Nodal solutions for some singularly perturbed Dirichlet problems

D’Aprile, Teresa; Pistoia, Angela

doi:10.1090/S0002-9947-2011-05221-9

Nodal solutions for some singularly perturbed Dirichlet problems
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by Teresa D’Aprile and Angela Pistoia PDF

Trans. Amer. Math. Soc. 363 (2011), 3601-3620 Request permission

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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