American Institute of Mathematical Sciences

Advanced
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. Google Scholar

[2]

V. Benci, Introduction to Morse theory. A new approach,, in, (1995), 37. 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. 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. Google Scholar

[5]

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

[6]

E. Dancer and S. Yan, Multipeak solutions for a singularly perturbed neumann problem,, Pacific J. Math., 189 (1999), 241. 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. 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. 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, (2007), 15. Google Scholar

[10]

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

[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. Google Scholar

[12]

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

[13]

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

[20]

F. Quinn, Transversal approximation on Banach manifolds,, in, (1968), 213. Google Scholar

[21]

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

[22]

L. Schwartz, "Functional Analysis,", Courant Institute, (1964). Google Scholar

[23]

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

[24]

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

[25]

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

[2]

V. Benci, Introduction to Morse theory. A new approach,, in, (1995), 37. 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. 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. Google Scholar

[5]

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

[6]

E. Dancer and S. Yan, Multipeak solutions for a singularly perturbed neumann problem,, Pacific J. Math., 189 (1999), 241. 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. 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. 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, (2007), 15. Google Scholar

[10]

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

[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. Google Scholar

[12]

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

[13]

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

[20]

F. Quinn, Transversal approximation on Banach manifolds,, in, (1968), 213. Google Scholar

[21]

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

[22]

L. Schwartz, "Functional Analysis,", Courant Institute, (1964). Google Scholar

[23]

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

[24]

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

[25]

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

[1]

Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia. The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2535-2560. doi: 10.3934/dcds.2014.34.2535
[2]

Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256
[3]

Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151
[4]

A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253
[5]

Andrés Ávila, Louis Jeanjean. A result on singularly perturbed elliptic problems. Communications on Pure & Applied Analysis, 2005, 4 (2) : 341-356. doi: 10.3934/cpaa.2005.4.341
[6]

Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565
[7]

Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098
[8]

Huxiao Luo, Xianhua Tang, Zu Gao. Sign-changing solutions for non-local elliptic equations with asymptotically linear term. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1147-1159. doi: 10.3934/cpaa.2018055
[9]

Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237
[10]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439
[11]

Gabriele Cora, Alessandro Iacopetti. Sign-changing bubble-tower solutions to fractional semilinear elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6149-6173. doi: 10.3934/dcds.2019268
[12]

Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737
[13]

Yuxin Ge, Monica Musso, A. Pistoia, Daniel Pollack. A refined result on sign changing solutions for a critical elliptic problem. Communications on Pure & Applied Analysis, 2013, 12 (1) : 125-155. doi: 10.3934/cpaa.2013.12.125
[14]

Norimichi Hirano, A. M. Micheletti, A. Pistoia. Existence of sign changing solutions for some critical problems on $\mathbb R^N$. Communications on Pure & Applied Analysis, 2005, 4 (1) : 143-164. doi: 10.3934/cpaa.2005.4.143
[15]

Filomena Pacella, Dora Salazar. Asymptotic behaviour of sign changing radial solutions of Lane Emden Problems in the annulus. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 793-805. doi: 10.3934/dcdss.2014.7.793
[16]

M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057
[17]

Luigi Montoro. On the shape of the least-energy solutions to some singularly perturbed mixed problems. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1731-1752. doi: 10.3934/cpaa.2010.9.1731
[18]

Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883
[19]

Wen Zhang, Xianhua Tang, Bitao Cheng, Jian Zhang. Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2161-2177. doi: 10.3934/cpaa.2016032
[20]

Ida De Bonis, Daniela Giachetti. Singular parabolic problems with possibly changing sign data. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2047-2064. doi: 10.3934/dcdsb.2014.19.2047

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences