For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation, $ -Δu+V_{μ,ν} u=(I_α\ast |u|^{\frac{N+α}{N}}){|u|}^{\fracα{N}-1}u$, in $\mathbb{R}^N$ where $N\ge 3$, $V_{μ, ν} : \mathbb{R}^N \to \mathbb{R}$ is an external potential defined for $μ,ν\in\mathbb{R}$ and $x \in \mathbb{R}^N$ by $V_{μ, ν} (x)=1-μ/(ν^2 + |x|^2)$ and $I_α: \mathbb{R}^N \to 0$ is the Riesz potential for $α\in (0,N)$, we exhibit two thresholds $μ_ν,μ^ν>0$ such that the equation admits a positive ground state solution if and only if $μ_ν<μ<μ^ν$ and no ground state solution exists for $μ<μ_ν$. Moreover, if $μ>\max\left\{μ^ν,\frac{N^2(N-2)}{4(N+1)}\right\}$, then equation still admits a sign changing ground state solution provided $N\ge4$ or in dimension $N=3$ if in addition $\frac{3}{2}<α<3$ and ker$(-Δ+ V_{μ, ν}) = \{0\}$, namely in the non-resonant case.
In this paper, an autonomous Choquard equation with the upper critical exponent is considered. By using the Pohožaev constraint method, the subcritical approximation method and the compactness lemma of Strauss, a groundstate solution in $H^1(\mathbb{R}^N)$ which is positive and radially symmetric is obtained. The result here extends and complements the earlier theorems.
We consider a semilinear elliptic problem [- Δu + u = (I_α\ast \abs{u}^p) \abs{u}^{p - 2} u \quad\text{in (\mathbb{R}^N),}] where (I_α) is a Riesz potential and (p>1). This family of equations includes the Choquard or nonlinear Schrödinger-Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groun…
We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equation $$ -Δu+u=(I_α*F(u))F'(u)\qquad\text{in }\mathbb{R}^2, $$ where $I_α$ is the Riesz potential of order $α$ on the plane $\mathbb{R}^2$ under general nontriviality, growth and subcriticality on the nonlinearity $F \in C^1 (\mathbb{R},\mathbb{R})$.
We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation $$ - Δu + a u = \frac{1}{2 π} \Bigl[\ln \frac{1}{|x|}* |u|^2 \Bigr] \ u \qquad \text{in $\mathbb{R}^2$} $$ and we establish its nondegeneracy. For the corresponding three-dimensional problem, the nondegeneracy property of the positive ground state to the Choquard equation was proved by E. Lenzmann (Analysis & PDE, 2009).
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.
We prove the existence of a nontrivial solution (u \in H^1 (\R^N)) to the nonlinear Choquard equation [- Δu + u = \bigl(I_α\ast F (u)\bigr) F' (u) \quad \text{in (\R^N),}] where (I_α) is a Riesz potential, under almost necessary conditions on the nonlinearity (F) in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover (F) is even and monotone on ((0,\infty)), then (u) is of constant sign and radially symmetric.
We consider a nonlinear Choquard equation $$ -Δu+u= (V * |u|^p )|u|^{p-2}u \qquad \text{in }\mathbb{R}^N, $$ when the self-interaction potential $V$ is unbounded from below. Under some assumptions on $V$ and on $p$, covering $p =2$ and $V$ being the one- or two-dimensional Newton kernel, we prove the existence of a nontrivial groundstate solution $u\in H^1 (\mathbb{R}^N)\setminus\{0\}$ by solving a relaxed problem by a constrained minimization and then proving the convergence of the relaxed solutions to a groundstate of the original equation.
We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -Δu+ u=\big(I_α*|u|^{\fracα{N}+1}\big)|u|^{\fracα{N}-1}u+f(x,u)\qquad \text{ in } \mathbb{R}^N \] where $N\geq 1$, $I_α$ is the Riesz potential of order $α\in (0, N)$, the exponent $\fracα{N}+1$ is critical with respect to the Hardy--Littlewood--Sobolev inequality and the nonlinear perturbation $f$ satisfies suitable growth and structural assumptions.
In this work, we have proved a version of the Hardy-Littlewood-Sobolev inequality for variable exponents. After we use the variational method to establish the existence of solution for a class of Choquard equations involving the $p(x)$-Laplacian operator.
We consider nonlinear Choquard equation $$ - Δu + V u = \bigl(I_α\ast |u|^{\fracα{N}+1}\bigr) |u|^{\fracα{N}-1} u\quad\text{in (\mathbb{R}^N)},$$ where $N \ge 3$, $V \in L^\infty (\mathbb{R}^N)$ is an external potential and $I_α(x)$ is the Riesz potential of order $α\in (0, N)$. The power $\fracα{N}+1$ in the nonlocal part of the equation is critical with respect to the Hardy-Littlewood-Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if $\liminf_{|x| \to \infty} \bigl(1 - V (x)\bigr)|x|^2 > \frac{N^2 (N - 2)}{4 (N + 1)…
We consider the following nonlinear Choquard equation with Dirichlet boundary condition $$-Δu =\left(\int_Ω\frac{|u|^{2_μ^{\ast}}}{|x-y|^μ}dy\right)|u|^{2_μ^{\ast}-2}u+λf(u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} Ω, $$ where $Ω$ is a smooth bounded domain of $\mathbb{R}^N$, $λ>0$, $N\geq3$, $0<μ<N$ and $2_μ^{\ast}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different types of nonlinearities $f(u)$, we are able to prove some existence and multiplicity results for the equation by variational methods.
We are concerned with the existence of ground states for nonlinear Choquard equations involving a critical nonlinearity in the sense of Hardy-Littlewood-Sobolev. Our result complements previous results by Moroz and Van Schaftingen where the subcritical case was considered. Then, we focus on the existence of semi-classical states and by using a truncation argument approach, we establish the existence and concentration of single peak solutions concentrating around minima of the Schrödinger potential, as the Planck constant goes to zero. The result is robust in the sense that the nonlinearity is …
In this paper we are concerned with the following nonlinear Choquard equation $$-Δu+V(x)u =\left(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^μ}dy\right)g(x,u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $N\geq4$, $0<μ<N$ and $G(x,u)=\displaystyle\int^u_0g(x,s)ds$. If $0$ lies in a gap of the spectrum of $-Δ+V$ and $g(x,u)$ is of critical growth due to the Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in \cite{AC, KS, MS2}.
In this paper, we consider the following magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^α}*|u|^{2_α^*}\right) |u|^{2_α^*-2} u + λf(u)\ \textrm{ in }\ \R^N,\] where $2_α^{*}=\frac{2N-α}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $λ>0$, $N\geq 3$, $0<α< N$, $A: \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is an $C^1$, $\mathbb{Z}^N$-periodic vector potential and $V$ is a continuous scalar potential given as a perturbation of a periodic potential. Under suitable assumptions on different types of nonlinearities $f$, n…
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.
In this paper, we study the nonlocal Choquard equation $$ -\varepsilon^2 Δu_\varepsilon + V u_\varepsilon= (I_α* |u_\varepsilon|^p)|u_\varepsilon|^{p-2}u_\varepsilon $$ where $N\geq 1$, $I_α$ is the Riesz potential of order $α\in (0, N)$ and $\varepsilon>0$ is a parameter. When the nonnegative potential $V\in C (\mathbb{R}^N)$ achieves $0$ with a homogeneous behaviour or on the closure of an open set but remains bounded away from $0$ at infinity, we show the existence of groundstate solutions for small $\varepsilon>0$ and exhibit the concentration behaviour as $\varepsilon\to 0$.
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 paper, we study the Brezis-Nirenberg type problem for Choquard equations in $\mathbb{R}^N$ \begin{equation*} -Δu+u=(I_α\ast|u|^{p})|u|^{p-2}u+λ|u|^{q-2}u \quad \mathrm{in}\ \mathbb{R}^N, \end{equation*} where $N\geq 3,\ α\in(0,N)$, $λ>0$, $q\in (2,\frac{2N}{N-2}]$, $p=\frac{N+α}{N}$ or $\frac{N+α}{N-2}$ are the critical exponents in the sense of Hardy-Littlewood-Sobolev inequality and $I_α$ is the Riesz potential. Based on the results of the subcritical problems, and by using the subcritical approximation and the Pohožaev constraint method, we obtain a positive and radially nonincreasi…
We investigate the following problem $$ -{\rm div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-2}u= \Big(|x|^{-θ}*\frac{|u|^{b}}{|x|^α}\Big)\frac{|u|^{b-2}}{|x|^α}u+λ\Big(|x|^{-γ}*\frac{|u|^{c}}{|x|^β}\Big)\frac{|u|^{c-2}}{|x|^β}u \quad\mbox{ in }\R^{N}, $$ where $b, c, α, β>0$, $θ,γ\in (0,N)$, $N\geq 3$, $2\leq m< \infty$ and $λ\in \R$. Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.
We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$-Δu =\left(\int_Ω\frac{|u|^{2_μ^{\ast}}}{|x-y|^μ}dy\right)|u|^{2_μ^{\ast}-2}u+λu\4.14mm\mbox{in}\1.14mm Ω, $$ where $Ω$ is a bounded domain of $\mathbb{R}^N$, with Lipschitz boundary, $λ$ is a real parameter, $N\geq3$, $2_μ^{\ast}=(2N-μ)/(N-2)$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.
We consider the nonlinear Choquard equation $$ -Δu+V u=(I_α\ast \vert u\vert ^p)\vert u\vert ^{p-2}u \qquad \text{ in } \mathbb{R}^N $$ where $N\geq 1$, $I_α$ is the Riesz potential integral operator of order $α\in (0, N)$ and $p > 1$. If the potential $ V \in C (\mathbb{R}^N; [0,+\infty)) $ satisfies the confining condition $$ \liminf\limits_{\vert x\vert \to +\infty}\frac{V(x)}{1+\vert x\vert ^{\frac{N+α}{p}-N}}=+\infty, $$ and $\frac{1}{p} > \frac{N - 2}{N + α}$, we show the existence of a groundstate, of an infinite sequence of solutions of unbounded energy and, when $p \ge 2$ the existenc…
In this paper we study the semiclassical limit for the singularly perturbed Choquard equation $$ -\vr^2Δu +V(x)u =\vr^{μ-3}\Big(\int_{\R^3} \frac{Q(y)G(u(y))}{|x-y|^μ}dy\Big)Q(x)g(u) \quad \mbox{in $\R^3$}, $$ where $0<μ<3$, $\vr$ is a positive parameter, $V,Q$ are two continuous real function on $\R^3$ and $G$ is the primitive of $g$ which is of critical growth due to the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on the nonlinearity $g$, we first establish the existence of ground states for the critical Choquard equation with constant coefficients in $\R^3$. Next we esta…
We study Choquard type equation of the form $$-Δu +\varepsilon u-(I_α*|u|^p)|u|^{p-2}u+|u|^{q-2}u=0\quad in \quad {\mathbb R}^N,\qquad\qquad(P_\varepsilon)$$ where $N\geq3$, $I_α$ is the Riesz potential with $α\in(0,N)$, $p>1$, $q>2$ and $\varepsilon\ge 0$. Equations of this type describe collective behaviour of self-interacting many-body systems. The nonlocal nonlinear term represents long-range attraction while the local nonlinear term represents short-range repulsion. In the first part of the paper for a nearly optimal range of parameters we prove the existence and study regularity and qual…
In this paper we consider the nonlinear Choquard equation $$ -Δu+V(x)u =\left(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^μ}dy\right)g(x,u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $0<μ<N$, $N\geq3$, $g(x,u)$ is of critical growth due to the Hardy--Littlewood--Sobolev inequality and $G(x,u)=\displaystyle\int^u_0g(x,s)ds$. Firstly, by assuming that the potential $V(x)$ might be sign-changing, we study the existence of Mountain-Pass solution via a concentration-compactness principle for the Choquard equation. Secondly, under the conditions introduced by Benci and Cerami \cite{BC1…
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 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…
In this paper we are interested in the existence of semiclassical states for the Choquard type equation $$ -\vr^2Δu +V(x)u =\Big(\int_{\R^N} \frac{G(u(y))}{|x-y|^μ}dy\Big)g(u) \quad \mbox{in $\R^N$}, $$ where $0<μ<N$, $N\geq3$, $\vr$ is a positive parameter and $G$ is the primitive of $g$ which is of critical growth due to the Hardy--Littlewood--Sobolev inequality. The potential function $V(x)$ is assumed to be nonnegative with $V(x)=0$ in some region of $\R^N$, which means it is of the critical frequency case. Firstly we study a Choquard equation with double critical exponents and prove the e…
In this paper, we study normalized ground state solutions for the following nonautonomous Choquard equation: $$-Δu-λu=\left(\frac{1}{|x|^μ}\ast A|u|^{p}\right)A|u|^{p-2}u,\quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c,\quad u\in H^1(\mathbb{R}^N,\mathbb{R}),$$ where $c>0$, $0<μ<N$, $λ\in\mathbb{R}$, $A\in C^1(\mathbb{R}^N,\mathbb{R})$. For $p\in(2_{*,μ}, \bar{p})$, we prove that the Choquard equation possesses ground state normalized solutions, and the set of ground states is orbitally stable. For $p\in (\bar{p},2^*_μ)$, we find a normalized solution, which is not a global minimizer. $2^*_μ$ and $2_{*…
In this paper, we study a class of the critical Choquard equations with axisymmetric potentials, $$ -Δu+ V(|x'|,x'')u =\Big(|x|^{-4}\ast |u|^{2}\Big)u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^6, $$ where $(x',x'')\in \mathbb{R}^2\times\mathbb{R}^{4}$, $V(|x'|, x'')$ is a bounded nonnegative function in $\mathbb{R}^{+}\times\mathbb{R}^{4}$, and $*$ stands for the standard convolution. The equation is critical in the sense of the Hardy-Littlewood-Sobolev inequality. By applying a finite dimensional reduction argument and developing novel local Pohožaev identities, we prove that if the f…
We study the Choquard equation with a local perturbation \begin{equation*} -Δu=λu+(I_α\ast|u|^p)|u|^{p-2}u+μ|u|^{q-2}u,\ x\in \mathbb{R}^{N} \end{equation*} having prescribed mass \begin{equation*} \int_{\mathbb{R}^N}|u|^2dx=a^2. \end{equation*} For a $L^2$-critical or $L^2$-supercritical perturbation $μ|u|^{q-2}u$, we prove nonexistence, existence and symmetry of normalized ground states, by using the mountain pass lemma, the Pohožaev constraint method, the Schwartz symmetrization rearrangements and some theories of polarizations. In particular, our results cover the Hardy-Littlewood-Sobolev …
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…
In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents (-Δ)_{p(\cdot)}^{s(\cdot)}u(x)&=λ|u(x)|^{α(x)-2}u(x)+ \left(\DD\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u(x)), x\in Ω, u(x)&=0, x\in \mathbb R^N\setminusΩ, where $Ω\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,μ$ and $α$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is Carathédory function. Under suitable assumption on $s,p,μ,α$ and $f(x,t)$, first we study the analogous Hardy-Sobolev-Littlewood…
In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: $$ E(u)=\frac{1}{2}\ds\int_{\R^N}|\nabla u|^2+\frac{1}{2}\ds\int_{\R^N}V(x)|u|^2-\frac{1}{2p}\ds\int_{\R^N}(I_\al*|u|^p)|u|^p $$ on $\widetilde{S}(c)=\{u\in H^1(\R^N)|\ \int_{\R^N}V(x)|u|^2<+\infty,\ |u|_2=c,c>0\},$ where $N\geq1$ $\al\in(0,N)$, $\frac{N+α}{N}\leq p<\frac{N+α}{(N-2)_+}$ and $I_\al:\R^N\rightarrow\R$ is the Riesz potential. We present sharp existence results for $E(u)$ constrained on $\widetilde{S}(c)$ when $V(x)\equiv0$ for all $\frac{N+α}{N}\leq p<\frac{…
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 this paper we are interested in the following nonlinear Choquard equation $$ -Δu+(λV(x)-β)u =\big(|x|^{-μ}\ast |u|^{2_μ^{\ast}}\big)|u|^{2_μ^{\ast}-2}u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $λ,β\in\mathbb{R}^+$, $0<μ<N$, $N\geq4$, $2_μ^{\ast}=(2N-μ)/(N-2)$ is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function $V\in \mathcal{C}(\mathbb{R}^N,\mathbb{R})$ such that $Ω:=\mbox{int} V^{-1}(0)$ is a nonempty bounded set with smooth boundary. If $β>0$ is a constant such that the operator $-Δ+λV(x)-β$ is non-degen…
In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of the form \begin{align*} -\mathcal{L}_{p,q}^{a}(u) + |u|^{p-2}u+ a(x) |u|^{q-2}u = \left( \int_{\mathbb{R}^N} \frac{F(y, u)}{|x-y|^μ}\,\mathrm{d} y\right)f(x,u) \quad\text{in } \mathbb{R}^N, \end{align*} where $\mathcal{L}_{p,q}^{a}$ is the double phase operator given by \begin{align*} \mathcal{L}_{p,q}^{a}(u):= \operatorname{div}\big(|\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \big), \quad u\in W^{1,\mathcal{H}}(\mathbb{R}^N), \end{align*} $0<μ<N$, $1<p<N$, $p<q<p…
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 paper we study the existence, multiplicity and regularity of positive weak solutions for the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \iint\limits_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dxdy\right) (-Δ)^s u = \fracλ{u^γ} + \left( \int\limits_Ω \frac{|u(y)|^{2^{*}_{μ,s}}}{|x-y|^ μ}\, dy\right) |u|^{2^{*}_{μ,s}-2}u \;\text{in} \; Ω, %\quad \quad u > 0\quad \text{in} \; Ω, \quad \quad u = 0\quad \text{in} \; \mathbb{R}^{N}\backslashΩ, \end{array} \end{equation*} where $Ω$ is open bounded domain of $\mathbb{R}^{N}$ with …
The paper is concerned with the existence and multiplicity of positive solutions of the nonhomogeneous Choquard equation over an annular type bounded domain. Precisely, we consider the following equation \[ -\De u = \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy\right)|u|^{2^*_μ-2}u+f \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a smooth bounded annular domain in $\mathbb{R}^N( N\geq 3)$, $2^*_μ=\frac{2N-μ}{N-2}$, $f \in L^{\infty}(\Om)$ and $f \geq 0$. We prove the existence of four positive solutions of the above problem using the Lusternik-Schnirelmann theory …
We consider the following Kirchhoff - Choquard equation \[ -M(\|\na u\|_{L^2}^{2})\De u = \la f(x)|u|^{q-2}u+ \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy\right)|u|^{2^*_μ-2}u \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a bounded domain in $\mathbb{R}^N( N\geq 3)$ with $C^2$ boundary, $2^*_μ=\frac{2N-μ}{N-2}$, $1<q\leq 2$, and $f$ is a continuous real valued sign changing function. When $1<q< 2$, using the method of Nehari manifold and Concentration-compactness Lemma, we prove the existence and multiplicity of positive solutions of the above problem. We also p…
In this article we are concern for the following Choquard equation \[ -Δu = λ|u|^{q-2}u +\left(\int_Ω\frac{|u(y)|^{2^*_μ}}{|x-y|^μ} dy \right)|u|^{2^*_μ-2} u \; \text{in}\; Ω,\quad u = 0 \; \text{ on } \partial Ω, \] where $Ω$ is an open bounded set with continuous boundary in $\mathbb{R}^N( N\geq 3)$, $2^*_μ=\frac{2N-μ}{N-2}$ and $q \in [2,2^*)$ where $2^*=\frac{2N}{N-2}$. Using Lusternik-Schnirelman theory, we associate the number of positive solutions of the above problem with the topology of $Ω$. Indeed, we prove if $λ< λ_1$ then problem has $\text{cat}_Ω(Ω)$ positive solutions whenever $q…
Compact embedding from variable-order Sobolev space to $L^{q(x)}(Ω)$ and its application to Choquard equation with variable order and variable critical exponent
