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 e 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 5:899–928, 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. Poincare Anal Non Lineaire 26:1423–1451, 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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