Critical Choquard Research Articles

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

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.

Existence and Asymptotical Behavior of Multiple Solutions for the Critical Choquard Equation

Existence and Asymptotical Behavior of Multiple Solutions for the Critical Choquard Equation

Normalized Solutions for Lower Critical Choquard Equations with Critical Sobolev Perturbation

We study normalized solutions for the following Choquard equations with lower critical exponent and a local perturbation $ -\Delta u+\lambda u=\gamma (I_{\alpha }\ast |u|^{\frac{\alpha }{N}+1})|u|^{\frac{\alpha }{N}-1}u+\mu |u|^{q-2}u \quad \text{in}\quad \mathbb{R}^{N}, \int_{\mathbb{R}^{N}}$ $|u|^{2}dx=c^{2},$ where $\gamma ,\,\mu,\,c$ are given positive numbers and $2<q\leq \frac{2N}{N-2}$. The frequency $\lambda $ appears as a real Lagrange parameter and is part of the unknowns. By introducing new arguments and under different assumptions on $q,\,c,\,\gamma $, and $\mu ,$ we prove several nonexistence and existence results. In particular, we consider the case $q=\frac{2N}{N-2},$ which corresponds to equations involving double critical exponents. We also describe some qualitative properties of the solutions with prescribed mass and of the associated Lagrange multipliers $\lambda$.

Sign-changing solutions to the critical Choquard equation

We consider the following Choquard problem −Δu=λu+f(x)(|x|−μ∗|u|2μ∗)|u|2μ∗−2uinΩ,u=0on∂Ω, where 0<λ<λ1(Ω),|Ω|<∞. 2μ∗ is the upper critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. We prove the existence of sign-changing solution by combining Nehari manifold method with the Ljusternik–Schnirelman theory. The main results extend and complement the earlier works (Guo et al., 2019; Moroz and Schaftingen, 2015).

Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability

AbstractIn this article, we consider the upper critical Choquard equation with a local perturbation−Δu=λu+(Iα∗∣u∣p)∣u∣p−2u+μ∣u∣q−2u,x∈RN,u∈H1(RN),∫RN∣u∣2=a,\left\{\begin{array}{l}-\Delta u=\lambda u+\left({I}_{\alpha }\ast | u\hspace{-0.25em}{| }^{p})| u\hspace{-0.25em}{| }^{p-2}u+\mu | u\hspace{-0.25em}{| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\hspace{1em}{\displaystyle \int }_{{{\mathbb{R}}}^{N}}| u\hspace{-0.25em}{| }^{2}=a,\end{array}\right.whereN≥3N\ge 3,μ&gt;0\mu \gt 0,a&gt;0a\gt 0,λ∈R\lambda \in {\mathbb{R}},α∈(0,N)\alpha \in \left(0,N),p=p¯≔N+αN−2p=\bar{p}:= \frac{N+\alpha }{N-2},q∈2,2+4Nq\in \left(2,2+\frac{4}{N}\right), andIα=C∣x∣N−α{I}_{\alpha }=\frac{C}{| x{| }^{N-\alpha }}withC&gt;0C\gt 0. Whenμaq(1−γq)2≤(2K)qγq−2p¯2(p¯−1)\mu {a}^{\tfrac{q\left(1-{\gamma }_{q})}{2}}\le {\left(2K)}^{\tfrac{q{\gamma }_{q}-2\bar{p}}{2\left(\bar{p}-1)}}withγq=N2−Nq{\gamma }_{q}=\frac{N}{2}-\frac{N}{q}andKKbeing some positive constant, we prove(1)Existence and orbital stability of the ground states.(2)Existence, positivity, radial symmetry, exponential decay, and orbital instability of the “second class” solutions.This article generalized and improved parts of the results obtained for the Schrödinger equation.

Multiple solutions for biharmonic critical Choquard equation involving sign-changing weight functions

Multiple solutions for biharmonic critical Choquard equation involving sign-changing weight functions

Ground states of coupled critical Choquard equations with weighted potentials

Ground states of coupled critical Choquard equations with weighted potentials

Ground state sign-changing solutions for critical Choquard equations with steep well potential

In this paper, we study sign-changing solution of the Choquard type equation − Δ u + ( λ V ( x ) + 1 ) u = ( I α ∗ | u | 2 α ∗ ) | u | 2 α ∗ − 2 u + μ | u | p − 2 u in R N , where N ≥ 3 , α ∈ ( ( N − 4 ) + , N ) , I α is a Riesz potential, p ∈ [ 2 α ∗ , 2 N N − 2 ) , 2 α ∗ := N + α N − 2 is the upper critical exponent in terms of the Hardy–Littlewood–Sobolev inequality, μ &gt; 0 , λ &gt; 0 , V ∈ C ( R N , R ) is nonnegative and has a potential well. By combining the variational methods and sign-changing Nehari manifold, we prove the existence and some properties of ground state sign-changing solution for λ , μ large enough. Further, we verify the asymptotic behaviour of ground state sign-changing solutions as λ → + ∞ and μ → + ∞ , respectively.

Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity

In this paper, we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: \begin{align*} (-\Delta)^s u & = -\lambda|u|^{q-2}u + au + b \Big ( \int_{\Omega} \frac{(u^{+}(y))^{2^{*}_{\mu ,s}}} {|x-y|^ \mu}\, dy \Big ) (u^{+})^{2^{*}_{\mu ,s}-2}u \quad\text{in} \; \Omega,\\ u & = 0\quad \text{in} \; \mathbb{R}^{N}\backslash\Omega, \end{align*} where $\Omega$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$. Here, $(-\Delta)^s$ is the fractional Laplace operator, $\lambda > 0$ is a real parameter, $q \in (1, 2)$, $a > 0$ and $b > 0$ are given constants, $0 < \mu < \min\{N,4s\}$ and $2^{*}_{\mu ,s} = \frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and the notation $u^{+} = \max \{u, 0\}$. We prove that the above problem has at least three nontrivial solutions using the Mountain pass Lemma and Linking theorem.

Semiclassical states for critical Choquard equations with critical frequency

We study the multiplicity of semiclassical states for the Choquard equation $$ -\varepsilon^2\Delta u +V(x)u =\varepsilon^{\mu-N}\bigg(\int_{\mathbb{R}^{N}} \frac{G(y,u(y))}{|x-y|^\mu}dy\bigg)g(x,u) \quad \mbox{in $\mathbb{R}^{N}$}, $$ where $0&lt; \mu&lt; N$, $N\geq3$, $\varepsilon$ 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 $V$ is assumed to be nonnegative with $V(x)=0$ in some region of $\mathbb{R}^{N}$. Using the genus theory we prove the multiplicity of semiclassical states for the critical Choquard equation.

A critical fractional choquard problem involving a singular nonlinearity and a radon measure

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. 0.1 $$\begin{aligned} \begin{aligned} (-\Delta )^su-\alpha \frac{u}{|x|^{2s}}&=\lambda u+ u^{-\gamma }+\beta \left( \int _{\Omega }\frac{u^{2_b^*}(y)}{|x-y|^b}dy\right) u^{2_b^*-1}+\mu ~\text {in}~\Omega ,\\ u&>0~\text {in}~\Omega ,\\ u&= 0~\text {in}~\mathbb {R}^N{\setminus }\Omega . \end{aligned} \end{aligned}$$ Here, $$\Omega $$ is a bounded domain of $$\mathbb {R}^N$$ , $$s\in (0,1)$$ , $$\alpha ,\lambda $$ and $$\beta $$ are positive real parameters, $$N>2s$$ , $$\gamma \in (0,1)$$ , $$0<b<\min \{N,4s\}$$ , $$2_b^*=\frac{2N-b}{N-2s}$$ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and $$\mu $$ is a bounded Radon measure in $$\Omega $$ .

Saddle solutions for the critical Choquard equation

We study the saddle type nodal solutions for the Choquard equation $$\begin{aligned} -\Delta u = (I_\alpha *|u|^{\frac{N+\alpha }{N-2}} )|u|^{\frac{N+\alpha }{N-2}-2}u \quad \text { in }\;{\mathbb {R}}^N, \end{aligned}$$ where $$N\ge 3$$ , $$I_\alpha $$ is the Riesz potential of order $$\alpha \in (0, N)$$ and $$\frac{N+\alpha }{N-2}$$ refers to the upper critical exponent with respect to the Hardy–Littlewood–Sobolev inequality. By introducing the symmetric groups of Coxeter, we give a unified framework to construct saddle solutions with prescribed symmetric nodal structures. To overcome the difficulties arising from the lack of compactness, we give a fine analyzes on the profile decompositions of the symmetric Palais–Smale sequence. These results further feature the nonlocal nature of the Choquard equation, in contrast, the counterpart Yamabe equation $$-\Delta u=|u|^{\frac{4}{N-2}}u$$ can not permit such type of nodal solutions.

Semiclassical states for critical Choquard equations

In this paper, by constructing an appropriate Byeon-Wang type penalization function, we consider the critical Choquard equation−ϵ2Δu+V(x)u=1ϵα(Iα⁎|u|N+αN−2)|u|N+αN−2−2u+λg(u)inRN, where N≥3, (N−4)+<α<N, ϵ,λ>0 are parameters, and Iα is the Riesz potential. We obtained the localized semiclassical states concentrating at local minimizers of potential. It is the first time to study Choquard equations using this method as far as we know.

Multiple positive bound state solutions for a critical Choquard equation

<p style='text-indent:20px;'>In this paper we consider the problem <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (P_{\lambda})\ \ \ \ \ \ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u = (I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N}, \\ u&gt;0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right. $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ V_{\lambda} = \lambda+V_{0} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ \lambda \geq 0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ V_0\in L^{N/2}({\mathbb{R}}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ I_{\mu} = \frac{1}{|x|^\mu} $\end{document}</tex-math></inline-formula> is the Riesz potential with <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\mu&lt;\min\{N, 4\} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ 2^{*}_{\mu} = \frac{2N-\mu}{N-2} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ N\geq 3 $\end{document}</tex-math></inline-formula>. Under some smallness assumption on <inline-formula><tex-math id="M8">\begin{document}$ V_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> we prove the existence of two positive solutions of <inline-formula><tex-math id="M10">\begin{document}$ (P_\lambda) $\end{document}</tex-math></inline-formula>. In order to prove the main results, we used variational methods combined with degree theory.

Semiclassical states for Choquard type equations with critical growth: critical frequency case * *Minbo Yang is the corresponding author who is partially supported by NSFC (11971436, 12011530199) and ZJNSF (LD19A010001); Yanheng Ding is partially supported by NSFC (11871242); Fashun Gao is partially supported by NSFC (11901155).

In this paper we are interested in the existence of semiclassical states for the Choquard type equation where 0 < μ < N, N ⩾ 3, ɛ 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 , which means it is of the critical frequency case. Firstly, we study a Choquard equation with double critical exponents and prove the existence and multiplicity of semiclassical states by the mountain-pass lemma and the genus theory. Secondly, we consider a class of critical Choquard equation without lower perturbation, by establishing a global compactness lemma for the nonlocal Choquard equation, we prove the multiplicity of high energy semiclassical states by the Lusternik–Schnirelman theory.

Existence of multiple semiclassical solutions for a critical Choquard equation with indefinite potential

We study the existence and multiplicity of semiclassical states for the Choquard equation with critical growth −ε2Δu+V(x)u=(∫RNG(y,u(y))|x−y|μdy)g(x,u)in RN,where N≥3, 0<μ<min{4,N}, V(x) is sign changing and G is the primitive of g which is of critical growth due to the well known Hardy–Littlewood–Sobolev inequality. Under suitable assumptions on V and g, we prove the existence and multiplicity of semiclassical states by critical point theory.

Infinitely many solutions for a class of critical Choquard equation with zero mass

In this paper we investigate the following nonlinear Choquard equation $$ -\Delta u =\bigg(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^{\mu}}dy\bigg)g(x,u)\quad \textrm{in} \mathbb{R}^N, $$ where $0< \mu< N$, $N\geq3$, $g(x,u)$ is of critical growth in the sense of the Hardy-Littlewood-Sobolev inequality and $G(x,u)=\int^u_0g(x,s)ds$. By applying minimax procedure and perturbation technique, we obtain the existence of infinitely many solutions.

Existence and concentrate behavior of ground state solutions for critical Choquard equations

In this paper, we investigate the following Choquard equation (P)−Δu+(1+μg(x))u=(Iα∗|u|p)|u|p−2u+u5inR3,where p∈(2,3),α∈(0,p−2), Iα is the Riesz potential and the function g≥0 has a potential well. By the Nehari manifold, we obtain the existence of ground state solutions for μ large enough. Moreover, we construct a family of ground state solutions which converges to a ground state solution of the limiting critical local equation as α→0.