scholarly journals Mass and Extremals Associated with the Hardy–Schrödinger Operator on Hyperbolic Space

2018 ◽  
Vol 18 (4) ◽  
pp. 671-689
Author(s):  
Hardy Chan ◽  
Nassif Ghoussoub ◽  
Saikat Mazumdar ◽  
Shaya Shakerian ◽  
Luiz Fernando de Oliveira Faria
Keyword(s):  
Hyperbolic Space ◽  
Schrödinger Operator ◽  
Ground State ◽  
Existence Of Solutions ◽  
Ground State Solution ◽  
Positive Definite ◽  
Schrodinger Operator ◽  
Hardy Potential ◽  
Critical Equation ◽  
Euclidean Setting

AbstractWe consider the Hardy–Schrödinger operator {L_{\gamma}:=-\Delta_{\mathbb{B}^{n}}-\gamma{V_{2}}} on the Poincaré ball model of the hyperbolic space {\mathbb{B}^{n}} ({n\geq 3}). Here {V_{2}} is a radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e., {V_{2}(r)\sim\frac{1}{r^{2}}}. As in the Euclidean setting, {L_{\gamma}} is positive definite whenever {\gamma<\frac{(n-2)^{2}}{4}}, in which case we exhibit explicit solutions for the critical equation {L_{\gamma}u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in {\mathbb{B}^{n},} where {0\leq s<2}, {2^{*}(s)=\frac{2(n-s)}{n-2}}, and {V_{2^{*}(s)}} is a weight that behaves like {\frac{1}{r^{s}}} around 0. In dimensions {n\geq 5}, the equation {L_{\gamma}u-\lambda u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in a domain Ω of {\mathbb{B}^{n}} away from the boundary but containing 0 has a ground state solution, whenever {0<\gamma\leq\frac{n(n-4)}{4}}, and {\lambda>\frac{n-2}{n-4}(\frac{n(n-4)}{4}-\gamma)}. On the other hand, in dimensions 3 and 4, the existence of solutions depends on whether the domain has a positive “hyperbolic mass” a notion that we introduce and analyze therein.

scholarly journals Ground State for the Schrödinger Operator with the Weighted Hardy Potential

2011 ◽  
Vol 2011 ◽  
pp. 1-26
Author(s):  
J. Chabrowski ◽  
K. Tintarev
Keyword(s):  
Laplace Operator ◽  
Schrödinger Operator ◽  
Ground State ◽  
Ground States ◽  
Schrodinger Operator ◽  
Hardy Potential ◽  
Higher Integrability ◽  
Hardy Type ◽  
The Laplace Operator ◽  
Type Potential

We establish the existence of ground states on for the Laplace operator involving the Hardy-type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.

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 SYMMETRY AND EXISTENCE OF SOLUTIONS OF SEMI-LINEAR ELLIPTIC SYSTEMS IN HYPERBOLIC SPACE

2014 ◽  
Vol 57 (3) ◽  
pp. 519-541
Author(s):  
HAIYANG HE
Keyword(s):  
Positive Solution ◽  
Hyperbolic Space ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Decay Estimates ◽  
Symmetry Properties ◽  
Formula Group ◽  
Solution Pair ◽  
Image Position

Abstract(0.1) \begin{equation}\label{eq:0.1} \left\{ \begin{array}{ll} \displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v x, \\ \displaystyle -\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, \\ \end{array} \right. \end{equation} in the whole Hyperbolic space ℍN. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space ℝN, we prove that there is a positive solution pair (u, v) ∈ H1(ℍN) × H1(ℍN) of problem (0.1), moreover a ground state solution is obtained. Furthermore, we also prove that the above problem has a radial positive solution.

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>

scholarly journals Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Zhang ◽  
Qiongfen Zhang
Keyword(s):  
Critical Exponent ◽  
Ground State ◽  
Sobolev Inequality ◽  
Existence Of Solutions ◽  
Riesz Potential ◽  
Ground State Solution ◽  
Pohozaev Identity ◽  
Choquard Equation ◽  
Minimax Methods ◽  
Variable Potential

AbstractIn this paper, we focus on the existence of solutions for the Choquard equation $$\begin{aligned} \textstyle\begin{cases} {-}\Delta {u}+V(x)u=(I_{\alpha }* \vert u \vert ^{\frac{\alpha }{N}+1}) \vert u \vert ^{ \frac{\alpha }{N}-1}u+\lambda \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$ { − Δ u + V ( x ) u = ( I α ∗ | u | α N + 1 ) | u | α N − 1 u + λ | u | p − 2 u , x ∈ R N ; u ∈ H 1 ( R N ) , where $\lambda >0$ λ > 0 is a parameter, $\alpha \in (0,N)$ α ∈ ( 0 , N ) , $N\ge 3$ N ≥ 3 , $I_{\alpha }: \mathbb{R}^{N}\to \mathbb{R}$ I α : R N → R is the Riesz potential. As usual, $\alpha /N+1$ α / N + 1 is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if $\lambda >\lambda _{*}$ λ > λ ∗ for some given number $\lambda _{*}$ λ ∗ in three cases: (i) $2< p<\frac{4}{N}+2$ 2 < p < 4 N + 2 , (ii) $p=\frac{4}{N}+2$ p = 4 N + 2 , and (iii) $\frac{4}{N}+2< p<2^{*}$ 4 N + 2 < p < 2 ∗ . Our result improves the previous related ones in the literature.

scholarly journals Ground-state positivity, negativity, and compactness for a Schrödinger operator in RN

2007 ◽  
Vol 245 (1) ◽  
pp. 213-248 ◽  
Author(s):  
Bénédicte Alziary ◽  
Jacqueline Fleckinger-Pellé ◽  
Peter Takáč
Keyword(s):  
Schrödinger Operator ◽  
Ground State ◽  
Schrodinger Operator

scholarly journals SCHRÖDINGER SECANT LOWER BOUNDS TO SEMIRELATIVISTIC EIGENVALUES

2007 ◽  
Vol 22 (10) ◽  
pp. 1899-1904 ◽  
Author(s):  
RICHARD L. HALL ◽  
WOLFGANG LUCHA
Keyword(s):  
Lower Bounds ◽  
Schrödinger Operator ◽  
Ground State ◽  
Schrodinger Operator ◽  
Bounded Below

It is shown that the ground-state eigenvalue of a semirelativistic Hamiltonian of the form [Formula: see text] is bounded below by the Schrödinger operator m + β p2 + V, for suitable β>0. An example is discussed.

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