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-Sobolev inequality) and $F$ is its primitive. Next, for $p\geq2,$ we discuss the Sobolev versus Hölder minimizers of the energy functional $J$ associated to the above problem, and using that we establish the existence of the local minimizer of $J$ in the fractional Sobolev space $W_0^{s,p}(Ω).$ Moreover, we discuss the aforementioned results by adding a local perturbation term (at most critical in the sense of Sobolev inequality) in the right-hand side in the above equation.
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.
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 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…
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.
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）
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 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 …
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.
In this article, we are study the following Dirichlet problem with Choquard type non linearity \[ -Δ_{\mathbb{H}} u = a u+ \left(\int_Ω\frac{|u(η)|^{Q^*_λ}}{|η^{-1}ξ|^λ}dη\right)|u|^{Q^*_λ-2}u \; \text{in}\; Ω,\quad u = 0 \; \text{ on } \partial Ω, \] where $Ω$ is a smooth bounded subset of the Heisenberg group $\mathbb{H}^N, N\in \mathbb N$ with $C^2$ boundary and $Δ_{\mathbb{H}}$ is the Kohn Laplacian on the Heisenberg group $\mathbb{H}^N$. Here, $Q^*_λ=\frac{2Q-λ}{Q-2},\; Q= 2N+2$ and $a$ is a positive real parameter. We derive the Brezis-Nirenberg type result for the above problem. Moreove…
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.
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.
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.
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…
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 …
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 …
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…
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…
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…
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 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. …
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…
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 …
Infinitely many solutions for $p$-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method
In this article, we study the following nonlinear Choquard equation with singular nonlinearity \begin{equation*} \quad -\De u = \la u^{-q} + \left( \int_{\Om}\frac{|u|^{2^*_μ}}{|x-y|^μ}\mathrm{d}y \right)|u|^{2^*_μ-2}u, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{on}\; \partial\Om, \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > 2,\; \la >0,\; 0 < q < 1, \; 0<μ<n$ and $2^*_μ=\frac{2n-μ}{n-2}$. Using variational approach and structure of associated Nehari manifold, we show the existence and multiplicity of positive weak solutions of …
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 …
We consider the $N$-Laplacian Schrödinger equation strongly coupled with higher order fractional Poisson's equations. When the order of the Riesz potential $α$ is equal to the Euclidean dimension $N$, and thus it is a logarithm, the system turns out to be equivalent to a nonlocal Choquard type equation. On the one hand, the natural function space setting in which the Schrödinger energy is well defined is the Sobolev limiting space $W^{1,N}(\mathbb{R}^N)$, where the maximal nonlinear growth is of exponential type. On the other hand, in order to have the nonlocal energy well defined and prove th…
We prove the existence of solutions for the following critical Choquard type problem with a variable-order fractional Laplacian and a variable singular exponent \begin{align*} \begin{split} a(-Δ)^{s(\cdot)}u+b(-Δ)u&=λ|u|^{-γ(x)-1}u+\left(\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u) & +ηH(u-α)|u|^{r(x)-2}u,~\text{in}~Ω, u&=0,~\text{in}~\mathbb{R}^N\setminusΩ. \end{split} \end{align*} where $a(-Δ)^{s(\cdot)}+b(-Δ)$ is a mixed operator with variable order $s(\cdot):\mathbb{R}^{2N}\rightarrow (0,1)$, $a, b\geq 0$ with $a+b>0$, $H$ is the Heaviside function (i.e., $H(t)=0$ if $t\leq0$, $H(…
For a one-phase free boundary problem involving a fractional Laplacian, we prove that "flat free boundaries" are $C^{1,α}$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.
We consider Orlicz-growth generalization to evolutionary $p$-Laplacian and to the evolutionary symmetric $p$-Laplacian. We derive the spatial second-order Caccioppoli estimate for a local weak solution to these systems. The result is new even for the $p$-case.
We prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack inequality.
In this paper we study the existence of positive smooth solutions for a class of singular (p(x),q(x))- Laplacian systems by using sub and supersolution methods.
We establish interior and up to the boundary Hölder regularity estimates for weak solutions of the Dirichlet problem for the fractional $g-$Laplacian with bounded right hand side and $g$ convex. These are the first regularity results available in the literature for integro-differential equations in the context of fractional Orlicz-Sobolev spaces.
We consider the IBVP in exterior domains for the p-Laplacian parabolic system. We prove regularity up to the boundary, extinction properties for p \in ( 2n/(n+2) , 2n/(n+1) ) and exponential decay for p= 2n/(n+1) .
We study the regularity of the solutions to initial-boundary value problems for N-systems of the p-Laplacian type, in $n\geq 3$ space variables, with square-integrable external forces in the space-time cylinder. So, the ellipticity coefficient remains unbounded. The singular case $μ=0 $ is covered.
We prove interior Lipschitz regularity result for weak and viscosity solutions of the pseudo $p$Laplacien $(p-1)\sum_i |\partial_i u|^{p-2} \partial_{ii} u = f$ for $p>2$ and $f$ bounded.
In this paper we study an evolution equation involving the normalized $p$-Laplacian and a bounded continuous source term. The normalized $p$-Laplacian is in non divergence form and arises for example from stochastic tug-of-war games with noise. We prove local $C^{α, \fracα{2}}$ regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration.
We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.
