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

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>

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).

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 Ground State Solution of Pohožaev Type for Quasilinear Schrödinger Equation Involving Critical Exponent in Orlicz Space

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 779 ◽  
Author(s):  
Jianqing Chen ◽  
Qian Zhang
Keyword(s):  
Schrödinger Equation ◽  
Critical Exponent ◽  
Orlicz Space ◽  
Ground State ◽  
Schrodinger Equation ◽  
Nonlinear Term ◽  
Ground State Solution ◽  
Existence Results ◽  
Quasilinear Schrödinger Equation ◽  
Ground State Solutions

We study the following quasilinear Schrödinger equation involving critical exponent - Δ u + V ( x ) u - Δ ( u 2 ) u = A ( x ) | u | p - 1 u + λ B ( x ) u 3 N + 2 N - 2 , u ( x ) > 0 for x ∈ R N and u ( x ) → 0 as | x | → ∞ . By using a monotonicity trick and global compactness lemma, we prove the existence of positive ground state solutions of Pohožaev type. The nonlinear term | u | p - 1 u for the well-studied case p ∈ [ 3 , 3 N + 2 N - 2 ) , and the less-studied case p ∈ [ 2 , 3 ) , and for the latter case few existence results are available in the literature. Our results generalize partial previous works.

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