## Topological Methods in Nonlinear Analysis

### Generic properties of critical points of the boundary mean curvature

#### Abstract

Given a bounded domain $\Omega\subset\mathbb{R}^N$ of class $C^k$ with $k\ge3$, we prove that for a generic deformation $I+\psi$, with $\psi$ small enough, all the critical points of the mean curvature of the boundary of the domain $(I+\psi)\Omega$ are non degenerate.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 41, Number 2 (2013), 323-334.

Dates
First available in Project Euclid: 21 April 2016

https://projecteuclid.org/euclid.tmna/1461245481

Mathematical Reviews number (MathSciNet)
MR3114311

Zentralblatt MATH identifier
1292.58006

#### Citation

Micheletti, Anna Maria; Pistoia, Angela. Generic properties of critical points of the boundary mean curvature. Topol. Methods Nonlinear Anal. 41 (2013), no. 2, 323--334. https://projecteuclid.org/euclid.tmna/1461245481

#### References

• P.W. Bates, E.N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn–Hilliard equation in higher-dimension and instability , Adv. Differential Equations, 4 (1999), 1–69 \ref\key 2
• E.N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem , Pacific J. Math., 189 (1999), 241–262 \ref\key 3
• T. D'Aprile and A. Pistoia, Nodal clustered solutions for some singularly perturbed Neumann problems , Comm. Partial Differential Equations, 35 , no. 8 (2010), 1355–1401 \ref\key 4
• 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 \ref\key 5
• M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory , Calc. Var. Partial Differential Equations, 11(2000), 143–175 \ref\key 6
• C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84(1996), 739–769 \ref\key 7
• C. Gui and J. Wei, Multiple interior spike solutions for some singular perturbed Neumann problems , J. Differential Equations, 158(1999), 1–27 \ref\key 8 ––––, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems , Canad. J. Math., 52(2000), 522–538 \ref\key 9
• 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), 249–289 \ref\key 10
• Y.Y. Li, On a singularly perturbed equation with Neumann boundary conditions , Commun. Partial Differential Equations, 23(1998), 487–545 \ref\key 11
• C. Lin, W.M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems , J. Differential Equations, 72(1988), 1–27 \ref\key 12
• A.M. Micheletti and A. Pistoia, On the multiplicity of nodal solutions to a singularly perturbed Neumann problem, Mediterrean J. Mathematics, 5(2008), 285–294 \ref\key 13
• W.M. Ni, I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem , Duke Math. J., 70(1993), 247–281 \ref\key 14 ––––, On the shape of least-energy solutions to a semi-linear Neumann problem , Comm. Pure Appl. Math., 44(1991), 819–851 \ref\key 15
• E. Noussair and J. Wei, On the existence and profile of nodal solutions of some singularly perturbed semilinear Neumann problem , Comm. Partial Differential Equations, 23(1998), 793–816 \ref\key 16
• F. Quinn, Transversal approximation on Banach manifolds (1970, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968)), 213–222, Amer. Math. Soc., Providence, R.I. \ref\key 17
• J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems , Comm. Partial Differential Equations, 4 (1979), 293–319 \ref\key 18
• K. Uhlenbeck, Generic properties of eigenfunctions , Amer. J. Math., 98 (1976), 1059–1078 \ref\key 19
• Z.Q. Wang, On the existence of multiple single-peak solutions for a semilinear Neumann problem , Arch. Rational Mech. Anal., 120(1992), 375–399 \ref\key 20
• J. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem , J. Differential Equations, 134 (1997), 104–133 \ref\key 21
• J. Wei and T. Weth, On the number of nodal solutions to a singularly perturbed Neumann problem , Manuscripta Math., 117 (2005), 333–344