Choquard Equation Research Articles

Bound states of fractional Choquard equations with Hardy-Littlewood-Sobolev critical exponent

We deal with the following fractional Choquard equation(−Δ)su+V(x)u=(Iμ⁎|u|2μ,s⁎)|u|2μ,s⁎−2u,x∈RN, where Iμ(x) is the Riesz potential, s∈(0,1), 2s<N≠4s, 0<μ<min⁡{N,4s} and 2μ,s⁎=2N−μN−2s is the fractional critical Hardy-Littlewood-Sobolev exponent. By combining variational methods and the Brouwer degree theory, we investigate the existence and multiplicity of positive bound solutions to this equation when V(x) is a positive potential bounded from below. The results obtained in this paper extend and improve some recent works in the case where the coefficient V(x) vanishes at infinity.

Multi-bump Solutions for the Quasilinear Choquard Equation in $\mathbb {R}^{N}$

Multi-bump Solutions for the Quasilinear Choquard Equation in $\mathbb {R}^{N}$

Multiplicity of Concentrating Solutions for Choquard Equation with Critical Growth

Multiplicity of Concentrating Solutions for Choquard Equation with Critical Growth

Qualitative properties of singular solutions of Choquard equations

Our aim of this paper is to study qualitative properties of isolated singular solutions to Choquard equation \begin{equation}\label{eq 0.1} -\Delta u+ u =I_\alpha[u^p] u^q+k\delta_0\quad {\rm in}\ \, \mathcal{D}'(\mathbb{R}^N), \tag{0.1} \end{equation} where $p, \, q\ge 1$, $N\ge2$, $\alpha\in(0,N)$, $k > 0$, $\delta_0$ is the Dirac mass concentrated at the origin and $I_\alpha[u^p](x)=\int_{\mathbb{R}^N} \frac{u(y)^p}{|x-y|^{N-\alpha}}\, dy.$ Multiple properties of very weak solutions of (0.1) are considered: (i) to obtain the existence of minimal solutions and extremal solutions for $N=2$, which are derived in [8] when $N\geq3$; (ii) to analyze the stability of minimal solutions and the semi stability of extremal solutions; (iii) to derive a second solution by the Mountain Pass theorem when $q=p-1$ and $N=2,3$; (iv) to obtain the radial symmetry of the positive singular solutions by the method of moving planes.

Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential

&lt;p style='text-indent:20px;'&gt;In this paper, we study a class of Choquard equations with critical exponent and Dipole potential. We prove the existence of radial ground state solutions for Choquard equations by using the refined Sobolev inequality with the Morrey norm, and show that any nonnegative weak solutions of Choquard equations have additional decay properties in terms of ground state representation.&lt;/p&gt;

Normalized solutions to lower critical Choquard equation with a local perturbation

In this paper, we study the existence and non-existence of normalized solutions to the lower critical Choquard equation with a local perturbation $ \begin{equation*} \begin{cases} -\Delta u+\lambda u = \gamma (I_{\alpha}\ast|u|^{\frac{N+\alpha}{N}})|u|^{\frac{N+\alpha}{N}-2}u+\mu |u|^{q-2}u, \quad {\rm{in}}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx = c^2, \end{cases} \end{equation*} $ where $ \gamma, \mu, c&gt;0 $, $ 2&lt;q\leq 2+\frac{4}{N} $, and $ \lambda\in \mathbb{R} $ is an unknown parameter that appears as a Lagrange multiplier. The results of this paper about this equation answer some questions proposed by Yao, Chen, Rǎdulescu and Sun [Siam J. Math. Anal., 54(3) (2022), 3696-3723]. Moreover, based on the results obtained, we study the multiplicity of normalized solutions to the non-autonomous Choquard equation$ \begin{equation*} \begin{cases} -\Delta u+\lambda u = (I_\alpha\ast [h(\epsilon x)|u|^{\frac{N+\alpha}{N}}])h(\epsilon x)|u|^{\frac{N+\alpha}{N}-2}u+\mu|u|^{q-2}u, \ x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx = c^2, \end{cases} \end{equation*} $where $ \epsilon&gt;0 $, $ 2&lt;q&lt;2+\frac{4}{N} $, and $ h $ is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of $ h $ when $ \epsilon $ is small enough.

Normalized ground states for the lower critical fractional Choquard equation with a focusing local perturbation

Normalized ground states for the lower critical fractional Choquard equation with a focusing local perturbation

Existence and Concentration of Sign-Changing Solutions of Quasilinear Choquard Equation

Existence and Concentration of Sign-Changing Solutions of Quasilinear Choquard Equation

Existence and asymptotical behavior of the ground state solution for the Choquard equation on lattice graphs

&lt;abstract&gt;&lt;p&gt;In this paper, we study the nonlinear Choquard equation&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation*} - \Delta u + V(x)u = \left( {\sum\limits_{y \ne x \atop y \in { \mathbb {Z} ^{N}} } {\frac{|u(y)|^p}{|x-y|^{N-\alpha}}} }\right )|u|^{p-2}u \end{equation*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;on lattice graph $ \mathbb {Z}^{N} $. Under some suitable assumptions, we prove the existence of a ground state solution of the equation on the graph when the function $ V $ is periodic or confining. Moreover, when the potential function $ V(x) = \lambda a(x)+1 $ is confining, we obtain the asymptotic properties of the solution $ u_\lambda $ which converges to a solution of a corresponding Dirichlet problem as $ \lambda\rightarrow \infty $.&lt;/p&gt;&lt;/abstract&gt;

Multiplicity and Concentration of Solutions for Choquard Equation with Competing Potentials via Pseudo-Index Theory

Multiplicity and Concentration of Solutions for Choquard Equation with Competing Potentials via Pseudo-Index Theory

Existence and multiplicity results for a class of Kirchhoff–Choquard equations with a generalized sign-changing potential

In the present work, we are concerned with the Kirchhoff–Choquard-type equation -M\big(\|\nabla u\|_2^2\big)\Delta u+Q(x)u+\mu\big(V\big(|\cdot|\big)\ast u^2\big)u = f(u)\quad\text{in }\mathbb{R}^2, for M:\mathbb{R}\rightarrow\mathbb{R} given by M(t)=a+bt , \mu&gt;0 , V a sign-changing and possible unbounded potential, Q a continuous external potential, and a nonlinearity f with exponential critical growth. We prove existence and multiplicity of solutions in the nondegenerate case and guarantee the existence of solutions in the degenerate case.

Multiplicity of solutions for a class of upper critical Choquard equation with steep potential well

Multiplicity of solutions for a class of upper critical Choquard equation with steep potential well

Ground state solutions for critical Choquard equation with singular potential: existence and regularity

Ground state solutions for critical Choquard equation with singular potential: existence and regularity

Normalized solutions for nonlinear Choquard equations with general nonlocal term

Normalized solutions for nonlinear Choquard equations with general nonlocal term

Analysis of a low linear perturbed Choquard equation with critical growth

Analysis of a low linear perturbed Choquard equation with critical growth

Normalized solutions to the nonlinear Choquard equations with Hardy-Littlewood-Sobolev upper critical exponent

We study the effect of lower order nonlocal perturbations in the existence of positive solutions to the following nonlinear Choquard equation−Δu=λu+μ(Iα⁎|u|p)|u|p−2u+(Iα⁎|u|2α⁎)|u|2α⁎−2ux∈RN, having prescribed L2−norm∫RN|u|2dx=a>0, where N≥3, α∈(0,N), N+αN<p<2α⁎, μ>0 and λ∈R. Under different assumptions on p, we establish several existence results and characterize the behavior of solutions as μ→0.

Existence of a Ground State Solution for the Choquard Equation with Nonperiodic Potentials

Existence of a Ground State Solution for the Choquard Equation with Nonperiodic Potentials

Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities

In this paper, we study a class of critical Choquard equations with axisymmetric potentials, $$\begin{aligned} -\Delta u+ V(|x'|,x'')u =\Big (|x|^{-4}*|u|^{2}\Big )u\text{ in } \mathbb {R}^6, \end{aligned}$$ 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 function $$r^2V(r,x'')$$ has a topologically nontrivial critical point then the problem admits infinitely many solutions with arbitrary large energies.

The existence of positive solutions to the Choquard equation with critical exponent and logarithmic term

We investigate the existence of solutions for the following nonlinear critical Choquard equation with the logarithmic term:{−Δu=(∫Ω|u(y)|2μ⁎|x−y|μdy)|u|2μ⁎−2u+βu+λulog⁡u2,x∈Ω,u=0,x∈∂Ω, where Ω is a bounded domain of RN with smooth boundary. λ, β are real parameters and 2μ⁎=2N−μN−2(N≥3) is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We will show that there exists at last one positive solution to the above problem under some appropriate assumptions of λ and β.

Semiclassical States of Fractional Choquard Equations with Exponential Critical Growth

Semiclassical States of Fractional Choquard Equations with Exponential Critical Growth