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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nodal solutions for some singularly perturbed Dirichlet problems


Authors: Teresa D’Aprile and Angela Pistoia
Journal: Trans. Amer. Math. Soc. 363 (2011), 3601-3620
MSC (2010): Primary 35B40, 35J20, 35J57
Published electronically: February 1, 2011
MathSciNet review: 2775820
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Abstract: We consider the equation $ -\varepsilon^2 \Delta u+u=f(u)$ in a bounded, smooth domain $ \Omega \subset \Bbb R^N$ with homogeneous Dirichlet boundary conditions. We prove the existence of nodal solutions with multiple peaks concentrating at different points of $ \Omega$. The nonlinearity $ f$ grows superlinearly and subcritically. We do not require symmetry conditions on the geometry of the domain.


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Additional Information

Teresa D’Aprile
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133 Roma, Italy
Email: daprile@mat.uniroma2.it

Angela Pistoia
Affiliation: Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, via Antonio Scarpa 16, 00161 Roma, Italy
Email: pistoia@dmmm.uniroma1.it

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05221-9
PII: S 0002-9947(2011)05221-9
Keywords: Nodal solutions, multiple peaks, finite-dimensional reduction.
Received by editor(s): December 16, 2008
Received by editor(s) in revised form: October 1, 2009
Published electronically: February 1, 2011
Additional Notes: The authors were supported by Mi.U.R. project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.
Article copyright: © Copyright 2011 American Mathematical Society