6.—Oscillation Theory for Semilinear Schrödinger Equations and Inequalities

1976 ◽  
Vol 75 (1) ◽  
pp. 67-81 ◽  
Author(s):  
E. S. Noussair ◽  
C. A. Swanson
Keyword(s):  
Schrödinger Equation ◽  
Positive Solution ◽  
Differential Inequality ◽  
Exterior Domain ◽  
Schrodinger Equation ◽  
Sufficient Conditions ◽  
Oscillation Theory ◽  
Oscillatory Behaviour ◽  
Spherical Mean ◽  
Fowler Equation

SynopsisSufficient conditions are derived for every solution of a nonlinear Schrödinger equation (or inequality) to be oscillatory in an exterior domain ofEn. Such results apply in particular to then-dimensional Emden-Fowler equation. The method involves oscillatory behaviour of solutions of a nonlinear ordinary differential inequality satisfied by the spherical mean of a positive solution of the Schrödinger equation.

scholarly journals Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Xiaowei An ◽  
Desheng Li ◽  
Xianfa Song
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Sufficient Conditions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Complex Valued ◽  
Sharp Thresholds

We consider the following Cauchy problem:-iut=Δu-V(x)u+f(x,|u|2)u+(W(x)⋆|u|2)u,x∈ℝN,t>0,u(x,0)=u0(x),x∈ℝN,whereV(x)andW(x)are real-valued potentials andV(x)≥0andW(x)is even,f(x,|u|2)is measurable inxand continuous in|u|2, andu0(x)is a complex-valued function ofx. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem.

scholarly journals A POSITIVE SOLUTION FOR A NONLOCAL SCHRÖDINGER EQUATION

2014 ◽  
Vol 90 (3) ◽  
pp. 469-475 ◽  
Author(s):  
YONGCHAO ZHANG ◽  
GAOSHENG ZHU
Keyword(s):  
Schrödinger Equation ◽  
Positive Solution ◽  
Lévy Process ◽  
Schrodinger Equation ◽  
Infinitesimal Generator ◽  
Existence Result ◽  
Nonlinear Schrodinger Equation ◽  
Levy Process ◽  
Classical Solutions ◽  
Rotationally Invariant

AbstractWe provide an existence result of radially symmetric, positive, classical solutions for a nonlinear Schrödinger equation driven by the infinitesimal generator of a rotationally invariant Lévy process.

Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials

2015 ◽  
Vol 145 (3) ◽  
pp. 445-465 ◽  
Author(s):  
Anouar Bahrouni ◽  
Hichem Ounaies ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Sufficient Conditions ◽  
Schrodinger Equations ◽  
Infinitely Many Solutions ◽  
Schrödinger Equations ◽  
Entire Solutions ◽  
Qualitative Properties

In this paper we are concerned with qualitative properties of entire solutions to a Schrödinger equation with sublinear nonlinearity and sign-changing potentials. Our analysis considers three distinct cases and we establish sufficient conditions for the existence of infinitely many solutions.

scholarly journals An Existence Result for a Class of Magnetic Problems in Exterior Domains

2021 ◽  
Author(s):  
Claudianor O. Alves ◽  
Vincenzo Ambrosio ◽  
César E. Torres Ledesma
Keyword(s):  
Schrödinger Equation ◽  
Vector Potential ◽  
Exterior Domain ◽  
Schrodinger Equation ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Exterior Domains ◽  
Continuous Vector ◽  
Semilinear Schrödinger Equation

AbstractIn this paper we deal with the existence of solutions for the following class of magnetic semilinear Schrödinger equation $$\begin{aligned} (P) \qquad \qquad \left\{ \begin{aligned}&(-i\nabla + A(x))^2u +u = |u|^{p-2}u,\;\;\text{ in }\;\;\Omega ,\\&u=0\;\;\text{ on }\;\;\partial \Omega , \end{aligned} \right. \end{aligned}$$ ( P ) ( - i ∇ + A ( x ) ) 2 u + u = | u | p - 2 u , in Ω , u = 0 on ∂ Ω , where $$N \ge 3$$ N ≥ 3 , $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is an exterior domain, $$p\in (2, 2^*)$$ p ∈ ( 2 , 2 ∗ ) with $$2^*=\frac{2N}{N-2}$$ 2 ∗ = 2 N N - 2 , and $$A: {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$$ A : R N → R N is a continuous vector potential verifying $$A(x) \rightarrow 0\;\;\text{ as }\;\;|x|\rightarrow \infty .$$ A ( x ) → 0 as | x | → ∞ .

Bifurcation and standing wave solutions for a quasilinear Schrödinger equation

2018 ◽  
Vol 149 (04) ◽  
pp. 939-968
Author(s):  
Guowei Dai
Keyword(s):  
Schrödinger Equation ◽  
Positive Solution ◽  
Standing Wave ◽  
Positive Solutions ◽  
Schrodinger Equation ◽  
Quasilinear Schrödinger Equation ◽  
Topological Methods ◽  
Wave Solutions ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

AbstractWe use bifurcation and topological methods to investigate the existence/nonexistence and the multiplicity of positive solutions of the following quasilinear Schrödinger equation$$\left\{ {\matrix{ {-\Delta u-\kappa \Delta \left( {u^2} \right)u = \beta u-\lambda \Phi \left( {u^2} \right)u{\mkern 1mu} {\mkern 1mu} } \hfill & {{\rm in}\;\Omega ,} \hfill \cr {u = 0} \hfill & {{\rm on}\;\partial \Omega } \hfill \cr } } \right.$$involving sublinear/linear/superlinear nonlinearities at zero or infinity with/without signum condition. In particular, we study the changes in the structure of positive solution withκas the varying parameter.

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