneous Choquard equations. The best constant of a Gagliardo–Nirenberg type inequality is obtained. Moreover, a sharp threshold of global existence versus blow-up dichotomy is obtained for mass super-critical and energy subcritical solutions.

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>.