Ground state solution for a class of Schrödinger equations involving general critical growth term

Nonlinearity ◽  
2017 ◽  
Vol 30 (3) ◽  
pp. 899-911 ◽  
Author(s):  
Jiu Liu ◽  
Jia-Feng Liao ◽  
Chun-Lei Tang
Keyword(s):  
Ground State ◽  
Ground State Solution ◽  
Critical Growth ◽  
State Solution ◽  
Schrodinger Equations ◽  
Schrödinger Equations

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>

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 - .

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