Positive solutions for singularly perturbed nonlinear elliptic problem on manifolds via Morse theory

Details Positive solutions for singularly perturbed nonlinear elliptic problem on manifolds via Morse theory Positive solutions for singularly perturbed nonlinear elliptic problem on manifolds via Morse theory

2010-12-27

arxiv.org/abs/1012.5672v1

Mathematics

Analysis of PDEs

Scientific paper

Given (M, g0) we consider the problem -{\epsilon}^2Delta_{g0+h}u + u = (u+)^{p-1} with ({\epsilon}, h) \in (0, {\epsilon}0) \times B{\rho}. Here B{\rho} is a ball centered at 0 with radius {\rho} in the Banach space of all Ck symmetric covariant 2-tensors on M. Using the Poincar\'e polynomial of M, we give an estimate on the number of nonconstant solutions with low energy for ({\epsilon}, h) belonging to a residual subset of (0, {\epsilon}0) \times B{\rho}, for ({\epsilon}0, {\rho}) small enough.

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