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A POSITIVE SOLUTION FOR A NONLOCAL SCHRÖDINGER EQUATION

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  15 July 2014

YONGCHAO ZHANG  and
GAOSHENG ZHU
YONGCHAO ZHANG*
Affiliation:
School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Taishan Road 143, Qinhuangdao 066004,PR China email [email protected]
GAOSHENG ZHU
Affiliation:
School of Science, Tianjin University, Weijin Road 92, Tianjin 300072,PR China email [email protected]
*
✉[email protected]
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Abstract

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We 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.

Keywords

nonlocal Schrödinger equationpositive solutionmountain pass theorem

MSC classification

Secondary: 35A01: Existence problems 35A15: Variational methods 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 3 , December 2014 , pp. 469 - 475
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Applebaum, D., Lévy Processes and Stochastic Calculus, 2nd edn (Cambridge University Press, Cambridge, 2009).Google Scholar
Byeon, J. and Wang, Z.-Q., ‘Standing waves with a critical frequency for nonlinear Schrödinger equations’, Arch. Ration. Mech. Anal. 165 (2002), 295–316.CrossRefGoogle Scholar
del Pino, M. and Felmer, P., ‘Local mountain passes for semilinear elliptic problems in unbounded domains’, Calc. Var. Partial Differential Equations 4 (1996), 121–137.Google Scholar
del Pino, M. and Felmer, P., ‘Semi-classical states of nonlinear Schrödinger equations: a variational reduction method’, Math. Ann. 324 (2002), 1–32.CrossRefGoogle Scholar
Dipierro, S., Palatucci, G. and Valdinoci, E., 2012. Existence and symmetry results for a Schödinger type problem involving the fractional Laplacian, arXiv:1202.0576v1.Google Scholar
Fall, M. M. and Valdinoci, E., 2013. Uniqueness and nondegeneracy of positive solutions of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}(-\Delta )^s u+u =u^p$ in $\mathbb{R}^n$ when $s$ is close to 1, arXiv:1301.4868v1.Google Scholar
Fall, M. M. and Weth, T., ‘Nonexistence results for a class of fractional elliptic boundary value problems’, J. Funct. Anal. 263 (2012), 2205–2227.CrossRefGoogle Scholar
Felmer, P., Quaas, A. and Tan, J., ‘Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian’, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237–1262.Google Scholar
Floer, A. and Weinstein, A., ‘Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential’, J. Funct. Anal. 69 (1986), 397–408.Google Scholar
Jeanjean, L. and Tanaka, K., ‘A positive solution for a nonlinear Schrödinger equation on ℝn’, Indiana Univ. Math. J. 54 (2005), 443–464.Google Scholar
Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, 3rd edn (Springer, Berlin, 1999).CrossRefGoogle Scholar
Struwe, M., Variational Methods, 4th edn (Springer, Berlin, 2008).Google Scholar
Willem, M., Minimax Theorems (Birkhäuser, Boston, MA, 1996).Google Scholar