Results 101 to 110 of about 446 (156)

Multiple solutions for nonhomogeneous Choquard equations

Electronic Journal of Differential Equations, 2018
In this article, we consider the multiple solutions for the nonhomogeneous Choquard equations $$ - \Delta u +u=\Big(\frac{1}{|x|^{\alpha}}\ast |u|^{p}\Big)|u|^{p-2}u+h(x), \quad x\in \mathbb{R}^N, $$ and $$ - \Delta u=\Big(\frac{1}{|x|^{\alpha ...
Lixia Wang
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Dual formulation for constraint solutions of the multi-state Choquard equation

The Choquard equation is a partial differential equation that has gained significant interest and attention in recent decades. It is a nonlinear equation that combines elements of both the Laplace and Schr\"odinger operators, and it arises frequently in ...
Wolansky, Gershon
core  

Existence of ground state solution for critical N-Laplacian Kirchhoff-type problem with convolution nonlinearity

Bulletin of Mathematical Sciences
In this paper, we investigate the following N-Laplacian Kirchhoff–Choquard-type equation involving the critical exponential growth nonlinearity of Trudinger–Moser-type: −1+b∫ℝN|∇u|NdxΔNu+V(x)|u|N−2u=(|x|−μ∗F(x,u))f(x,u),x∈ℝN,N≥2,u∈W1,N(ℝN), where ...
Lizhen Lai, Dongdong Qin, Die Hu, Jing Zhang   +3 more
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Existence and qualitative properties of solutions for a Choquard-type equation with Hardy potential

, 2023
In this paper, we study the existence and qualitative properties of positive solutions to a Choquard-type equation with Hardy potential. We develop a nonlocal version of concentration-compactness principle involving the Hardy potential to study the ...
Guo, Ting, Tang, Xianhua
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Semiclassical states for critical Choquard equations with critical frequency

, 2021
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< \mu ...
Zhou, Jiazheng, Gao, Fashun
core  

Multiple concentrating solutions for a fractional (p, q)-Choquard equation

Advanced Nonlinear Studies
We focus on the following fractional (p, q)-Choquard problem: (−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=1|x|μ*F(u)f(u) in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0 in RN, $\begin{cases}{\left(-{\Delta}\right)}_{p}^{s}u+{\left(-{\Delta}\right)}_{q}^{s}u+V\left(\varepsilon ...
Ambrosio Vincenzo
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Ground state solutions for Choquard type equations with a singular potential

Electronic Journal of Differential Equations, 2017
This article concerns the Choquard type equation $$ -\Delta u+V(x)u=\Big(\int_{\mathbb{R}^N}\frac{|u(y)|^p}{|x-y|^{N-\alpha}}dy\Big) |u|^{p-2}u,\quad x\in \mathbb{R}^N, $$ where $N\geq3$, $\alpha\in ((N-4)_+,N)$, $2\leq p
Tao Wang
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Ground state solution for Schrödinger-Choquard equation: doubly critical case

Boundary Value Problems
In this paper, we investigate the following Schrödinger-Choquard equation: − Δ u + u = ( I α ∗ | u | 2 α ♯ ) | u | 2 α ♯ − 2 u + | u | q − 2 u + | u | r − 2 u , x ∈ R N , $$ -\Delta u+u = (I_{\alpha }*|u|^{2_{\alpha }^{\sharp }})|u|^{2_{\alpha }^{ \sharp
Yusheng Shen, Zhiwei Zou, You Gao
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Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities

Advances in Nonlinear Analysis
In this article, we study the following quasilinear equation with nonlocal nonlinearity −Δu−κuΔ(u2)+λu=(∣x∣−μ*F(u))f(u),inRN,-\Delta u-\kappa u\Delta \left({u}^{2})+\lambda u=\left({| x| }^{-\mu }* F\left(u))f\left(u),\hspace{1em}\hspace{0.1em}\text{in ...
Jia Yue, Yang Xianyong
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Some existence results of multiple normalized solutions to critical Kirchhoff–Choquard equation in RN ${\mathbb{R}}^{N}$

Demonstratio Mathematica
The paper intends to prove the existence of multiple normalized solutions for the critical Kirchhoff–Choquard equation, and the main feature of our paper is the simultaneous appearance of critical term, nonlocal term, and potential function, which will ...
Liang Sihua, Pu Hongling, Zhang Xin
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