Ground state solutions of Nehari-Pankov type for a superlinear elliptic system on RN

2016 ◽  
Vol 40 (3) ◽  
pp. 729-740 ◽  
Author(s):  
Xianhua Tang ◽  
Jian Zhang
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Ground State Solutions

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).

Ground State Solutions of Nehari–Pankov Type for a Superlinear Hamiltonian Elliptic System on ℝN

2015 ◽  
Vol 58 (3) ◽  
pp. 651-663 ◽  
Author(s):  
Xianhua Tang
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Direct Approach ◽  
Ground State Solutions ◽  
Hamiltonian Elliptic System

AbstractThis paper is concerned with the following elliptic system of Hamiltonian typewhere the potential V is periodic and 0 lies in a gap of the spectrum of −Δ + V, W(x, u, v) is periodic in x and superlinear in u and v at infinity. We develop a direct approach to ûnding ground state solutions of Nehari–Pankov type for the above system. Our method is especially applicable to the case whenwhere with , and and hj(x, t) are nondecreasing in t ∊ ℝ+ for every x ∊ ℝN and gi(x, 0) = hj(x, 0) = 0.

scholarly journals Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation

2021 ◽  
Vol 19 (1) ◽  
pp. 297-305
Author(s):  
Yuting Zhu ◽  
Chunfang Chen ◽  
Jianhua Chen ◽  
Chenggui Yuan
Keyword(s):  
Bounded Domain ◽  
Ground State ◽  
Multiple Solutions ◽  
Nontrivial Solutions ◽  
Ground State Solutions ◽  
Petviashvili Equation

Abstract In this paper, we study the following generalized Kadomtsev-Petviashvili equation u t + u x x x + ( h ( u ) ) x = D x − 1 Δ y u , {u}_{t}+{u}_{xxx}+{\left(h\left(u))}_{x}={D}_{x}^{-1}{\Delta }_{y}u, where ( t , x , y ) ∈ R + × R × R N − 1 \left(t,x,y)\in {{\mathbb{R}}}^{+}\times {\mathbb{R}}\times {{\mathbb{R}}}^{N-1} , N ≥ 2 N\ge 2 , D x − 1 f ( x , y ) = ∫ − ∞ x f ( s , y ) d s {D}_{x}^{-1}f\left(x,y)={\int }_{-\infty }^{x}f\left(s,y){\rm{d}}s , f t = ∂ f ∂ t {f}_{t}=\frac{\partial f}{\partial t} , f x = ∂ f ∂ x {f}_{x}=\frac{\partial f}{\partial x} and Δ y = ∑ i = 1 N − 1 ∂ 2 ∂ y i 2 {\Delta }_{y}={\sum }_{i=1}^{N-1}\frac{{\partial }^{2}}{{\partial }_{{y}_{i}}^{2}} . We get the existence of infinitely many nontrivial solutions under certain assumptions in bounded domain without Ambrosetti-Rabinowitz condition. Moreover, by using the method developed by Jeanjean [13], we establish the existence of ground state solutions in R N {{\mathbb{R}}}^{N} .

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