×

Bifurcation and admissible solutions for the Hessian equation. (English) Zbl 1373.34039

J. Funct. Anal. 273, No. 10, 3200-3240 (2017); corrigendum ibid. 277, No. 6, 1999-2001 (2019).
Summary: We study the following eigenvalue problem of \(k\)-Hessian equation \[ \begin{cases} S_k(D^2 u) = \lambda^k f(- u) & \text{in}\, B, \\ u = 0 & \text{on}\, \partial B . \end{cases} \] Global bifurcation result is established for this problem. As applications of the bifurcation result, we determine the intervals of \(\lambda\) for the existence, nonexistence, uniqueness and multiplicity of radially symmetric \(k\)-admissible solutions for this problem.

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
35J60 Nonlinear elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Admas, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York
[2] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 8, 620-709 (1976) · Zbl 0345.47044
[3] Ambrosetti, A.; Calahorrano, R. M.; Dobarro, F. R., Global branching for discontinuous problems, Comment. Math. Univ. Carolin., 31, 213-222 (1990) · Zbl 0732.35101
[4] Ambrosetti, A.; Malchiodi, A., Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, vol. 104 (2007), Cambridge Univ. Press · Zbl 1125.47052
[5] Amick, C. J.; Turner, R. E.L., A global branch of steady vortex rings, J. Reine Angew. Math., 384, 1-23 (1988) · Zbl 0628.76032
[6] Arcoya, D.; Diaz, J. I.; Tello, L., \(S\)-shaped bifurcation branch in a quasilinear multivalued model arising in climatology, J. Differential Equations, 150, 215-225 (1998) · Zbl 0921.35198
[7] Berestycki, H.; Lions, P. L., Some applications of the method of super and subsolutions, (Bifurcation and Nonlinear Eigenvalue Problems. Bifurcation and Nonlinear Eigenvalue Problems, Lecture Notes in Mathematics, vol. 782 (1980), Springer-Verlag: Springer-Verlag Berlin), 16-41
[8] Brezis, H., Operateurs Maximaux Monotone et Semigroup de Contractions dans les Espase de Hilbert, Math. Stud., vol. 5 (1973), North-Holland: North-Holland Amsterdam
[9] Caffarelli, L.; Nirenberg, L.; Spruck, J., Dirichlet problem for nonlinear second order elliptic equations III, functions of the eigenvalues of the Hessian, Acta Math., 155, 261-301 (1985) · Zbl 0654.35031
[10] Chou, K. S.; Geng, D.; Yan, S. S., Critical dimension of a Hessian equation involving critical exponent and a related asymptotic result, J. Differential Equations, 129, 79-110 (1996) · Zbl 0864.35037
[11] Chou, K. S.; Wang, X. J., Variational theory for Hessian equations, Comm. Pure Appl. Math., 54, 1029-1064 (2001) · Zbl 1035.35037
[12] Clarke, F. H., Optimization and Nonsmooth Analysis (1983), Wiley: Wiley New York · Zbl 0582.49001
[13] Dai, G., Bifurcation and one-sign solutions of the \(p\)-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst., 36, 10, 5323-5345 (2016) · Zbl 06638709
[14] Dai, G., Two Whyburn type topological theorems and its applications to Monge-Ampère equations, Calc. Var. Partial Differential Equations, 55, 97 (2016) · Zbl 1403.34034
[15] Dai, G., Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var. Partial Differential Equations, 55, 72 (2016) · Zbl 1356.35117
[16] Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag New York · Zbl 0559.47040
[17] Del Pino, M.; Manásevich, R., Global bifurcation from the eigenvalues of the \(p\)-Laplacian, J. Differential Equations, 92, 226-251 (1991) · Zbl 0781.35017
[18] Dong, H., Hessian equations with elementary symmetric functions, Comm. Partial Differential Equations, 31, 1005-1025 (2006) · Zbl 1237.35051
[19] Gidas, B.; Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 8, 883-901 (1981) · Zbl 0462.35041
[20] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (2001), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1042.35002
[21] Guan, B., The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. Partial Differential Equations, 8, 45-69 (1999) · Zbl 0981.58020
[22] Idogawa, T.; Ôtani, M., The first eigenvalues of some abstact elliptic operator, Funkcial. Ekvac., 38, 1-9 (1995) · Zbl 0837.35108
[23] Ivochkina, N. M.; Trudinger, N. S.; Wang, X. J., The Dirichlet problem for degenerate Hessian equations, Comm. Partial Differential Equations, 29, 219-235 (2005)
[24] Jian, H. Y., Hessian equations with infinite Dirichlet boundary value, Indiana Univ. Math. J., 55, 1045-1062 (2006) · Zbl 1126.35026
[25] Labutin, D., Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111, 1-49 (2002) · Zbl 1100.35036
[26] Lions, P. L., On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24, 441-467 (1982) · Zbl 0511.35033
[27] López-Gómez, J., Spectral Theory and Nonlinear Functional Analysis (2001), Chapman and Hall/CRC: Chapman and Hall/CRC Boca Raton · Zbl 0978.47048
[28] Ouyang, T.; Shi, J., Exact multiplicity of positive solutions for a class of semilinear problem, J. Differential Equations, 146, 1, 121-156 (1998) · Zbl 0918.35049
[29] Ouyang, T.; Shi, J., Exact multiplicity of positive solutions for a class of semilinear problem II, J. Differential Equations, 158, 94-151 (1999) · Zbl 0947.35067
[30] Phuc, N. C.; Verbisky, I. E., Local integral estimates and removable singularities for quasilinear and Hessian equations with nonlinear source terms, Comm. Partial Differential Equations, 31, 1779-1791 (2006) · Zbl 1215.35071
[31] Phuc, N. C.; Verbisky, I. E., Quasilinear and Hessian equations of Lane-Emden type, Ann. of Math., 168, 859-914 (2008) · Zbl 1175.31010
[32] Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, 487-513 (1971) · Zbl 0212.16504
[33] Reilly, R. C., On the Hessian of a function and the curvatures of its graph, Michigan Math. J., 20, 373-383 (1973) · Zbl 0267.53003
[34] Swanson, C. A., Picone’s identity, Trends Math., 8, 373-397 (1975) · Zbl 0327.34028
[35] Sheng, W. M.; Trudinger, N. S.; Wang, X. J., The Yamabe problem for higher order curvatures, J. Differential Geom., 77, 3, 515-553 (2007) · Zbl 1133.53035
[36] Trudinger, N. S., Isoperimetric inequalities for quermassintegrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11, 411-425 (1994) · Zbl 0859.52001
[37] Trudinger, N. S., On the Dirichlet problem for Hessian equations, Acta Math., 175, 151-164 (1995) · Zbl 0887.35061
[38] Trudinger, N. S., On new isoperimetric inequalities and symmetrizations, J. Reine Angew. Math., 488, 203-220 (1997) · Zbl 0883.52006
[39] Trudinger, N. S., Weak solutions of Hessian equations, Comm. Partial Differential Equations, 22, 1251-1261 (1997) · Zbl 0883.35035
[40] Trudinger, N. S.; Wang, X. J., Hessian measures I, Topol. Methods Nonlinear Anal., 10, 225-239 (1997) · Zbl 0915.35039
[41] Trudinger, N. S.; Wang, X. J., A Poincaré type inequality for Hessian integrals, Calc. Var. Partial Differential Equations, 6, 315-328 (1998) · Zbl 0927.58013
[42] Trudinger, N. S.; Wang, X. J., Hessian measures II, Ann. of Math., 150, 579-604 (1999) · Zbl 0947.35055
[43] Trudinger, N. S.; Wang, X. J., Hessian measures III, J. Funct. Anal., 193, 1-23 (2002) · Zbl 1119.35325
[44] Trudinger, N. S.; Wang, X. J., The weak continuity of elliptic operators and applications in potential theory, Amer. J. Math., 551, 11-32 (2002)
[45] Tso, K., On symmetrization and Hessian equations, J. Anal. Math., 52, 94-106 (1989) · Zbl 0675.35040
[46] Tso, K., Remarks on critical exponents for Hessian operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7, 113-122 (1990) · Zbl 0715.35031
[47] Wang, X. J., A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J., 43, 25-54 (1994) · Zbl 0805.35036
[48] Warren, M.; Yuan, Y., Hessian estimates for the \(\sigma_2\)-equation in dimension three, Comm. Pure Appl. Math., 62, 3, 305-321 (2009) · Zbl 1173.35388
[49] Whyburn, G. T., Topological Analysis (1958), Princeton University Press: Princeton University Press Princeton
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.