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) $.
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.
In this paper, we study a class of nonlinear Choquard type equations involving a general nonlinearity. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which concentrate at the minimum points of the potential $V$. Moreover, the monotonicity of $f(s)/s$ and the so-called Ambrosetti-Rabinowitz condition are not required.
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.
This article deals with a survey of recent developments and results on Choquard equations where we focus on the existence and multiplicity of solutions of the partial differential equations which involve the nonlinearity of convolution type. Because of its nature, these equations are categorized under the nonlocal problems. We give a brief survey on the work already done in this regard following which we illustrate the problems we have addressed. Seeking the help of variational methods and asymptotic estimates, we prove our main results.
We consider the nonlinear Choquard equation $$\begin{cases} & - Δu = (I_α\ast F(u))F'(u) -μu \ \text{in}\ \mathbb{R}^N, & u \in \ H^1(\mathbb{R}^N), \ \int_{\mathbb{R}^N} |u|^2 dx=m, \end{cases} $$ where $α\in(0,N)$, $m>0$ is prescribed, $μ\in \mathbb{R}$ is a Lagarange multiplier, and $I_α$ is the Riesz potential. Under general assumptions on the nonlinearity $F,$ we prove the existence and multiplicity of normalized solutions.
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.
We study the problem \[ -\De u = \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy\right)|u|^{2^*_μ-2}u, \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a smooth bounded domain in $\mathbb{R}^N( N\geq 3)$, $2^*_μ=\frac{2N-μ}{N-2}$. we prove the existence of a positive solution of the above problem in an annular type domain when the inner hole is sufficiently small.
In the spirit of Berestycki and Lions, we prove the existence of saddle type nodal solutions for the Choquard equation \[ -Δu + u= \big(I_α\ast F(u)\big)F'(u)\qquad \text{ in }\;\mathbb{R}^N \] where $N\geq 2$ and $I_α$ is the Riesz potential of order $α\in (0,N)$. Without any compact setting, we construct saddle solutions in a unified way for the Choquard equation whose nodal domains are of cone shapes demonstrating Coxeter's symmetric configurations in $\mathbb R^N$. Moreover, if $F'$ is odd and has constant sign on $(0,+\infty)$, then the saddle solution maintains signed on the fundamental …
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).
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 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 …
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.
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.…
This article is concerned with the existence of positive weak solutions for the following quasilinear Schrödinger Choquard equation: \begin{equation*} \begin{array}{cc} \displaystyle -div(g^2(u)\nabla u) + g(u)g'(u)\nabla u + a(x) u = k(x, u) \;\text{in} \; \mathbb{R}^N, \end{array} \end{equation*} where $N \geq 3$, $\displaystyle k(x,u) := h(x,u) + (I_{\vartheta}*|u|^{α\cdot2^*_μ})|u|^{α\cdot2^*_μ-2}u$, $g : \mathbb{R} \to \mathbb{R}^+$ is a differentiable even function with $g(0) = 1$ and $g'(t) \geq 0$ for all $t \geq 0$; $h\in C( \mathbb{R}^N \times\mathbb{R}, \mathbb{R})$ and the potentia…
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 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 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…
The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $(P_λ)$(See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of non-smooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.
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 …
We are interested in the general Choquard equation \begin{multline*} \sqrt{\strut -Δ+ m^2} \ u - mu + V(x)u - \fracμ{|x|} u = \left( \int_{\mathbb{R}^N} \frac{F(y,u(y))}{|x-y|^{N-α}} \, dy \right) f(x,u) - K (x) |u|^{q-2}u \end{multline*} under suitable assumptions on the bounded potential \(V\) and on the nonlinearity \(f\). Our analysis extends recent results by the second and third author on the problem with $μ= 0$ and pure-power nonlinearity $f(x,u)=|u|^{p-2}u$. We show that, under appropriate assumptions on the potential, whether the ground state does exist or not. Finally, we study the a…
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 …
In the present paper, we consider the following singularly perturbed problem: \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^2\triangle u+V(x)u=\varepsilon^{-α}(I_α*F(u))f(u), & x\in \R^N; u\in H^1(\R^N), \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $N\ge 3$, $α\in (0, N)$, $F(t)=\int_{0}^{t}f(s)\mathrm{d}s$ and $I_α: \R^N\rightarrow \R$ is the Riesz potential. By introducing some new tricks, we prove that the above problem admits a semiclassical ground state solution ($\varepsilon\in (0,\varepsilon_0)$) and a ground state solution ($\varepsilon=1$) under…
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 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 $F \in C^1 (\mathbb{R},\mathbb{R})$.
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 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.
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 discuss the existence of ground state solutions for the Choquard equation $$-Δu=(I_α*F(u))F'(u)\quad\quad\quad\text{in }\mathbb R^N.$$ We prove the existence of solutions under general hypotheses, investigating in particular the case of a homogeneous nonlinearity $F(u)=\frac{|u|^p}p$. The cases $N=2$ and $N\ge3$ are treated differently in some steps. The solutions are found through a variational mountain pass strategy. The result presented are contained in the papers with arXiv ID 1212.2027 and 1604.03294
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(…
We consider the positive solutions for the nonlocal Choquard equation $- Δu + u - (|\cdot|^{-α} * |u|^p) |u|^{p-2} u = 0$ in $\mathbb{R}^d$. Compared with ground states, positive solutions form a larger class of solutions and lack variational information. Within the range of parameters of Ma-Zhao's result [Ma-Zhao, 2010] on symmetry, we prove a priori estimates for positive solutions, generalizing the classical method of De Figueiredo-Lions-Russbaum [De Figueiredo-Lions-Nussbaum, 1982] to the unbounded domain and the nonlocal nonlinearity in our model. As an application, we show uniqueness and…
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 propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in $\mathbb{R}^N$ \begin{equation*} \sqrt{\strut -Δ+ m^2} u - mu + V(x)u = \left( \int_{\mathbb{R}^N} \frac{|u(y)|^p}{|x-y|^{N-α}} \, dy \right) |u|^{p-2}u - Γ(x) |u|^{q-2}u, \end{equation*} where $m > 0$ and the potential $V$ is decomposed as the sum of a $\mathbb{Z}^N$-periodic term and of a bounded term that decays at infinity. The result is proved by variational methods applied to an auxiliary problem in the half-space $\mathbb{R}_{+}^{N+1}$.
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…
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 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…
