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Singularly perturbed nonlinear elliptic problems on manifolds. (English) Zbl 1126.58007
Summary: Let $\cal M$ be a connected compact smooth Riemannian manifold of dimension $n \ge 3$ with or without smooth boundary $\partial\cal M$. We consider the following singularly perturbed nonlinear elliptic problem on $\cal M$ $$\varepsilon^2\Delta_{\cal M}u - u + f(u) =0, \quad\text{on}\quad\cal M, \quad \frac{\partial u}{\partial \nu}=0\text{ on }\partial\cal M$$ where $\Delta_{\cal M}$ is the Laplace-Beltrami operator on $\cal M$, $\nu$ is an exterior normal to $\partial\cal M$ and a nonlinearity $f$ of subcritical growth. For certain $f$, there exists a mountain pass solution $u_\varepsilon$ 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 $\partial\cal M = \emptyset$ ($\partial\cal M \ne \emptyset$), the peak point $x_\varepsilon$ of the solution $u_\varepsilon$ converges to a maximum point of the scalar curvature $S$ on $\cal M$ (the mean curvature $H$ on $\partial\cal M$) as $\varepsilon \to 0$, respectively.

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