# American Institute of Mathematical Sciences

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 & 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. [2] V. Benci, Introduction to Morse theory. A new approach,, in, (1995), 37. [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. [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. [5] J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds,, Calc. Var. Partial Differential Equations, 24 (2005), 459. [6] E. Dancer and S. Yan, Multipeak solutions for a singularly perturbed neumann problem,, Pacific J. Math., 189 (1999), 241. [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. [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. [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, (2007), 15. [10] C. Gui, Multipeak solutions for a semilinear neumann problem,, Duke Math J., 84 (1996), 739. [11] C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed neumann problems,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 17 (2000), 47. [12] N. Hirano, Multiple existence of solutions for a nonlinear elliptic problem in a Riemannian manifold, , Nonlinear Anal., 70 (2009), 671. [13] Y. Y. Li, On a singularly perturbed equation with neumann boundary condition,, Comm. Partial Differential Equations, 23 (1998), 487. [14] A. M. Micheletti and A. Pistoia, Generic properties of singularly perturbed nonlinear elliptic problems on Riemannian manifolds, , Adv. Nonlinear Stud., 9 (2009), 803. [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. [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. [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). [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. [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. [20] F. Quinn, Transversal approximation on Banach manifolds,, in, (1968), 213. [21] J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems,, Comm. Partial Differential Equations, 4 (1979), 293. [22] L. Schwartz, "Functional Analysis,", Courant Institute, (1964). [23] K. Uhlenbeck, Generic properties of eigenfunctions,, Amer. J. Math., 98 (1976), 1059. [24] D. Visetti, Multiplicity of solutions of a zero mass nonlinear equation in a Riemannian manifold, , J. Differential Equations, 245 (2008), 2397. [25] J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem,, J. Differential Equations, 134 (1997), 104. [26] J. Wei and T. Weth, On the number of nodal solutions to a singularly perturbed Neumann problem,, Manuscripta Math., 117 (2005), 333. [27] J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems,, J. London Math. Soc., 59 (1999), 585.

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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. [2] V. Benci, Introduction to Morse theory. A new approach,, in, (1995), 37. [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. [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. [5] J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds,, Calc. Var. Partial Differential Equations, 24 (2005), 459. [6] E. Dancer and S. Yan, Multipeak solutions for a singularly perturbed neumann problem,, Pacific J. Math., 189 (1999), 241. [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. [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. [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, (2007), 15. [10] C. Gui, Multipeak solutions for a semilinear neumann problem,, Duke Math J., 84 (1996), 739. [11] C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed neumann problems,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 17 (2000), 47. [12] N. Hirano, Multiple existence of solutions for a nonlinear elliptic problem in a Riemannian manifold, , Nonlinear Anal., 70 (2009), 671. [13] Y. Y. Li, On a singularly perturbed equation with neumann boundary condition,, Comm. Partial Differential Equations, 23 (1998), 487. [14] A. M. Micheletti and A. Pistoia, Generic properties of singularly perturbed nonlinear elliptic problems on Riemannian manifolds, , Adv. Nonlinear Stud., 9 (2009), 803. [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. [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. [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). [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. [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. [20] F. Quinn, Transversal approximation on Banach manifolds,, in, (1968), 213. [21] J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems,, Comm. Partial Differential Equations, 4 (1979), 293. [22] L. Schwartz, "Functional Analysis,", Courant Institute, (1964). [23] K. Uhlenbeck, Generic properties of eigenfunctions,, Amer. J. Math., 98 (1976), 1059. [24] D. Visetti, Multiplicity of solutions of a zero mass nonlinear equation in a Riemannian manifold, , J. Differential Equations, 245 (2008), 2397. [25] J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem,, J. Differential Equations, 134 (1997), 104. [26] J. Wei and T. Weth, On the number of nodal solutions to a singularly perturbed Neumann problem,, Manuscripta Math., 117 (2005), 333. [27] J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems,, J. London Math. Soc., 59 (1999), 585.
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