Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems |
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Authors: | Teresa D??Aprile David Ruiz |
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Institution: | 1. Dipartimento di Matematica, Universit?? di Roma ??Tor Vergata??, via della Ricerca Scientifica 1, 00133, Rome, Italy 2. Departamento de An??lisis Matem??tico, Facultad de Ciencias, Universidad de Granada, Avda. Fuentenueva s/n, 18071, Granada, Spain
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Abstract: | We study the existence and asymptotic behavior of positive and sign-changing multipeak solutions for the equation $$ -\varepsilon^2\Delta v+V(x)v=f(v)\quad{\rm in}\,\,\,\mathbb{R}^N, $$ where ?? is a small positive parameter, f a superlinear, subcritical and odd nonlinearity, V a uniformly positive potential. No symmetry on V is assumed. It is known (Kang and Wei in Adv Differ Equ 5:899?C928, 2000) that this equation has positive multipeak solutions with all peaks approaching a local maximum of V. It is also proved that solutions alternating positive and negative spikes exist in the case of a minimum (see D??Aprile and Pistoia in Ann Inst H. Poincaré Anal Non Linéaire 26:1423?C1451, 2009). The aim of this paper is to show the existence of both positive and sign-changing multipeak solutions around a nondegenerate saddle point of V. |
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