Positive ground state solutions for an elliptic system with Hardy-Sobolev critical exponent growth

2019 ◽  
Vol 60 (10) ◽  
pp. 101502
Author(s):  
Zhenyu Guo ◽  
Wenming Zou
Keyword(s):  
Critical Exponent ◽  
Elliptic System ◽  
Ground State ◽  
Ground State Solutions ◽  
Sobolev Critical Exponent

scholarly journals Existence of groundstates for Choquard type equations with Hardy–Littlewood–Sobolev critical exponent

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xiaowei Li ◽  
Feizhi Wang
Keyword(s):  
Critical Exponent ◽  
Ground State ◽  
Concentration Compactness ◽  
Ground State Solutions ◽  
Whole Space ◽  
Sobolev Critical Exponent

AbstractIn this paper, we consider a class of Choquard equations with Hardy–Littlewood–Sobolev lower or upper critical exponent in the whole space $\mathbb{R}^{N}$ R N . We combine an argument of L. Jeanjean and H. Tanaka (see (Proc. Am. Math. Soc. 131:2399–2408, 2003) with a concentration–compactness argument, and then we obtain the existence of ground state solutions, which extends and complements the earlier results.

Ground state solutions of Hamiltonian elliptic systems in dimension two

2019 ◽  
Vol 150 (4) ◽  
pp. 1737-1768 ◽  
Author(s):  
Djairo G. de Figueiredo ◽  
João Marcos do Ó ◽  
Jianjun Zhang
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Elliptic Systems ◽  
Nehari Manifold ◽  
Existence Results ◽  
Ground State Solutions ◽  
Variational Framework ◽  
Hamiltonian Elliptic System ◽  
Schrödinger Systems ◽  
Image Position

AbstractThe aim of this paper is to study Hamiltonian elliptic system of the form 0.1$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$ where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane 0.2$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).

scholarly journals Existence of Nontrivial Solutions for Perturbed Elliptic System inℝN

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Juan Jiang
Keyword(s):  
Variational Method ◽  
Critical Exponent ◽  
Elliptic System ◽  
Existence Result ◽  
Nonlinear Elliptic System ◽  
Nontrivial Solutions ◽  
Nonlinear Elliptic ◽  
Sobolev Critical Exponent

We consider the perturbed nonlinear elliptic system-ε2Δu+V(x)u=K(x)|u|2*-2u+Hu(u,v),  x∈ℝN,-ε2Δv+V(x)v=K(x)|v|2*-2v+Hv(u,v),  x∈ℝN, whereN≥3,2*=2N/(N-2)is the Sobolev critical exponent. Under proper conditions onV,H, andK, the existence result and multiplicity of the system are obtained by using variational method providedεis small enough.

Concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system involving critical exponent

2019 ◽  
Vol 21 (06) ◽  
pp. 1850027 ◽  
Author(s):  
Zhipeng Yang ◽  
Yuanyang Yu ◽  
Fukun Zhao
Keyword(s):  
Small Parameter ◽  
Critical Exponent ◽  
Ground State ◽  
Local Minimum ◽  
Fractional Laplacian ◽  
Ground State Solution ◽  
Critical Nonlinearity ◽  
Poisson System ◽  
Ground State Solutions ◽  
Concentration Behavior

We are concerned with the existence and concentration behavior of ground state solutions of the fractional Schrödinger–Poisson system with critical nonlinearity [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text], [Formula: see text], [Formula: see text] denotes the fractional Laplacian of order [Formula: see text] and satisfies [Formula: see text]. The potential [Formula: see text] is continuous and positive, and has a local minimum. We obtain a positive ground state solution for [Formula: see text] small, and we show that these ground state solutions concentrate around a local minimum of [Formula: see text] as [Formula: see text].

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