scholarly journals Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data

2016 ◽  
Vol 15 (6) ◽  
pp. 2023-2058 ◽  
Author(s):  
Kenji Nakanishi ◽  
Tristan Roy
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Global Dynamics

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

Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

2021 ◽  
pp. 1-19
Author(s):  
Jing Zhang ◽  
Lin Li
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Critical Sobolev Exponent ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Hardy Potential ◽  
Asymptotically Periodic ◽  
Sobolev Exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

scholarly journals Ground state solutions for fractional Schrödinger equation with variable potential and Berestycki–Lions type nonlinearity

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jing Chen ◽  
Zu Gao
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Nonlocal Problem ◽  
Minimization Method ◽  
Fractional Schrödinger Equation ◽  
Ground State Solutions ◽  
The One ◽  
Least Energy Solutions ◽  
Fractional Schrodinger Equation

Abstract We consider the following nonlinear fractional Schrödinger equation: $$ (-\triangle )^{s} u+V(x)u=g(u) \quad \text{in } \mathbb{R} ^{N}, $$ ( − △ ) s u + V ( x ) u = g ( u ) in  R N , where $s\in (0, 1)$ s ∈ ( 0 , 1 ) , $N>2s$ N > 2 s , $V(x)$ V ( x ) is differentiable, and $g\in C ^{1}(\mathbb{R} , \mathbb{R} )$ g ∈ C 1 ( R , R ) . By exploiting the minimization method with a constraint over Pohoz̆aev manifold, we obtain the existence of ground state solutions. With the help of Pohoz̆aev identity we also process the existence of the least energy solutions for the above equation. Our results improve the existing study on this nonlocal problem with Berestycki–Lions type nonlinearity to the one that does not need the oddness assumption.

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