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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds, Calc. Var. Partial Differential Equations, 24 (2005), 459-477.  Google Scholar

[6]

E. Dancer and S. Yan, Multipeak solutions for a singularly perturbed neumann problem, Pacific J. Math., 189 (1999), 241-262.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. Gui, Multipeak solutions for a semilinear neumann problem, Duke Math J., 84 (1996), 739-769.  Google Scholar

[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.  Google Scholar

[12]

N. Hirano, Multiple existence of solutions for a nonlinear elliptic problem in a Riemannian manifold, Nonlinear Anal., 70 (2009), 671-692.  Google Scholar

[13]

Y. Y. Li, On a singularly perturbed equation with neumann boundary condition, Comm. Partial Differential Equations, 23 (1998), 487-545.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations, 4 (1979), 293-319.  Google Scholar

[22]

L. Schwartz, "Functional Analysis," Courant Institute, Lecture Notes, New York 1964. Google Scholar

[23]

K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078.  Google Scholar

[24]

D. Visetti, Multiplicity of solutions of a zero mass nonlinear equation in a Riemannian manifold, J. Differential Equations, 245 (2008), 2397-2439.  Google Scholar

[25]

J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem, J. Differential Equations, 134 (1997), 104-133.  Google Scholar

[26]

J. Wei and T. Weth, On the number of nodal solutions to a singularly perturbed Neumann problem, Manuscripta Math., 117 (2005), 333-344.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds, Calc. Var. Partial Differential Equations, 24 (2005), 459-477.  Google Scholar

[6]

E. Dancer and S. Yan, Multipeak solutions for a singularly perturbed neumann problem, Pacific J. Math., 189 (1999), 241-262.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. Gui, Multipeak solutions for a semilinear neumann problem, Duke Math J., 84 (1996), 739-769.  Google Scholar

[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.  Google Scholar

[12]

N. Hirano, Multiple existence of solutions for a nonlinear elliptic problem in a Riemannian manifold, Nonlinear Anal., 70 (2009), 671-692.  Google Scholar

[13]

Y. Y. Li, On a singularly perturbed equation with neumann boundary condition, Comm. Partial Differential Equations, 23 (1998), 487-545.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations, 4 (1979), 293-319.  Google Scholar

[22]

L. Schwartz, "Functional Analysis," Courant Institute, Lecture Notes, New York 1964. Google Scholar

[23]

K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078.  Google Scholar

[24]

D. Visetti, Multiplicity of solutions of a zero mass nonlinear equation in a Riemannian manifold, J. Differential Equations, 245 (2008), 2397-2439.  Google Scholar

[25]

J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem, J. Differential Equations, 134 (1997), 104-133.  Google Scholar

[26]

J. Wei and T. Weth, On the number of nodal solutions to a singularly perturbed Neumann problem, Manuscripta Math., 117 (2005), 333-344.  Google Scholar

[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.  Google Scholar

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