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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

Marco Ghimenti 1, and  A. M. Micheletti 2,

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.
Keywords: non degenerate sign changing solutions, Symmetric Riemannian manifolds, singularly perturbed nonlinear elliptic problems..
Mathematics Subject Classification: 58G03, 58E3.
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.

show all references

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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