scholarly journals Ground State Solutions to a Critical Nonlocal Integrodifferential System

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Min Liu ◽  
Zhijing Wang ◽  
Zhenyu Guo
Keyword(s):  
Lower Order ◽  
Ground State ◽  
Existence Result ◽  
Ground State Solution ◽  
Lipschitz Boundary ◽  
Critical Coupling ◽  
Coupling Term ◽  
Integrodifferential System ◽  
Integrodifferential Operator ◽  
Sobolev Critical Exponent

Consider the following nonlocal integrodifferential system: LKu+λ1u+μ1u2⁎-2u+Gu(x,u,v)=0  in  Ω,  LKv+λ2v+μ2v2⁎-2v+Gv(x,u,v)=0  in  Ω,  u=0,  v=0  in  RN∖Ω, where LK is a general nonlocal integrodifferential operator, λ1,λ2,μ1,μ2>0, 2⁎≔2N/N-2s is a fractional Sobolev critical exponent, 0<s<1, N>2s, G(x,u,v) is a lower order perturbation of the critical coupling term, and Ω is an open bounded domain in RN with Lipschitz boundary. Under proper conditions, we establish an existence result of the ground state solution to the nonlocal integrodifferential system.

scholarly journals Ground state solutions to a class of critical Schrödinger problem

2021 ◽  
Vol 11 (1) ◽  
pp. 96-127
Author(s):  
Anmin Mao ◽  
Shuai Mo
Keyword(s):  
Ground State ◽  
Global Minimum ◽  
Existence Result ◽  
Ground State Solution ◽  
State Solution ◽  
Monotonicity Condition ◽  
Ground State Solutions ◽  
Decay Assumption

Abstract We consider the following critical nonlocal Schrödinger problem with general nonlinearities − ε 2 a + ε b ∫ R 3 | ∇ u | 2 Δ u + V ( x ) u = f ( u ) + u 5 , x ∈ R 3 , u ∈ H 1 ( R 3 ) , $$\begin{array}{} \displaystyle \left\{\begin{array}{} &-\left(\varepsilon^{2}a+\varepsilon b\displaystyle\int\limits_{\mathbb{R}^{3}}|\nabla u|^{2}\right){\it\Delta} u+V(x)u=f(u)+u^{5}, &x \in \mathbb{R}^{3},\\ &u\in H^{1}(\mathbb{R}^{3}), \end{array}\right. \end{array}$$ (SKε ) and study the existence of semiclassical ground state solutions of Nehari-Pohožaev type to (SK ε ), where f(u) may behave like |u| q–2 u for q ∈ (2, 4] which is seldom studied. With some decay assumption on V, we establish an existence result which improves some exiting works which only handle q ∈ (4, 6). With some monotonicity condition on V, we also get a ground state solution v̄ ε and analysis its concentrating behaviour around global minimum x ε of V as ε → 0. Our results extend some related works.

scholarly journals Existence result for critical Klein-Gordon-Maxwell system involving steep potential well

2021 ◽  
Author(s):  
Canlin Gan
Keyword(s):  
Ground State ◽  
Existence Result ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Potential Well ◽  
Maxwell System ◽  
Cerami Sequence ◽  
Klein Gordon ◽  
Steep Potential Well ◽  
Existing Result

This paper deals with the following system \begin{equation*} \left\{\begin{aligned} &{-\Delta u+ (\lambda A(x)+1)u-(2\omega+\phi) \phi u=\mu f(u)+u^{5}}, & & {\quad x \in \mathbb{R}^{3}}, \\ &{\Delta \phi=(\omega+\phi) u^{2}}, & & {\quad x \in \mathbb{R}^{3}}, \end{aligned}\right. \end{equation*} where $\lambda, \mu>0$ are positive parameters. Under some suitable conditions on $A$ and $f$, we show the boundedness of Cerami sequence for the above system by adopting Poho\v{z}aev identity and then prove the existence of ground state solution for the above system on Nehari manifold by using Br\’{e}zis-Nirenberg technique, which improve the existing result in the literature.

scholarly journals Nonlinear evolution problems with singular coefficients in the lower order terms

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Fernando Farroni ◽  
Luigi Greco ◽  
Gioconda Moscariello ◽  
Gabriella Zecca
Keyword(s):  
Lower Order ◽  
Time Horizon ◽  
Existence Result ◽  
Nonlinear Evolution ◽  
Order Term ◽  
Time Behavior ◽  
Infinite Time ◽  
Singular Coefficient ◽  
Singular Coefficients ◽  
Lower Order Terms

AbstractWe consider a Cauchy–Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the solution in the case of the infinite–time horizon.

Exact Ground State Solution for Anyons with spin-$\frac {1}{2}$

1992 ◽  
Vol 17 (1) ◽  
pp. 107-110
Author(s):  
Shun-Qing Shen ◽  
J. Cai ◽  
Rui-Bao Tao
Keyword(s):  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Exact Ground State

Existence of a Ground State Solution for a Quasilinear Schrödinger Equation

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Xiang-dong Fang ◽  
Zhi-qing Han
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Ground State Solution ◽  
State Solution ◽  
Quasilinear Schrödinger Equation

AbstractIn this paper we are concerned with the quasilinear Schrödinger equation−Δu + V(x)u − Δ(uwhere N ≥ 3, 4 < p < 4N/(N − 2), and V(x) and q(x) go to some positive limits V

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