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Singularly perturbed nonlinear elliptic problems on manifolds. (English) Zbl 1126.58007

Summary: Let be a connected compact smooth Riemannian manifold of dimension n3 with or without smooth boundary . We consider the following singularly perturbed nonlinear elliptic problem on

ε 2 Δ u-u+f(u)=0,on,u ν=0on

where Δ is the Laplace-Beltrami operator on , ν is an exterior normal to and a nonlinearity f of subcritical growth. For certain f, there exists a mountain pass solution u ε of above problem which exhibits a spike layer. We are interested in the asymptotic behaviour of the spike layer. Without any non-degeneracy condition and monotonicity of f(t)/t, we show that if = (), the peak point x ε of the solution u ε converges to a maximum point of the scalar curvature S on (the mean curvature H on ) as ε0, respectively.

MSC:
58E05Abstract critical point theory
35B25Singular perturbations (PDE)
35J20Second order elliptic equations, variational methods
35J60Nonlinear elliptic equations
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