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Multiple existence of solutions for a nonlinear elliptic problem on a Riemannian manifold. (English) Zbl 1157.58006

Let \((M,g)\) be a compact, connected and orientable smooth \(N\)-dimensional Riemannian manifold without boundary, and let \(p\in(2,2^*)\) with \(2^*=2N/(N-2).\)
The author derives the rexistence of multiple solutions to the problem
\[ -\varepsilon^2\Delta_gu+u=| u| ^{p-2}u\quad \text{on}\;M \] with \(\Delta_g\) standing for the Laplacian operator over \(M.\)

MSC:

58J05 Elliptic equations on manifolds, general theory
35J65 Nonlinear boundary value problems for linear elliptic equations
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