Non degeneracy for solutions of singularly perturbed nonlinear elliptic problemson symmetric Riemannian manifolds

Marco Ghimenti; A. M. Micheletti

doi:10.3934/cpaa.2013.12.679

Article Contents

2013, Volume 12, Issue 2: 679-693. Doi: 10.3934/cpaa.2013.12.679

This issue Previous Article Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$ Next Article Quasilinear systems involving multiple critical exponents and potentials

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
2.
Dipartimento di Matematica Applicata "U.Dini", Università di Pisa, Via Bonanno 25B - 56126 Pisa

Received: May 31, 2011

Revised: November 30, 2011

Published: February 2013

Abstract Related Papers Cited by

Abstract

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:

Symmetric Riemannian manifolds,
non degenerate sign changing solutions,
singularly perturbed nonlinear elliptic problems.

Mathematics Subject Classification: 58G03, 58E30.

Citation:

Related Papers

Cited by

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.