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Entire subsolutions of Monge-Ampère type equations
Infinitely many solutions and Morse index for non-autonomous elliptic problems
Department of Mathematical Sciences, University of Cincinnati, Cincinnati Ohio 45221-0025 USA |
This paper deals with changes of variables, and the exact bifurcation diagrams for a class of self-similar equations. Our first result is a change of variables which transforms radial $ k $-Hessian equations into radial $ p $-Laplace equations. Then, in another direction, we generalize the classical results of D.D. Joseph and T.S. Lundgren [
References:
[1] |
I. Bihari,
A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar., 7 (1956), 81-94.
doi: 10.1007/BF02022967. |
[2] |
C. Budd and J. Norbury,
Semilinear elliptic equations and supercritical growth, J. Differential Equations, 68 (1987), 169-197.
doi: 10.1016/0022-0396(87)90190-2. |
[3] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second-order elliptic equations Ⅲ. Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.
doi: 10.1007/BF02392544. |
[4] |
S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications Inc., New York, 1957. |
[5] |
J. Dávila, M. del Pino, M. Musso and J. Wei,
Fast and slow decay solutions for supercritical elliptic problems in exterior domains, Calc. Var. Partial Differential Equations, 32 (2008), 453-480.
doi: 10.1007/s00526-007-0154-1. |
[6] |
B. Gidas, W.-M. Ni and L. Nirenberg,
Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.
|
[7] |
E. Hopf,
On Emden's differential equation, Monthly Notices of the Royal Astronomical Society, 91 (1931), 653-663.
|
[8] |
J. Jacobsen,
Global bifurcation problems associated with k-Hessian operators, Topol. Methods Nonlinear Anal., 14 (1999), 81-130.
doi: 10.12775/TMNA.1999.023. |
[9] |
J. Jacobsen and K. Schmitt,
The Liouville-Bratu-Gelfand problem for radial operators, J. Differential Equations, 184 (2002), 283-298.
doi: 10.1006/jdeq.2001.4151. |
[10] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.
doi: 10.1007/BF00250508. |
[11] |
P. Korman,
Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems, J. Differential Equations, 244 (2008), 2602-2613.
doi: 10.1016/j.jde.2008.02.014. |
[12] |
P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific, Hackensack, NJ, 2012.
doi: 10.1142/8308. |
[13] |
P. Korman,
Global solution curves for self-similar equations, J. Differential Equations, 257 (2014), 2543-2564.
doi: 10.1016/j.jde.2014.05.045. |
[14] |
P. Korman,
Infinitely many solutions for three classes of self-similar equations, with the $p$-Laplace operator, Proc. Roy. Soc. Edinburgh Sect. A, 147A (2017), 1-16.
doi: 10.1017/S0308210517000038. |
[15] |
P. Korman,
Explicit solutions and multiplicity results for some equations with the $p$-Laplacian, Quart. Appl. Math., 75 (2017), 635-647.
doi: 10.1090/qam/1471. |
[16] |
C. S. Lin and W.-M. Ni,
A counterexample to the nodal domain conjecture and related semilinear equation, Proc. Amer. Math. Soc., 102 (1988), 271-277.
doi: 10.2307/2045874. |
[17] |
F. Merle and L. A. Peletier,
Positive solutions of elliptic equations involving supercritical growth, Proc. Roy. Soc. Edinburgh Sect. A, 118 (1991), 49-62.
doi: 10.1017/S0308210500028882. |
[18] |
K. Nagasaki and T. Suzuki,
Spectral and related properties about the Emden-Fowler equation $-\Delta u=\lambda e^ u$ on circular domains, Math. Ann., 299 (1994), 1-15.
doi: 10.1007/BF01459770. |
[19] |
L. A. Peletier and J. Serrin,
Uniqueness of positive solutions of semilinear equations in $ R^{n}$, Arch. Rational Mech. Anal., 81 (1983), 181-197.
doi: 10.1007/BF00250651. |
[20] |
J. A. Pelesko,
Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908.
doi: 10.1137/S0036139900381079. |
[21] |
J. Sánchez and V. Vergara,
Bounded solutions of a $k$-Hessian equation in a ball, J. Differential Equations, 261 (2016), 797-820.
doi: 10.1016/j.jde.2016.03.021. |
[22] |
N. S. Trudinger and X.-J. Wang,
Hessian measures Ⅰ, Topol. Methods Nonlinear Anal., 10 (1997), 225-239.
doi: 10.12775/TMNA.1997.030. |
show all references
References:
[1] |
I. Bihari,
A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar., 7 (1956), 81-94.
doi: 10.1007/BF02022967. |
[2] |
C. Budd and J. Norbury,
Semilinear elliptic equations and supercritical growth, J. Differential Equations, 68 (1987), 169-197.
doi: 10.1016/0022-0396(87)90190-2. |
[3] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second-order elliptic equations Ⅲ. Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.
doi: 10.1007/BF02392544. |
[4] |
S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications Inc., New York, 1957. |
[5] |
J. Dávila, M. del Pino, M. Musso and J. Wei,
Fast and slow decay solutions for supercritical elliptic problems in exterior domains, Calc. Var. Partial Differential Equations, 32 (2008), 453-480.
doi: 10.1007/s00526-007-0154-1. |
[6] |
B. Gidas, W.-M. Ni and L. Nirenberg,
Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.
|
[7] |
E. Hopf,
On Emden's differential equation, Monthly Notices of the Royal Astronomical Society, 91 (1931), 653-663.
|
[8] |
J. Jacobsen,
Global bifurcation problems associated with k-Hessian operators, Topol. Methods Nonlinear Anal., 14 (1999), 81-130.
doi: 10.12775/TMNA.1999.023. |
[9] |
J. Jacobsen and K. Schmitt,
The Liouville-Bratu-Gelfand problem for radial operators, J. Differential Equations, 184 (2002), 283-298.
doi: 10.1006/jdeq.2001.4151. |
[10] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.
doi: 10.1007/BF00250508. |
[11] |
P. Korman,
Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems, J. Differential Equations, 244 (2008), 2602-2613.
doi: 10.1016/j.jde.2008.02.014. |
[12] |
P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific, Hackensack, NJ, 2012.
doi: 10.1142/8308. |
[13] |
P. Korman,
Global solution curves for self-similar equations, J. Differential Equations, 257 (2014), 2543-2564.
doi: 10.1016/j.jde.2014.05.045. |
[14] |
P. Korman,
Infinitely many solutions for three classes of self-similar equations, with the $p$-Laplace operator, Proc. Roy. Soc. Edinburgh Sect. A, 147A (2017), 1-16.
doi: 10.1017/S0308210517000038. |
[15] |
P. Korman,
Explicit solutions and multiplicity results for some equations with the $p$-Laplacian, Quart. Appl. Math., 75 (2017), 635-647.
doi: 10.1090/qam/1471. |
[16] |
C. S. Lin and W.-M. Ni,
A counterexample to the nodal domain conjecture and related semilinear equation, Proc. Amer. Math. Soc., 102 (1988), 271-277.
doi: 10.2307/2045874. |
[17] |
F. Merle and L. A. Peletier,
Positive solutions of elliptic equations involving supercritical growth, Proc. Roy. Soc. Edinburgh Sect. A, 118 (1991), 49-62.
doi: 10.1017/S0308210500028882. |
[18] |
K. Nagasaki and T. Suzuki,
Spectral and related properties about the Emden-Fowler equation $-\Delta u=\lambda e^ u$ on circular domains, Math. Ann., 299 (1994), 1-15.
doi: 10.1007/BF01459770. |
[19] |
L. A. Peletier and J. Serrin,
Uniqueness of positive solutions of semilinear equations in $ R^{n}$, Arch. Rational Mech. Anal., 81 (1983), 181-197.
doi: 10.1007/BF00250651. |
[20] |
J. A. Pelesko,
Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908.
doi: 10.1137/S0036139900381079. |
[21] |
J. Sánchez and V. Vergara,
Bounded solutions of a $k$-Hessian equation in a ball, J. Differential Equations, 261 (2016), 797-820.
doi: 10.1016/j.jde.2016.03.021. |
[22] |
N. S. Trudinger and X.-J. Wang,
Hessian measures Ⅰ, Topol. Methods Nonlinear Anal., 10 (1997), 225-239.
doi: 10.12775/TMNA.1997.030. |
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