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.


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
Full Text: DOI


[1] Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973). · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[2] Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82, 313–345 (1983). · Zbl 0533.35029
[3] Byeon, J.: Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains. Comm. in P. D. E. 22, 1731–1769 (1997). · Zbl 0883.35040 · doi:10.1080/03605309708821317
[4] Chavel, I.: Eigenvalues in Riemannian Geometry. Academic Press (1984). · Zbl 0551.53001
[5] Del Pino, M., Felmer, P.L.: Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting. Indiana Univ. Math. J. 48, 883–898 (1999). · Zbl 0932.35080
[6] Escobar, F.: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Annal. Math. 136, 1–50 (1992). · Zbl 0766.53033 · doi:10.2307/2946545
[7] Gidas, B., Ni, W.N., Nirenberg, L.: Symmetry and related properties via teh maximum principle. Comm. Math. Phys. 68, 209–243 (1979). · Zbl 0425.35020 · doi:10.1007/BF01221125
[8] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order; second edition. Grundlehren 224, Springer, Berlin, Heidelberg, New York and Tokyo (1983). · Zbl 0562.35001
[9] Jeanjean, L., Tanaka, K.: A remark on least energy solutions in RN. Proc. Amer. Math. Soc. 131, 2399–2408 (2003). · Zbl 1094.35049 · doi:10.1090/S0002-9939-02-06821-1
[10] Kwong, K.M.: Uniqueness of positive solutions of {\(\Delta\)} u+up = 0 in Rn. Arch. Rat. Mech. Anal. 105, 243–266 (1989). · Zbl 0676.35032 · doi:10.1007/BF00251502
[11] Li, Y.Y., Nirenberg, L.: The Dirichlet problem for singularly perturbed elliptic equations. Comm. Pure Appl. Math. 51, 1445–1490 (1998). · Zbl 0933.35083 · doi:10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z
[12] Lin, C.S., Ni, W.M., Takagi, I.: Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations 72, 1–27 (1988). · Zbl 0676.35030 · doi:10.1016/0022-0396(88)90147-7
[13] Ni, W.M., Takagi, I.: On the shape of least-energy solutions to a semilinear Neumann problem. Comm. Pure Appl. Math. 44, 819–851 (1991). · Zbl 0754.35042 · doi:10.1002/cpa.3160440705
[14] Ni, W.M., Takagi, I.: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70, 247–281 (1993). · Zbl 0796.35056 · doi:10.1215/S0012-7094-93-07004-4
[15] Ni, W.M., Takagi, I., Yanagida, E.: Stability of least energy patterns of the shadow system for an activator-inhibitor model. Japan J. Indust. Appl. Math. 18, 259–272 (2001). · Zbl 1200.35172 · doi:10.1007/BF03168574
[16] Ni, W.M., Wei, J.: On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. Comm. Pure Appl. Math. 48, 731–768 (1995). · Zbl 0838.35009 · doi:10.1002/cpa.3160480704
[17] Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer-Verlag, New York, Berlin, Heidelberg and Tokyo (1984). · Zbl 0549.35002
[18] Struwe, M.: Variational Methods; Application to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer-Verlag (1990). · Zbl 0746.49010
[19] Wei, J., Winter, M.: Higher-order energy expansions and spike locations. Calculus of Variations and PDE 20, 403–430 (2004). · Zbl 1154.35353
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.