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 establish the existence of solutions to the fractional $p-$Kirchhoff type equations with a generalized Choquard nonlinearities without assuming the Ambrosetti-Rabinowitz 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.
Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} -Δu +u & = &(I_α*F(u))F'(u) \quad \text{in } \mathbb{R}^N, \\ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation*} where $I_α$ denotes Riesz potential and $α\in (0, N)$. In this paper, we show that when $F$ is doubly critical, i.e. $F(u) = \frac{N}{N+α}|u|^{\frac{N+α}{N}}+\frac{N-2}{N+α}|u|^{\frac{N+α}{N-2}}$, the nonlinear Choquard equation admits a nontrivial solution if $N \geq 5$ and $α+ 4 < N$.
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…
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 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 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…
It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form $$ \begin{array}{rcl} -Δu +V(x) u &=& (I_α* |u|^p)|u|^{p-2}u+ λ|u|^{q-2}u, \, u \in H^1(\mathbb{R}^{N}), \end{array} $$ where $λ> 0, N \geq 3, α\in (0, N)$. The potential $V$ is a continuous function and $I_α$ denotes the standard Riesz potential. Assume also that $1 < q < 2,~2_α < p < 2^*_α$ where $2_α=(N+α)/N$, $2_α=(N+α)/(N-2)$. Our main contribution is to consider a specific condition on the parameter $λ> 0$ taking into account the nonlinear …
Infinitely many solutions for $p$-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method
We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation $$ -Δu+\varepsilon u=(I_α\ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u \quad {\rm in} \ \mathbb R^N, $$ where $N\ge 3$ is an integer, $p\in [\frac{N+α}{N}, \frac{N+α}{N-2}]$, $q\in (2,\frac{2N}{N-2}]$, $I_α$ is the Riesz potential and $\varepsilon>0$ is a parameter. We show that as $\varepsilon\to 0$ (resp. $\varepsilon\to \infty$), after a suitable rescaling the ground state solutions of $(P_\varepsilon)$ converge in $H^1(\mathbb R^N)$ to a particular solution of some limit equations. We also establish a…
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 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 prove decay with respect to some Lebesgue norms for a class of Schrödinger equations with non-local nonlinearities by showing new Morawetz inequalities and estimates. As a byproduct, we obtain large-data scattering in the energy space for the solutions to the systems of $N$ defocusing Schrödinger-Choquard equations with mass-energy intercritical nonlinearities in any space dimension and of defocusing Hartree-Fock equations, for any dimension $d\geq3$.
In this paper, the positive solutions to Choquard equation involving fully nonlinear nonlocal operator are shown to be symmetric and monotone by using the moving plane method which has been introduced by Chen, Li and Li in 2015. The key ingredients are to obtain the "narrow region principle" and "decay at infinity" for the corresponding problems. Similar ideas can be easily applied to various nonlocal problems with more general nonlinearities.
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…
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 paper deals with existence of solutions to the following fractional $p$-Laplacian system of equations \begin{equation*} %\tag{$\mathcal P$}\label{MAT1} \begin{cases} (-Δ_p)^s u =|u|^{p^*_s-2}u+ \frac{γα}{p_s^*}|u|^{α-2}u|v|^β\;\;\text{in}\;Ω, (-Δ_p)^s v =|v|^{p^*_s-2}v+ \frac{γβ}{p_s^*}|v|^{β-2}v|u|^α\;\;\text{in}\;Ω, % % u,\;v\in\wsp, \end{cases} \end{equation*} where $s\in(0,1)$, $p\in(1,\infty)$ with $N>sp$, $α,\,β>1$ such that $α+β= p^*_s:=\frac{Np}{N-sp}$ and $Ω=\mathbb{R}^N$ or smooth bounded domains in $\mathbb{R}^N$. For $Ω=\mathbb{R}^N$ and $γ=1$, we show that any ground state so…
In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents \begin{equation*} \begin{array}{rl} (-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)),\\ &~\hspace{6cm} x\in Ω, \\ u(x)&=0 ,\hspace{20mm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $\Om\subset\mathbb R^N$ 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 continuous function with…
This paper is concerned with the existence and uniqueness of the solution to a doubly nonlinear parabolic problem which arises directly from a circuit model of microwave heating. Beyond the relevance from a physical point of view, the problem is very interesting also in a mathematical approach: in fact, it consists of a nonlinear partial differential equation with a further nonlinearity in the boundary condition. Actually, we are going to prove a general result: the two nonlinearities are allowed to be maximal monotone operators and then an existence result will be shown for the resulting prob…
In this paper we discuss a family of viscous Cahn-Hilliard equations with a non-smooth viscosity term. This system may be viewed as an approximation of a "forward-backward" parabolic equation. The resulting problem is highly nonlinear, coupling in the same equation two nonlinearities with the diffusion term. In particular, we prove existence of solutions for the related initial and boundary value problem. Under suitable assumptions, we also state uniqueness and continuous dependence on data.
We consider equations of the form $Δu +λ^2 V(x)e^{\,u}=ρ$ in various two dimensional settings. We assume that $V>0$ is a given function, $λ>0$ is a small parameter and $ρ=\mathcal O(1)$ or $ρ\to +\infty$ as $λ\to 0$. In a recent paper we prove the existence of the maximal solutions for a particular choice $V\equiv 1$, $ρ=0$ when the problem is posed in doubly connected domains under Dirichlet boundary conditions. We related the maximal solutions with a novel free boundary problem. The purpose of this note is to derive the corresponding free boundary problems in other settings. Solvability of s…
We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean field equation of the equilibrium turbulance with arbitrarily signed vortices. For the first time, we consider the problem with both supercritical parameters and we give an existence result by using variational methods. In doing this, we present a new Moser-Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of both e^u and e^{-u}, where u is the unknown function in the equation.
In this article we study global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type nonlinearities, $u_{tt}-Lu_{xx}=B(- |u|^{p-1}u)_{xx}, ~(p>1)$, where the nonlocality enters through two pseudo-differential operators $L$ and $B$. We establish thresholds for global existence versus blow-up using the potential well method which relies essentially on the ideas suggested by Payne and Sattinger. Our results improve the global existence and blow-up results given in the literature for the present class of nonlocal nonlinear wave equations and c…
In this paper we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity: \begin{eqnarray*} \begin{cases} \varepsilon^{2s}M([u]_{s,A}^2)(-Δ)_{A}^su + V(x)u = (|x|^{-α}*F(|u|^2))f(|u|^2)u + |u|^{2_s^\ast-2}u,\ \ \ x\in \mathbb{R}^N,\\ u(x) \rightarrow 0,\ \ \quad \mbox{as}\ |x| \rightarrow \infty, \end{cases} \end{eqnarray*} where $(-Δ)_{A}^s$ is the fractional magnetic operator with $0<s<1$, $2_s^\ast = 2N/(N-2s)$, $α< \min\{N, 4s\}$, $M : \mathbb{R}^{+}_{0}\rightarrow …
Study the following two-component elliptic system% \begin{equation*} \left\{\aligned&Δu-(λa(x)+a_0)u+u^3+βv^2u=0\quad&\text{in }\bbr^4,\\% &Δv-(λb(x)+b_0)v+v^3+βu^2v=0\quad&\text{in }\bbr^4,\\% &(u,v)\in\h\times\h,\endaligned\right.% \end{equation*} where $a_0,b_0\in\bbr$ are constants; $λ>0$ and $β\in\bbr$ are parameters and $a(x), b(x)\geq0$ are potential wells which are not necessarily to be radial symmetric. By using the variational method, we investigate the existence of ground state solutions and general ground state solutions (i.e., possibly semi-trivial) to this system. Indeed, to the …
Let $N\ge 1$ and let $f\in C[0,\infty)$ be a nonnegative nondecreasing function and $u_0$ be a possibly singular nonnegative initial function. We are concerned with existence and nonexistence of a local in time nonnegative solution in a uniformly local Lebesgue space of a semilinear heat equation \[ \begin{cases} \partial_tu=Δu+f(u) & \textrm{in}\ \mathbb{R}^N\times(0,T),\\ u(x,0)=u_0(x) & \textrm{in}\ \mathbb{R}^N \end{cases} \] under mild assumptions on $f$. A relationship between a growth of $f$ and an integrability of $u_0$ is studied in detail. Our existence theorem gives a sharp integrab…
In this article, we study the following nonlinear doubly nonlocal problem involving the fractional Laplacian in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{\begin{aligned} (-Δ)^s u & = au+bv+\frac{2p}{p+q}\int_Ω\frac{|v(y)|^q}{|x-y|^μ}dy|u|^{p-2}u+2ξ_1\int_Ω\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy|u|^{2^*_μ-2}u,&& \text{in } Ω;\\ (-Δ)^s v & = bu+cv+\frac{2q}{p+q}\int_Ω\frac{|u(y)|^p}{|x-y|^μ}dy|v|^{q-2}v+2ξ_2\int_Ω\frac{|v(y)|^{2^*_μ}}{|x-y|^μ}dy|v|^{2^*_μ-2}v,&& \text{in } Ω;\\ u &=v=0,\text{ in } \R^N\setminusΩ, \end{aligned}\right. \end{equation*} where $Ω$ is a smooth…
We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form $dX_t-\text{div}\,γ(\nabla X_t)\,dt+β(X_t)\,dt\ni B(t,X_t)\,dW_t$, where $γ$ and $β$ are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on $\mathbb{R}^d$ and $\mathbb{R}$ respectively, and $W$ is a cylindrical Wiener process. Using variational techniques, suitable uniform estimates (both pathwise and in expectation) and some compactness results, well-posedness is proved under the classical Leray-Lions conditions on $γ$ and with no restric…
