scholarly journals A NOTE ON THE EXISTENCE OF A GROUND STATE SOLUTION TO A FRACTIONAL SCHRÖDINGER EQUATION

2013 ◽  
Vol 67 (1) ◽  
pp. 227-236 ◽  
Author(s):  
Yonggeun CHO ◽  
Tohru OZAWA
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Ground State Solution ◽  
State Solution ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation

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.

On the Vanishing of the Cosmological Vacuum Energy

1997 ◽  
Vol 12 (16) ◽  
pp. 1127-1130 ◽  
Author(s):  
M. D. Pollock
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Vacuum Energy ◽  
Ground State Solution ◽  
Space Time ◽  
State Solution ◽  
Superstring Theory ◽  
The Universe

By demanding the existence of a globally invariant ground-state solution of the Wheeler–De Witt equation (Schrödinger equation) for the wave function of the Universe Ψ, obtained from the heterotic superstring theory, in the four-dimensional Friedmann space-time, we prove that the cosmological vacuum energy has to be zero.

scholarly journals Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type

2012 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Die Hu ◽  
Xianhua Tang ◽  
Qi Zhang
Keyword(s):  
Schrödinger Equation ◽  
Nontrivial Solution ◽  
Ground State ◽  
Schrodinger Equation ◽  
Existence Of Solutions ◽  
Ground State Solution ◽  
State Solution ◽  
Quasilinear Schrödinger Equation ◽  
Kirchhoff Type

<p style='text-indent:20px;'>In this paper, we discuss the generalized quasilinear Schrödinger equation with Kirchhoff-type:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1a"> \begin{document}$\left (1\!+\!b\int_{\mathbb{R}^{3}}g^{2}(u)|\nabla u|^{2} dx \right) \left[-\mathrm{div} \left(g^{2}(u)\nabla u\right)\!+\!g(u)g'(u)|\nabla u|^{2}\right] \!+\!V(x)u\! = \!f( u),(\rm P)$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ b&gt;0 $\end{document}</tex-math></inline-formula> is a parameter, <inline-formula><tex-math id="M2">\begin{document}$ g\in \mathbb{C}^{1}(\mathbb{R},\mathbb{R}^{+}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ V\in \mathbb{C}^{1}(\mathbb{R}^3,\mathbb{R}) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ f\in \mathbb{C}(\mathbb{R},\mathbb{R}) $\end{document}</tex-math></inline-formula>. Under some "Berestycki-Lions type assumptions" on the nonlinearity <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> which are almost necessary, we prove that problem <inline-formula><tex-math id="M6">\begin{document}$ (\rm P) $\end{document}</tex-math></inline-formula> has a nontrivial solution <inline-formula><tex-math id="M7">\begin{document}$ \bar{u}\in H^{1}(\mathbb{R}^{3}) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M8">\begin{document}$ \bar{v} = G(\bar{u}) $\end{document}</tex-math></inline-formula> is a ground state solution of the following problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1b"> \begin{document}$-\left(1+b\int_{\mathbb{R}^{3}} |\nabla v|^{2} dx \right) \triangle v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} = \frac{f(G^{-1}(v))}{g(G^{-1}(v))},(\rm \bar{P})$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M9">\begin{document}$ G(t): = \int_{0}^{t} g(s) ds $\end{document}</tex-math></inline-formula>. We also give a minimax characterization for the ground state solution <inline-formula><tex-math id="M10">\begin{document}$ \bar{v} $\end{document}</tex-math></inline-formula>.</p>

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