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

Summary: Let \(\mathcal M\) be a connected compact smooth Riemannian manifold of dimension \(n \geq 3\) with or without smooth boundary \(\partial\mathcal M\). We consider the following singularly perturbed nonlinear elliptic problem on \(\mathcal M\) \[ \varepsilon^2\Delta_{\mathcal M}u - u + f(u) =0, \quad\text{on}\quad\mathcal M, \quad \frac{\partial u}{\partial \nu}=0\text{ on }\partial\mathcal M \] where \(\Delta_{\mathcal M}\) is the Laplace-Beltrami operator on \(\mathcal M\), \(\nu\) is an exterior normal to \(\partial\mathcal M\) 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 \(\partial\mathcal M = \emptyset\) (\(\partial\mathcal M \neq \emptyset\)), the peak point \(x_\varepsilon\) of the solution \(u_\varepsilon\) converges to a maximum point of the scalar curvature \(S\) on \(\mathcal M\) (the mean curvature \(H\) on \(\partial\mathcal M\)) as \(\varepsilon \to 0\), respectively.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B25 Singular perturbations in context of PDEs
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
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