An existence result for a nonlinear boundary value problem via topological arguments

Khadijah Sharaf

DOI: http://dx.doi.org/10.12775/TMNA.2016.034

Abstract


We investigate a nonlinear PDE related to the prescribing mean curvature problem on the boundary of the unit ball. We use variational and topological methods to prove the existence of at least one solution when the function to be prescribed satisfies at its critical points a non-degeneracy condition.

Keywords


Conformal metric; boundary mean curvature; lack of compactness; critical points at infinity; stable and unstable manifolds; retracts by deformation

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References


W. Abdelhedi and H. Chtioui, The Prescribed boundary mean curvature problem on standard n-dimensional ball, Nonlinear Anal. 67 (2007), 668–686.

W. Abdelhedi, H. Chtioui and M. Ould Ahmedou, A Morse theoretical approach for boundary mean Curvature Problem on B4 , J. Funct. Anal. 254 (2008), 1307–1341.

W. Abdelhedi, H. Chtioui and M. Ould Ahmedou, Conformal metrics with prescribed boundary mean curvature on balls, Ann. Global Anal. Geom. 36 (2009), Number 4, 327–362.

M.A. Al-Ghamdi, H. Chtioui and K. Sharaf, Topological methods for boundary mean curvature problem on B n , Adv. Nonlinear Stud. 14 (2) (2014), 445–461.

T. Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs Math., Springer Verlag, Berlin 1998.

A. Bahri, Critical point at infinity in some variational problems, Pitman Res. Notes Math. Ser. 182, Longman Sci. Tech. Harlow 1989.

A. Bahri, An invariant for yamabe-type flows with applications to scalar curvature problems in high dimensions, A celebration of J. F. Nash Jr., Duke Math. J. 81 (1996), 323–466. Notes Math. Ser. 182, Longman Sci. Tech. Harlow 1989.

A. Bahri and J. M. Coron, The scalar curvature problem on the standard three dimensional spheres, J. Funct. Anal. 95, (1991), 106–172.

A. Bahri and P. Rabinowitz, Periodic orbits of hamiltonian systems of three body type, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 561–649.

S.A. Chang, M.J. Gursky and P.C. Yang, The scalar curvature equation on 2 and 3 spheres, Calc. Var. 1 (1993), 205–229.

S.A. Chang, X. Xu and P. C. Yang, A perturbation result for prescribing mean curvature, Math. Ann 310 (3) (1998), 473–496.

P. Cherrier, Problèmes de Neumann non linéaires sur les variétés Riemanniennes, J. Funct. Anal. 57 (1984), 154–207.

Z. Djadli, A. Malchiodi and M. Ould Ahmedou, The prescribed boundary mean curvature problems on B 4 , J. Differential Equations 206 (2004), 373–398.

J.F. Escobar, Conformal deformation of Riemannian metric to scalar flat metric with constant mean curvature on the boundary, Ann. of Math. 136 (1992), 1–50.

J.F. Escobar, Conformal metric with prescribed mean curvature on the boundary, Calc. Var. Partial Differential Equations 4 (1996), 559–592.

J.F. Escobar and G. Garcia, Conformal metric on the ball with zero scalare and prescribed mean curvature on the boundary, J. Funct. Anal. 211:1 (2004), 71–152.

V. Felli and M. Ould Ahmedou, On a geometric equation with critical nonlinearity on the boundary, Pacific J. Math. 219 (2005), 1–25.

Z.C. Han and Y.Y. Li, The Yamabe problem on manifolds with boundary existence and compactness results, Duke Math. J. 99, (1999), 489–542.

Z.C. Han and Y.Y. Li, The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature, Comm. Anal. Geom. 8, (2000), 809–869.

J. Kazdan and F. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math (2) 101 (1975), 317–331.

Y.Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995), 383–417.

K. Sharaf, H. Alharthy and S. Altharwi, Conformal transformation of metrics on the n-ball, Nonlinear Anal. 95 (2014), 246–262.


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