# American Institute of Mathematical Sciences

January  2020, 19(1): 31-46. doi: 10.3934/cpaa.2020003

## Infinitely many solutions and Morse index for non-autonomous elliptic problems

 Department of Mathematical Sciences, University of Cincinnati, Cincinnati Ohio 45221-0025 USA

Received  February 2018 Revised  April 2019 Published  July 2019

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 [10] by using the method we developed in [13] and [14]. We provide a considerably simpler approach, which yields additional information on the Morse index of solutions.

Citation: Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure & Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications Inc., New York, 1957.  Google Scholar [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.  Google Scholar [6] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.   Google Scholar [7] E. Hopf, On Emden's differential equation, Monthly Notices of the Royal Astronomical Society, 91 (1931), 653-663.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific, Hackensack, NJ, 2012. doi: 10.1142/8308.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications Inc., New York, 1957.  Google Scholar [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.  Google Scholar [6] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.   Google Scholar [7] E. Hopf, On Emden's differential equation, Monthly Notices of the Royal Astronomical Society, 91 (1931), 653-663.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific, Hackensack, NJ, 2012. doi: 10.1142/8308.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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