scholarly journals A positive ground state solution for a class of asymptotically periodic Schrödinger equations with critical exponent

2016 ◽  
Vol 72 (7) ◽  
pp. 1851-1864 ◽  
Author(s):  
Jiu Liu ◽  
Jia-Feng Liao ◽  
Chun-Lei Tang
Keyword(s):  
Critical Exponent ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Asymptotically Periodic

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>

scholarly journals Existence of ground state for coupled system of biharmonic Schrödinger equations

2021 ◽  
Vol 7 (3) ◽  
pp. 3719-3730
Author(s):  
Yanhua Wang ◽  
◽  
Min Liu ◽  
Gongming Wei ◽  
Keyword(s):  
Ground State ◽  
Coupling Parameter ◽  
Coupled System ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Schrodinger Equations ◽  
Concentration Compactness ◽  
Schrödinger Equations ◽  
Whole Space

<abstract><p>In this paper we consider the following system of coupled biharmonic Schrödinger equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \ \left\{ \begin{aligned} \Delta^{2}u+\lambda_{1}u = u^{3}+\beta u v^{2}, \\ \Delta^{2}v+\lambda_{2}v = v^{3}+\beta u^{2}v, \end{aligned} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ (u, v)\in H^{2}({\mathbb{R}}^{N})\times H^2(\mathbb R^N) $, $ 1\leq N\leq7 $, $ \lambda_{i} &gt; 0 (i = 1, 2) $ and $ \beta $ denotes a real coupling parameter. By Nehari manifold method and concentration compactness theorem, we prove the existence of ground state solution for the coupled system of Schrödinger equations. Previous results on ground state solutions are obtained in radially symmetric Sobolev space $ H_r^2(\mathbb R^N)\times H_r^2(\mathbb R^N) $. When $ \beta $ satisfies some conditions, we prove the existence of ground state solution in the whole space $ H^2(\mathbb R^N)\times H^2(\mathbb R^N) $.</p></abstract>

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 Non-local to local transition for ground states of fractional Schrödinger equations on $$\mathbb {R}^N$$

2020 ◽  
Vol 22 (3) ◽  
Author(s):  
Bartosz Bieganowski ◽  
Simone Secchi
Keyword(s):  
Ground State ◽  
Ground States ◽  
Ground State Solution ◽  
State Solution ◽  
Local Problem ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Fractional Schrödinger Equations ◽  
Non Local ◽  
Local Transition

Abstract We consider the nonlinear fractional problem $$\begin{aligned} (-\Delta )^{s} u + V(x) u = f(x,u)&\quad \hbox {in } \mathbb {R}^N \end{aligned}$$ ( - Δ ) s u + V ( x ) u = f ( x , u ) in R N We show that ground state solutions converge (along a subsequence) in $$L^2_{\mathrm {loc}} (\mathbb {R}^N)$$ L loc 2 ( R N ) , under suitable conditions on f and V, to a ground state solution of the local problem as $$s \rightarrow 1^-$$ s → 1 - .

scholarly journals Schrödinger equations with asymptotically periodic terms

2015 ◽  
Vol 145 (4) ◽  
pp. 745-757 ◽  
Author(s):  
Reinaldo de Marchi
Keyword(s):  
Mountain Pass Theorem ◽  
Ground State Solution ◽  
Schrodinger Equations ◽  
Periodic Case ◽  
Local Version ◽  
Concentration Compactness ◽  
Schrödinger Equations ◽  
Concentration Compactness Principle ◽  
Asymptotically Periodic ◽  
Compactness Principle

We study the existence of non-trivial solutions for a class of asymptotically periodic semilinear Schrödinger equations in ℝN. By combining variational methods and the concentration-compactness principle, we obtain a non-trivial solution for the asymptotically periodic case and a ground state solution for the periodic one. In the proofs we apply the mountain pass theorem and its local version.

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