scholarly journals Nehari-Type Ground State Positive Solutions for Superlinear Asymptotically Periodic Schrödinger Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Xiaoyan Lin ◽  
X. H. Tang
Keyword(s):  
Positive Solutions ◽  
Ground State ◽  
Schrodinger Equation ◽  
Direct Method ◽  
Nonlinear Schrodinger Equation ◽  
Periodic Case ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Asymptotically Periodic ◽  
Existence Criteria

We deal with the existence of Nehari-type ground state positive solutions for the nonlinear Schrödinger equation-Δu+Vxu=fx, u, x ∈ RN, u ∈ H1 RN. Under a weaker Nehari condition, we establish some existence criteria to guarantee that the above problem has Nehari-type ground state solutions by using a more direct method in two cases: the periodic case and the asymptotically periodic case.

scholarly journals Positive ground state solutions for asymptotically periodic generalized quasilinear Schrödinger equations

2021 ◽  
Vol 7 (1) ◽  
pp. 1015-1034
Author(s):  
Shulin Zhang ◽  
◽  
Keyword(s):  
Variational Method ◽  
Ground State ◽  
Periodic Potential ◽  
Ground State Solution ◽  
State Solution ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Asymptotically Periodic

<abstract><p>In this paper, we study the existence of a positive ground state solution for a class of generalized quasilinear Schrödinger equations with asymptotically periodic potential. By the variational method, a positive ground state solution is obtained. Compared with the existing results, our results improve and generalize some existing related results.</p></abstract>

Ground States of some Fractional Schrödinger Equations on ℝN

2014 ◽  
Vol 58 (2) ◽  
pp. 305-321 ◽  
Author(s):  
Xiaojun Chang
Keyword(s):  
Variational Methods ◽  
Ground State ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Ground States ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Fractional Schrödinger Equations ◽  
Image Position ◽  
Fractional Schrodinger Equation

AbstractIn this paper, we study a time-independent fractional Schrödinger equation of the form (−Δ)su + V(x)u = g(u) in ℝN, where N ≥, s ∈ (0,1) and (−Δ)s is the fractional Laplacian. By variational methods, we prove the existence of ground state solutions when V is unbounded and the nonlinearity g is subcritical and satisfies the following geometry condition:

AN IMPROVED RESULT ON GROUND STATE SOLUTIONS OF QUASILINEAR SCHRÖDINGER EQUATIONS WITH SUPER-LINEAR NONLINEARITIES

2018 ◽  
Vol 99 (2) ◽  
pp. 231-241
Author(s):  
SITONG CHEN ◽  
ZU GAO
Keyword(s):  
Schrödinger Equation ◽  
Ground State Energy ◽  
Ground State ◽  
Schrodinger Equation ◽  
State Energy ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Quasilinear Schrödinger Equation ◽  
Ground State Solutions

By using variational and some new analytic techniques, we prove the existence of ground state solutions for the quasilinear Schrödinger equation with variable potentials and super-linear nonlinearities. Moreover, we establish a minimax characterisation of the ground state energy. Our result improves and extends the existing results in the literature.

Standing waves of modified Schrödinger equations coupled with the Chern–Simons gauge theory

2019 ◽  
Vol 150 (4) ◽  
pp. 1915-1936 ◽  
Author(s):  
Pietro d'Avenia ◽  
Alessio Pomponio ◽  
Tatsuya Watanabe
Keyword(s):  
Gauge Theory ◽  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Ground State Solutions ◽  
Chern Simons ◽  
Constraint Minimization

AbstractWe are interested in standing waves of a modified Schrödinger equation coupled with the Chern–Simons gauge theory. By applying a constraint minimization of Nehari-Pohozaev type, we prove the existence of radial ground state solutions. We also investigate the nonexistence for nontrivial solutions.

Existence and Asymptotic Behavior of Positive Solutions for a Class of Quasilinear Schrödinger Equations

2018 ◽  
Vol 18 (1) ◽  
pp. 131-150 ◽  
Author(s):  
Youjun Wang ◽  
Yaotian Shen
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Positive Solutions ◽  
Ground State ◽  
Schrodinger Equation ◽  
Radial Solution ◽  
Schrödinger Equations ◽  
Quasilinear Schrödinger Equation ◽  
Content Type ◽  
Graphic Content

AbstractIn this paper, we study the quasilinear Schrödinger equation{-\Delta u+V(x)u-\frac{\gamma}{2}(\Delta u^{2})u=|u|^{p-2}u},{x\in\mathbb{R}^{N}}, where{V(x):\mathbb{R}^{N}\to\mathbb{R}}is a given potential,{\gamma>0}, and either{p\in(2,2^{*})},{2^{*}=\frac{2N}{N-2}}for{N\geq 4}or{p\in(2,4)}for{N=3}. If{\gamma\in(0,\gamma_{0})}for some{\gamma_{0}>0}, we establish the existence of a positive solution{u_{\gamma}}satisfying{\max_{x\in\mathbb{R}^{N}}|\gamma^{\mu}u_{\gamma}(x)|\to 0}as{\gamma\to 0^{+}}for any{\mu>\frac{1}{2}}. Particularly, if{V(x)=\lambda>0}, we prove the existence of a positive classical radial solution{u_{\gamma}}and up to a subsequence,{u_{\gamma}\to u_{0}}in{H^{2}(\mathbb{R}^{N})\cap C^{2}(\mathbb{R}^{N})}as{\gamma\to 0^{+}}, where{u_{0}}is the ground state of the problem{-\Delta u+\lambda u=|u|^{p-2}u},{x\in\mathbb{R}^{N}}.

scholarly journals New results on the existence of ground state solutions for generalized quasilinear Schrödinger equations coupled with the Chern–Simons gauge theory

2021 ◽  
pp. 1-17
Author(s):  
Yingying Xiao ◽  
Chuanxi Zhu
Keyword(s):  
Gauge Theory ◽  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Quasilinear Schrödinger Equation ◽  
Ground State Solutions ◽  
Chern Simons ◽  
Constraint Minimization

In this paper, we study the following quasilinear Schrödinger equation − Δ u + V ( x ) u − κ u Δ ( u 2 ) + μ h 2 ( | x | ) | x | 2 ( 1 + κ u 2 ) u + μ ( ∫ | x | + ∞ h ( s ) s ( 2 + κ u 2 ( s ) ) u 2 ( s ) d s ) u = f ( u ) in   R 2 , κ > 0 V ∈ C 1 ( R 2 , R ) and f ∈ C ( R , R ) By using a constraint minimization of Pohožaev–Nehari type and analytic techniques, we obtain the existence of ground state solutions.

Sign in / Sign up

Export Citation Format

Share Document