scholarly journals Existence of solutions for critical Choquard equations via the concentration-compactness method

2019 ◽  
Vol 150 (2) ◽  
pp. 921-954 ◽  
Author(s):  
Fashun Gao ◽  
Edcarlos D. da Silva ◽  
Minbo Yang ◽  
Jiazheng Zhou
Keyword(s):  
Sobolev Inequality ◽  
Existence Of Solutions ◽  
High Energy ◽  
Mountain Pass ◽  
Concentration Compactness ◽  
Choquard Equation ◽  
Concentration Compactness Principle ◽  
Nonlinear Choquard Equation ◽  
Compactness Principle ◽  
Image Position

AbstractIn this paper, we consider the nonlinear Choquard equation $$-\Delta u + V(x)u = \left( {\int_{{\open R}^N} {\displaystyle{{G(u)} \over { \vert x-y \vert ^\mu }}} \,{\rm d}y} \right)g(u)\quad {\rm in}\;{\open R}^N, $$ where 0 < μ < N, N ⩾ 3, g(u) is of critical growth due to the Hardy–Littlewood–Sobolev inequality and $G(u)=\int ^u_0g(s)\,{\rm d}s$. Firstly, by assuming that the potential V(x) might be sign-changing, we study the existence of Mountain-Pass solution via a nonlocal version of the second concentration- compactness principle. Secondly, under the conditions introduced by Benci and Cerami , we also study the existence of high energy solution by using a nonlocal version of global compactness lemma.

Multiplicity of solutions to the weighted critical quasilinear problems

2012 ◽  
Vol 55 (1) ◽  
pp. 181-195 ◽  
Author(s):  
Sihua Liang ◽  
Jihui Zhang
Keyword(s):  
Bounded Domain ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Concentration Compactness ◽  
Positive Parameter ◽  
Symmetric Mountain Pass Theorem ◽  
Concentration Compactness Principle ◽  
Compactness Principle ◽  
Quasilinear Problems ◽  
Image Position

AbstractWe consider a class of critical quasilinear problemswhere 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain and 1 < p < N, a < N/p, a ≤ b < a + 1, λ is a positive parameter, 0 ≤ μ < $\bar{\mu}$ ≡ ((N − p)/p − a)p, q = q*(a, b) ≡ Np/[N − pd] and d ≡ a+1 − b. Infinitely many small solutions are obtained by using a version of the symmetric Mountain Pass Theorem and a variant of the concentration-compactness principle. We deal with a problem that extends some results involving singularities not only in the nonlinearities but also in the operator.

scholarly journals Multiple solutions for critical Choquard-Kirchhoff type equations

2020 ◽  
Vol 10 (1) ◽  
pp. 400-419 ◽  
Author(s):  
Sihua Liang ◽  
Patrizia Pucci ◽  
Binlin Zhang
Keyword(s):  
Critical Exponent ◽  
Critical Exponents ◽  
Sobolev Inequality ◽  
Multiple Solutions ◽  
Concentration Compactness ◽  
Positive Real ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Concentration Compactness Principle ◽  
Compactness Principle

Abstract In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

Multiple solutions for a class of p ( x )-Laplacian equations in involving the critical exponent

2010 ◽  
Vol 466 (2118) ◽  
pp. 1667-1686 ◽  
Author(s):  
Yongqiang Fu ◽  
Xia Zhang
Keyword(s):  
Critical Exponent ◽  
Multiple Solutions ◽  
Existence Of Solutions ◽  
Concentration Compactness ◽  
Concentration Compactness Principle ◽  
Radially Symmetric Solutions ◽  
Compactness Principle ◽  
Laplacian Equations

In this paper, we first establish a principle of concentration compactness in . Then, based on this concentration compactness principle, we study the existence of solutions for a class of p ( x )-Laplacian equations in involving the critical exponent. Under suitable assumptions, we obtain a sequence of radially symmetric solutions associated with a sequence of positive energies going towards infinity.

scholarly journals Multiple solutions for a quasilinear Schrödinger system of Kirchhoff type with critical exponents

2018 ◽  
Vol 37 (4) ◽  
pp. 187-203
Author(s):  
Mohammed Massar ◽  
Ahmed Hamydy ◽  
Najib Tsouli
Keyword(s):  
Critical Exponents ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Type Systems ◽  
Concentration Compactness ◽  
Kirchhoff Type ◽  
Concentration Compactness Principle ◽  
Genus Theory ◽  
Compactness Principle ◽  
Schrödinger System

This paper is devoted to the existence of solutions for a class of Kirchhoff type systems involving critical exponents. The proof of the main results is based on  concentration compactness principle related to critical elliptic systems due to Kang combined with genus theory.

scholarly journals EXISTENCE OF SOLUTIONS FOR CRITICAL SYSTEMS WITH VARIABLE EXPONENTS

2018 ◽  
Vol 23 (4) ◽  
pp. 596-610 ◽  
Author(s):  
Hadjira Lalilia ◽  
Saadia Tas ◽  
Ali Djellit
Keyword(s):  
Existence Result ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Growth Conditions ◽  
Critical Growth ◽  
Concentration Compactness ◽  
Variable Exponents ◽  
Critical Systems ◽  
Concentration Compactness Principle ◽  
Compactness Principle

In this work, we deal with elliptic systems under critical growth conditions on the nonlinearities. Using a variant of concentration-compactness principle, we prove an existence result.

scholarly journals Two Types of Solutions to a Class of (p,q)-Laplacian Systems with Critical Sobolev Exponents in RN

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Jing Li ◽  
Caisheng Chen
Keyword(s):  
Elliptic System ◽  
Existence Of Solutions ◽  
Potential Functions ◽  
Mountain Pass Lemma ◽  
Compact Embedding ◽  
Concentration Compactness ◽  
Critical Sobolev Exponents ◽  
Concentration Compactness Principle ◽  
Genus Theory ◽  
Compactness Principle

We focus on the following elliptic system with critical Sobolev exponents:  -div⁡∇up-2∇u+m(x)up-2u=λup⁎-2u+(1/η)Gu(u,v),  x∈RN; -div⁡∇vq-2∇v+n(x)vq-2v=μvq⁎-2v+(1/η)Gv(u,v),  x∈RN; u(x)>0,v(x)>0,  x∈RN, where μ,λ>0,1<p≤q<N, either η∈(1,p) or η∈(q,p⁎), and critical Sobolev exponents p⁎=pN/(N-p) and q⁎=qN/(N-q). Conditions on potential functions m(x),n(x) lead to no compact embedding. Relying on concentration-compactness principle, mountain pass lemma, and genus theory, the existence of solutions to the elliptic system with η∈(q,p⁎) or η∈(1,p) will be established.

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