scholarly journals Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential

2013 ◽  
Vol 33 (1) ◽  
pp. 7-26 ◽  
Author(s):  
Thomas Bartsch ◽  
◽  
Zhongwei Tang ◽  
Keyword(s):  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Potential Well ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Indefinite Potential ◽  
Steep Potential Well

NONLINEAR SCHRÖDINGER EQUATIONS WITH STEEP POTENTIAL WELL

2001 ◽  
Vol 03 (04) ◽  
pp. 549-569 ◽  
Author(s):  
THOMAS BARTSCH ◽  
ALEXANDER PANKOV ◽  
ZHI-QIANG WANG
Keyword(s):  
Essential Spectrum ◽  
Model Equation ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Potential Well ◽  
Local Minima ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Steep Potential Well

We investigate nonlinear Schrödinger equations like the model equation [Formula: see text] where the potential Vλ has a potential well with bottom independent of the parameter λ > 0. If λ → ∞ the infimum of the essential spectrum of -Δ + Vλ in L2(ℝN) converges towards ∞ and more and more eigenvalues appear below the essential spectrum. We show that as λ→∞ more and more solutions of the nonlinear Schrödinger equation exist. The solutions lie in H1(ℝN) and are localized near the bottom of the potential well, but not near local minima of the potential. We also investigate the decay rate of the solutions as |x|→∞ as well as their behaviour as λ→∞.

scholarly journals SIGN-CHANGING SOLUTIONS FOR A CLASS OF NONLINEAR SCHRÖDINGER EQUATIONS

2009 ◽  
Vol 80 (2) ◽  
pp. 294-305 ◽  
Author(s):  
XIANGQING LIU ◽  
YISHENG HUANG
Keyword(s):  
Variational Methods ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Potential Well ◽  
Asymptotically Linear ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Sign Changing Solutions

AbstractUsing variational methods, we obtain the existence of sign-changing solutions for a class of asymptotically linear Schrödinger equations with deepening potential well.

Ground States for Nonlinear Schrödinger Equations with a Sign-changing Potential Well

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Zhi-Qiang Wang ◽  
Jiankang Xia
Keyword(s):  
Ground State ◽  
Ground States ◽  
Nehari Manifold ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Potential Well ◽  
Nonlinear Schrödinger ◽  
Variational Functional ◽  
Nonlinear Schrodinger Equations

AbstractIn this paper, we consider the ground state solutions for a class of nonlinear Schrödinger equationswhere 2 < p < 2*. We investigate the case λ > inf σ (−Δ + V), i.e., an indefinite problem.We characterize the ground states as minimizers of the variational functional on a modified Nehari manifold.

scholarly journals Infinitely many positive solutions of nonlinear Schrödinger equations

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Riccardo Molle ◽  
Donato Passaseo
Keyword(s):  
Positive Solutions ◽  
Radial Symmetry ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

AbstractThe paper deals with the equation $$-\Delta u+a(x) u =|u|^{p-1}u $$ - Δ u + a ( x ) u = | u | p - 1 u , $$u \in H^1({\mathbb {R}}^N)$$ u ∈ H 1 ( R N ) , with $$N\ge 2$$ N ≥ 2 , $$p> 1,\ p< {N+2\over N-2}$$ p > 1 , p < N + 2 N - 2 if $$N\ge 3$$ N ≥ 3 , $$a\in L^{N/2}_{loc}({\mathbb {R}}^N)$$ a ∈ L loc N / 2 ( R N ) , $$\inf a> 0$$ inf a > 0 , $$\lim _{|x| \rightarrow \infty } a(x)= a_\infty $$ lim | x | → ∞ a ( x ) = a ∞ . Assuming that the potential a(x) satisfies $$\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta > 0$$ lim | x | → ∞ [ a ( x ) - a ∞ ] e η | x | = ∞ ∀ η > 0 , $$ \lim _{\rho \rightarrow \infty } \sup \left\{ a(\rho \theta _1) - a(\rho \theta _2) \ :\ \theta _1, \theta _2 \in {\mathbb {R}}^N,\ |\theta _1|= |\theta _2|=1 \right\} e^{\tilde{\eta }\rho } = 0 \quad \text{ for } \text{ some } \ \tilde{\eta }> 0$$ lim ρ → ∞ sup a ( ρ θ 1 ) - a ( ρ θ 2 ) : θ 1 , θ 2 ∈ R N , | θ 1 | = | θ 2 | = 1 e η ~ ρ = 0 for some η ~ > 0 and other technical conditions, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers because we do not require that a(x) has radial symmetry, or that $$N=2$$ N = 2 , or that $$|a(x)-a_\infty |$$ | a ( x ) - a ∞ | is uniformly small in $${\mathbb {R}}^N$$ R N , etc. ....

scholarly journals N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 733
Author(s):  
Yu-Shan Bai ◽  
Peng-Xiang Su ◽  
Wen-Xiu Ma
Keyword(s):  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Lax Pairs ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
The Given

In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.

Sign in / Sign up

Export Citation Format

Share Document