# American Institute of Mathematical Sciences

March  2013, 12(2): 679-693. doi: 10.3934/cpaa.2013.12.679

## Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds

 1 Dipartimento di Matematica Applicata, Università di Pisa, via Buonarroti 1c, 56127, Pisa, Italy 2 Dipartimento di Matematica Applicata "U.Dini", Università di Pisa, Via Bonanno 25B - 56126 Pisa, Italy

Received  June 2011 Revised  December 2011 Published  September 2012

Given a symmetric Riemannian manifold $(M,g)$, we show some results of genericity for non degenerate sign changing solutions of singularly perturbed nonlinear elliptic problems with respect to the parameters: the positive number $\varepsilon$ and the symmetric metric $g$. Using these results we obtain a lower bound on the number of non degenerate solutions which change sign exactly once.
Citation: Marco Ghimenti, A. M. Micheletti. Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds. Communications on Pure and Applied Analysis, 2013, 12 (2) : 679-693. doi: 10.3934/cpaa.2013.12.679
##### References:
 [1] N. Aronszajn, A. Krzywicki and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat., 4 (1962), 417-453. [2] V. Benci, Introduction to Morse theory. A new approach, in "Topological Nonlinear Analysis: Degree, Singularity, and Variations" (Michele Matzeu and Alfonso Vignoli, eds.), Progress in Nonlinear Differential Equations and their Applications, no. 15, Birkhäuser, Boston, 1995, pp. 37-177. [3] V. Benci, C. Bonanno and A. M. Micheletti, On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds, J. Funct. Anal., 252 (2007), 464-489. [4] V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48. [5] J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds, Calc. Var. Partial Differential Equations, 24 (2005), 459-477. [6] E. Dancer and S. Yan, Multipeak solutions for a singularly perturbed neumann problem, Pacific J. Math., 189 (1999), 241-262. [7] E. N. Dancer, A. M. Micheletti and A. Pistoia, Multipeak solutions for some singularly perturbed nonlinear elliptic problems in a {Riemannian manifold}, Manuscripta Math., 128 (2009), 163-193. [8] M. Del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79. [9] M. Ghimenti and A. M. Micheletti, On the number of nodal solutions for a nonlinear elliptic problem on symmetric Riemannian manifolds, Proceedings of the 2007 Conference on Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems (San Marcos, TX), Electron. J. Differ. Equ. Conf., vol. 18, Southwest Texas State Univ., 2010, pp. 15-22. [10] C. Gui, Multipeak solutions for a semilinear neumann problem, Duke Math J., 84 (1996), 739-769. [11] C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 47-82. [12] N. Hirano, Multiple existence of solutions for a nonlinear elliptic problem in a Riemannian manifold, Nonlinear Anal., 70 (2009), 671-692. [13] Y. Y. Li, On a singularly perturbed equation with neumann boundary condition, Comm. Partial Differential Equations, 23 (1998), 487-545. [14] A. M. Micheletti and A. Pistoia, Generic properties of singularly perturbed nonlinear elliptic problems on Riemannian manifolds, Adv. Nonlinear Stud., 9 (2009), 803-815. [15] A. M. Micheletti and A. Pistoia, Nodal solutions for a singularly perturbed nonlinear elliptic problem in a Riemannian manifold, Adv. Nonlinear Stud., 9 (2009), 565-577. [16] A. M. Micheletti and A. Pistoia, The role of the scalar curvature in a nonlinear elliptic problem in a Riemannian manifold, Calc. Var. Partial Differential Equations, 34 (2009), 233-265. [17] A. M. Micheletti and A. Pistoia, On the existence of nodal solutions for a nonlinear elliptic problem on symmetric Riemannian manifolds, Int. J. Differ. Equ., (2010), Art. ID 432759, 11 pp. [18] W. N. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851. [19] W. N. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear neumann problem, Duke Math. J., 70 (1993), 247-281. [20] F. Quinn, Transversal approximation on Banach manifolds, in "Global Analysis (Proc. Sympos. Pure Math.," Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 213-222. [21] J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations, 4 (1979), 293-319. [22] L. Schwartz, "Functional Analysis," Courant Institute, Lecture Notes, New York 1964. [23] K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078. [24] D. Visetti, Multiplicity of solutions of a zero mass nonlinear equation in a Riemannian manifold, J. Differential Equations, 245 (2008), 2397-2439. [25] J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem, J. Differential Equations, 134 (1997), 104-133. [26] J. Wei and T. Weth, On the number of nodal solutions to a singularly perturbed Neumann problem, Manuscripta Math., 117 (2005), 333-344. [27] J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems, J. London Math. Soc., 59 (1999), 585-606.

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##### References:
 [1] N. Aronszajn, A. Krzywicki and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat., 4 (1962), 417-453. [2] V. Benci, Introduction to Morse theory. A new approach, in "Topological Nonlinear Analysis: Degree, Singularity, and Variations" (Michele Matzeu and Alfonso Vignoli, eds.), Progress in Nonlinear Differential Equations and their Applications, no. 15, Birkhäuser, Boston, 1995, pp. 37-177. [3] V. Benci, C. Bonanno and A. M. Micheletti, On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds, J. Funct. Anal., 252 (2007), 464-489. [4] V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48. [5] J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds, Calc. Var. Partial Differential Equations, 24 (2005), 459-477. [6] E. Dancer and S. Yan, Multipeak solutions for a singularly perturbed neumann problem, Pacific J. Math., 189 (1999), 241-262. [7] E. N. Dancer, A. M. Micheletti and A. Pistoia, Multipeak solutions for some singularly perturbed nonlinear elliptic problems in a {Riemannian manifold}, Manuscripta Math., 128 (2009), 163-193. [8] M. Del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79. [9] M. Ghimenti and A. M. Micheletti, On the number of nodal solutions for a nonlinear elliptic problem on symmetric Riemannian manifolds, Proceedings of the 2007 Conference on Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems (San Marcos, TX), Electron. J. Differ. Equ. Conf., vol. 18, Southwest Texas State Univ., 2010, pp. 15-22. [10] C. Gui, Multipeak solutions for a semilinear neumann problem, Duke Math J., 84 (1996), 739-769. [11] C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 47-82. [12] N. Hirano, Multiple existence of solutions for a nonlinear elliptic problem in a Riemannian manifold, Nonlinear Anal., 70 (2009), 671-692. [13] Y. Y. Li, On a singularly perturbed equation with neumann boundary condition, Comm. Partial Differential Equations, 23 (1998), 487-545. [14] A. M. Micheletti and A. Pistoia, Generic properties of singularly perturbed nonlinear elliptic problems on Riemannian manifolds, Adv. Nonlinear Stud., 9 (2009), 803-815. [15] A. M. Micheletti and A. Pistoia, Nodal solutions for a singularly perturbed nonlinear elliptic problem in a Riemannian manifold, Adv. Nonlinear Stud., 9 (2009), 565-577. [16] A. M. Micheletti and A. Pistoia, The role of the scalar curvature in a nonlinear elliptic problem in a Riemannian manifold, Calc. Var. Partial Differential Equations, 34 (2009), 233-265. [17] A. M. Micheletti and A. Pistoia, On the existence of nodal solutions for a nonlinear elliptic problem on symmetric Riemannian manifolds, Int. J. Differ. Equ., (2010), Art. ID 432759, 11 pp. [18] W. N. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851. [19] W. N. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear neumann problem, Duke Math. J., 70 (1993), 247-281. [20] F. Quinn, Transversal approximation on Banach manifolds, in "Global Analysis (Proc. Sympos. Pure Math.," Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 213-222. [21] J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations, 4 (1979), 293-319. [22] L. Schwartz, "Functional Analysis," Courant Institute, Lecture Notes, New York 1964. [23] K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078. [24] D. Visetti, Multiplicity of solutions of a zero mass nonlinear equation in a Riemannian manifold, J. Differential Equations, 245 (2008), 2397-2439. [25] J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem, J. Differential Equations, 134 (1997), 104-133. [26] J. Wei and T. Weth, On the number of nodal solutions to a singularly perturbed Neumann problem, Manuscripta Math., 117 (2005), 333-344. [27] J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems, J. London Math. Soc., 59 (1999), 585-606.
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