In this paper we study the following class of fractional Choquard--type equations \[ (-Δ)^{1/2}u + u=\Big( I_μ\ast F(u)\Big)f(u), \quad x\in\mathbb{R}, \] where $(-Δ)^{1/2}$ denotes the $1/2$--Laplacian operator, $I_μ$ is the Riesz potential with $0<μ<1$ and $F$ is the primitive function of $f$. We use Variational Methods and minimax estimates to study the existence of solutions when $f$ has critical exponential growth in the sense of Trudinger--Moser inequality.
We investigate a class of nonlinear Schrodinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.
In this article, we establish the existence of solutions to the fractional $p-$Kirchhoff type equations with a generalized Choquard nonlinearities without assuming the Ambrosetti-Rabinowitz condition.
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.
This paper deals with Strauss and Lions-type theorems for fractional Sobolev spaces with variable exponent $W^{s,p(.),\tilde{p}(.,.)}(Ω)$, when $p$ and $\tilde{p}$ satisfies some conditions. As application, we study the existence of solutions for a class of Kirchhoff-Choquard problem in $\mathbb{R}^N$.
We study the saddle solutions for the fractional Choquard equation \begin{align*} (-Δ)^{s}u+ u=(K_α\ast|u|^{p})|u|^{p-2}u, \quad x\in \mathbb{R}^N \end{align*} where $s\in(0,1)$, $N\geq 3$ and $K_α$ is the Riesz potential with order $α\in (0,N)$. For every Coxeter group $G$ with rank $1\leq k\leq N$ and $p\in[2,\frac{N+α}{N-2s})$, 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 th…
This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy-Littlewood-Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik-Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.
In this article we study the existence of weak solution, existence of ground state solution using Nehari manifold and existence of infinitely many solutions using Fountain theorem and Dual fountain theorem for a class of doubly nonlocal Kirchhoff-Choquard type equations involving the variable-order fractional $p(\cdot)-$ Laplacian operator. Here the nonlinearity does not satisfy the well known Ambrosetti-Rabinowitz type condition.
We study the following nonlinear Choquard equation driven by a fractional Laplacian: $$ (-Δ)^{s}u+ u =(|x|^{-μ}\ast F(u))f(u)|{4.14mm}{in}|{1.14mm} \mathbb{R}^N, $$ with $N\geq3$, $s\in(0,1)$ and $μ\in(0,N)$. By Supposing that the nonlinearities satisfy the general Berestycki-Lions type conditions \cite{BL}, we are able to prove the existence of groundstates for this equation by variational methods.
This paper is devoted to study the following Choquard equation \begin{eqnarray*}\left\{ \begin{array}{lll} (-\triangle)^{α/2}u=(|x|^{β-n}\ast u^p)u^{p-1},~~~&x\in R^n, u\geq0,\,\,&x\in R^n, \end{array} \right. \end{eqnarray*} where $0<α,β<2$, $1\leq p<\infty$, 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.
In this paper, we are concerned with the existence and asymptotic behavior of least energy solutions for following nonlinear Choquard equation driven by fractional Laplacian $$(-Δ)^{s} u+λV(x)u=(I_α\ast F(u))f(u) \ \ in \ \ R^{N},$$ where $N> 2s$, $ (N-4s)^{+}<α< N$, $λ$ is a positive parameter and the nonnegative potential function $V(x)$ is continuous. By variational methods, we prove the existence of least energy solution which localize near the potential well $int (V^{-1}(0))$ as $λ$ large enough.
This paper is devoted to the study of the following fractional Choquard equation $$ \varepsilon^{2s}(-Δ)^{s} u + V(x)u = \varepsilon^{μ-N}\left(\frac{1}{|x|^μ}*F(u)\right)f(u) \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-Δ)^{s}$ is the fractional Laplacian, $V$ is a positive continuous potential with local minimum, $0<μ<2s$, and $f$ is a superlinear continuous function with subcritical growth. By using the penalization method and the Ljusternik-Schnirelmann theory, we investigate the multiplicity and concentration of positive solutions for the …
We consider the following nonlinear fractional Choquard equation, \begin{equation}\label{e:introduction} \begin{cases} (-Δ)^{s} u + u = (1 + a(x))(I_α\ast (|u|^{p}))|u|^{p - 2}u\quad\text{ in }\mathbb{R}^N,\\ u(x)\to 0\quad\text{ as }|x|\to \infty, \end{cases} \end{equation} here $s\in (0, 1)$, $α\in (0, N)$, $p\in [2, \infty)$ and $\frac{N - 2s}{N + α} < \frac{1}{p} < \frac{N}{N + α}$. Assume $\lim_{|x|\to\infty}a(x) = 0$ and satisfying suitable assumptions but not requiring any symmetry property on $a(x)$, we prove the existence of ground state solutions.
This paper deals with the following fractional Schr$ \ddot{\textrm{o}}$dinger equations with Choquard-type nonlinearities \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} (-Δ)^{\fracα{2}}u + u - C_{n,-β} \,(|x|^{β-n}\ast u^{p})\, u^{p-1}& = &0 & \mbox{in}\ \ \mathbb{R}^{n}\,, \\[0.05cm] u & > & 0 & \mbox{on}\ \ \mathbb{R}^{n}, \end{array}\right. \end{equation*} where $ 0< α,β< 2, 1\leq p <\infty \,\,and\,\, n\geq 2. $ First we construct a decay result at infinity and a narrow region principle for related equations. Then we establish the radial symmetry of positive solutions for the a…
We consider the following nonlinear fractional Choquard equation $$ \varepsilon^{2s}(-Δ)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{μ-N}\left(\frac{1}{|x|^μ}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $0<μ<2s$, $N\geq 3$, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a smooth magnetic potential, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive potential with a local minimum and $f$ is a continuous nonlinearity with subcritical growth. By using variational methods…
In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian \[ (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{μ,s}}}{|x-y|^μ}\mathrm{d}y \right)|u|^{2^*_{μ,s}-2}u +\la u \; \text{in } \Om,\] where $\Om $ is a bounded domain in $\mathbb R^n$ with Lipschitz boundary, $\la $ is a real parameter, $s \in (0,1)$, $n >2s$ and $2^*_{μ,s}= (2n-μ)/(n-2s)$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation …
In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation $(-Δ)^{\frac{1}{2}} u + au + λ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}$, for $ a>0 $, $ λ>0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under exponential critical and subcritical growth. Morever, when $ f $ has subcritical growth we guarantee the existence of infinitely many solutions, via genus theory.
In this paper we deal with the following fractional Choquard equation \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{sp}(-Δ)^{s}_{p} u + V(x)|u|^{p-2}u = \varepsilon^{μ-N}\left(\frac{1}{|x|^μ}*F(u)\right)f(u) \mbox{ in } \mathbb{R}^{N},\\ u\in W^{s,p}(\R^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $p\in (1, \infty)$, $N>sp$, $(-Δ)^{s}_{p}$ is the fractional $p$-Laplacian, $V$ is a positive continuous potential, $0<μ<sp$, and $f$ is a continuous superlinear function with subcritical growth. …
We study the equation \begin{equation} (-Δ)^{s}u+V(x)u= (I_α*|u|^{p})|u|^{p-2}u+λ(I_β*|u|^{q})|u|^{q-2}u \quad\mbox{ in } \R^{N}, \end{equation} where $I_γ(x)=|x|^{-γ}$ for any $γ\in (0,N)$, $p, q >0$, $α,β\in (0,N)$, $N\geq 3$ and $ λ\in R$. First, the existence of a groundstate solutions using minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.
Infinitely many solutions for $p$-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method
In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: \begin{equation*} \begin{array}{rl} (-Δ)^s u &= \displaystyle-λ|u|^{q-2}u + au + b\left( \int\limits_Ω \frac{(u^{+}(y))^{2^{*}_{μ,s}}}{|x-y|^ μ}\, dy\right) (u^{+})^{2^{*}_{μ,s}-2}u \quad\text{in} \; Ω, u &= 0\quad \text{in} \; \mathbb{R}^{N}\backslashΩ, \end{array} \end{equation*} where $Ω$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$. Here $(-Δ)^s$ is the fractional Laplace operator, $λ> 0$ is a …
In the present work we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation $(-Δ)_{p}^{s}u + |u|^{p-2}u + (\ln|\cdot|\ast |u|^{p})|u|^{p-2}u = f(u) \textrm{ \ in \ } \mathbb{R}^N $ , where $ N=sp $, $ s\in (0, 1) $, $ p>2 $, $ a>0 $, $ λ>0 $ and $f: \mathbb{R}\rightarrow \mathbb{R} $ a continuous nonlinearity with exponential critical and subcritical growth. We guarantee the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under critical and subcritical growth. Morever, when $ f $ has subcritical…
In this work, we study the higher order Kirchhoff type Choquard equation $(KC)$ involving a critical exponential non-linearity and singular weights. We prove the existence of solution to $(KC)$ using Mountain pass Lemma in light of Moser-Trudinger and singular Adams-Moser inequalities. In the second part of the paper, using the Nehari manifold technique and minimization over its suitable subsets, we prove the existence of at least two solutions to the Kirchhoff type Choquard equation $(\mathcal{P_{\la,M}})$ involving convex-concave type non-linearity.
In the present work we are concerned with the Choquard Logarithmic equation $-Δu + au + λ(\ln|\cdot|\ast |u|^{2})u = f(u)$ in $\mathbb{R}^2$, for $ a>0 $, $ λ>0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution. Also, we provide these results under a symmetric setting, taking into account subgroups of $ O(2) $.
In this article, first we address the regularity of weak solution for a class of $p$-fractional Choquard equations: \begin{equation*} \;\;\; \left.\begin{array}{rl} (-Δ)_p^su&=\left(\displaystyle\int_Ω\frac{F(y,u)}{|x-y|^μ}dy\right)f(x,u),\hspace{5mm}x\in Ω, u&=0,\hspace{35mm}x\in \mathbb R^N\setminus Ω, \end{array} \right\} \end{equation*} where $Ω\subset\mathbb R^N$ is a smooth bounded domain, $1<p<\infty$ and $0<s<1$ such that $sp<N,$ $0<μ<\min\{N,2sp\}$ and $f:Ω\times\mathbb R\to\mathbb R$ is a continuous function with at most critical growth condition (in the sense of Hardy-Littlewood-Sob…
In this article, we deal with the existence of non-negative solutions of the class of following non local problem $$ \left\{ \begin{array}{lr} \quad - M\left(\displaystyle\int_{\mathbb R^n}\int_{\mathbb R^{n}} \frac{|u(x)-u(y)|^{\frac{n}{s}}}{|x-y|^{2n}}~dxdy\right) (-Δ)^{s}_{n/s} u=\left(\displaystyle\int_Ω\frac{G(y,u)}{|x-y|^μ}~dy\right)g(x,u) \; \text{in}\; Ω,\\ \quad \quad u =0\quad\text{in} \quad \mathbb R^n \setminus Ω, \end{array} \right. $$ where $(-Δ)^{s}_{n/s}$ is the $n/s$-fractional Laplace operator, $n\geq 1$, $s\in(0,1)$ such that $n/s\geq 2$, $Ω\subset \mathbb R^n$ is a bounded …
This paper deals with the following fractional Choquard equation $$\varepsilon^{2s}(-Δ)^su +Vu=\varepsilon^{-α}(I_α*|u|^p)|u|^{p-2}u\ \ \ \mathrm{in}\ \mathbb{R}^N,$$ where $\varepsilon>0$ is a small parameter, $(-Δ)^s$ is the fractional Laplacian, $N>2s$, $s\in(0,1)$, $α\in\big((N-4s)_{+}, N\big)$, $p\in[2, \frac{N+α}{N-2s})$, $I_α$ is a Riesz potential, $V\in C\big(\mathbb{R}^N, [0, +\infty)\big)$ is an electric potential. Under some assumptions on the decay rate of $V$ and the corresponding range of $p$, we prove that the problem has a family of solutions $\{u_\varepsilon\}$ concentrating a…
In this paper, we study the following biharmonic Choquard type equation \begin{align*} \begin{split} \left\{ \begin{array}{ll} γΔ^2u-βΔu=λu+(I_μ*F(u))f(u), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=c^2>0,\quad u\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $γ>0$, $β\geq0$, $λ\in \mathbb{R}$, $I_μ=\frac{1}{|x|^μ}$ with $μ\in (0,4)$, $F(u)$ is the primitive function of $f(u)$, and $f$ is a continuous function with exponential critical growth. We can prove the existence of ground state normalized solutions for the above problem when the no…
In present paper, we study the fractional Choquard equation $$\varepsilon^{2s}(-Δ)^s u+V(x)u=\varepsilon^{μ-N}(\frac{1}{|x|^μ}\ast F(u))f(u)+|u|^{2^\ast_s-2}u$$ where $\varepsilon>0$ is a parameter, $s\in(0,1),$ $N>2s,$ $2^*_s=\frac{2N}{N-2s}$ and $0<μ<\min\{2s,N-2s\}$. 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.
Concentrating solutions of the fractional $(p,q)$-Choquard equation with exponential growth
College of Mathematics, Changchun Normal University（长春师范大学数学学院）；Faculty of Education, University of Ljubljana（卢布尔雅那大学教育学院）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）
In the present work we briefly explain how to adapt techniques already used in fractional and $p$-fractional Laplacian cases to obtain the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution, for the critical case, and the existence of infinitely many solutions, for the subcritical case, to the Choquard Logarithmic equation, $-Δ_N u + a(x)|u|^{N-2}u + λ(\ln|\cdot|\ast |u|^{N})|u|^{N-2}u = f(u) \textrm{ in } \mathbb{R}^N $, where $ a:\mathbb{R}^N \rightarrow \mathbb{R} $, $ λ>0 $, $ N \geq 3 $ and $f: \mathbb{R} \rightarrow [0, \infty) $ is conti…
In this paper, we consider the normalized ground state solution for the following biharmonic Choquard type problem \begin{align*} \begin{split} \left\{ \begin{array}{ll} Δ^2u-βΔu=λu+(I_μ*F(u))f(u), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=c^2,\quad u\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $β\geq0$, $c>0$, $λ\in \mathbb{R}$, $I_μ=\frac{1}{|x|^μ}$ with $μ\in (0,4)$, $F(u)$ is the primitive function of $f(u)$, and $f$ is a continuous function with exponential critical growth in the sense of the Adams inequality. By using a minimax p…
In this paper, we investigate the following fractional Choquard type equation: \[ (- Δ)_p^s\, u = λ\frac{|u|^{r-2}u}{|x|^α}\,+γ\big(\int_Ω\frac{|u|^q}{|x-y|^μ}dy\big) |u|^{q-2}u \ \ \text{in } Ω,\ \ u = 0 \ \text{in } \R^N \setminus Ω, \] where $Ω$ is a bounded domain in $\R^N$ with Lipschitz boundary, $p>1$, $0<s<1$, $N>sp$, $0\leqα\leq sp$, $0<μ<N$,$λ, γ>0$, $p\leq r\leq p^*_α$, $p\leq 2q\leq 2p_{μ,s}^*$, $p_α^*=\frac{(N-α)p}{N-sp}$ and $p_{μ,s}^*=\frac{(N-\fracμ{2})p}{N-sp}$ are the fractional critical Hardy-Sobolev and the critical exponents in the sense of Hardy-Littlewood-Sobolev inequal…
The present work is concerned with the following version of Choquard Logarithmic equations $ -Δ_p u -Δ_N u + a|u|^{p-2}u + b|u|^{N-2}u + λ(\ln|\cdot|\ast G(u))g(u) = f(u) \textrm{ in } \mathbb{R}^N $ , where $ a, b, λ>0 $, $ \max\{\frac{N}{2}, 2 \} < p< N $, $f, g: \mathbb{R} \rightarrow \mathbb{R} $ are continuous functions that behave like $ \exp(α|u|^{\frac{N}{N-1}}) $ at infinity, for $ α>0 $, and that has polynomial growth, respectively, and $ G(s)=\int\limits_{0}^{s}g(τ)dτ$. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution.…
On Coron problems with Choquard term and mixed operator
Laboratoire de Mathématiques et de leurs Applications, Université de Pau et des Pays de l’Adour（波城及阿杜尔地区大学应用数学与数学实验室）；Department of Mathematics, Indian Institute of Technology Jodhpur（焦特普尔印度理工学院数学系）；Department of Mathematics, Indian Institute of Technology Delhi（德里印度理工学院数学系）
In this paper, we study the nonlinear Choquard equation \begin{eqnarray*} Δ^{2}u-Δu+(1+λa(x))u=(R_α\ast|u|^{p})|u|^{p-2}u \end{eqnarray*} on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension $N\geq 2$, where $α\in(0,N),\,p>\frac{N+α}{N},\,λ$ is a positive parameter and $R_α$ stands for the Green's function of the discrete fractional Laplacian, which has same asymptotics as the Riesz potential. Under some assumptions on $a(x)$, we establish the existence and asymptotic behavior of ground state solutions for the nonlinear Choquard equation by the method of Ne…
Normalized Solutions and Semiclassical Concentration for Upper-Critical Fractional Choquard Equations
Ghent University（根特大学）；Guangxi University of Science and Technology（广西科技大学）；Queen Mary University of London（伦敦大学玛丽女王学院）；Yunnan Normal University（云南师范大学）
In this article, we first employ the concentration compactness techniques to prove existence and stability results of standing waves for nonlinear fractional Schrödinger-Choquard equation \[ i\partial_tΨ+ (-Δ)^αΨ= a |Ψ|^{s-2}Ψ+λ\left( \frac{1}{|x|^{N-β}} \star |Ψ|^p \right)|Ψ|^{p-2}Ψ \ \ \mathrm{in}\ \mathbb{R}^{N+1}, \] where $N\geq 2$, $α\in (0,1)$, $β\in (0, N)$, $s\in (2, 2+\frac{4α}{N})$, $p\in [2, 1+\frac{2α+β}{N})$, and the constants $a, λ$ are nonnegative satisfying $a+λ> 0.$ We then extend the arguments to establish similar results for coupled standing waves of nonlinear fractional Sc…
This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see $(KC)$ below). We use the variational method in the light of Moser-Trudinger inequality to show the existence of weak solutions to $(KC)$. Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex-concave problem $(\mathcal{P}_{\la,M})$ below.
In this paper, let $G$ be a Cayley graph of a discrete group of polynomial growth with homogeneous dimension $N\geq3$. We study the Choquard type equation on $G$: \begin{equation} Δu+(R_α\ast\mid u\mid^{p})\mid u\mid^{p-2}u=0, \end{equation} where $α\in(0,N)$, $p>\frac{N+α}{N-2}$ and $R_α$ stands for the Green's function of the discrete fractional Laplace operator, which has same asymptotics as the Riesz potential. We prove the discrete Hardy-Littlewood-Sobolev inequality on such Cayley graphs, and by the discrete Concentration-Compactness principle we prove the existence of extremal functions…
