# American Institute of Mathematical Sciences

January  2014, 13(1): 1-73. doi: 10.3934/cpaa.2014.13.1

## High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems

 1 Department of Applied Mathematics, Complutense University of Madrid, Madrid, 28040, Spain 2 Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain 3 Dipartimento di Matematica e Informatica, Università, Via Delle Scienze 206, I-33100 Udine

Received  October 2012 Revised  May 2013 Published  July 2013

This paper analyzes the existence and structure of the positive solutions of a very simple superlinear indefinite semilinear elliptic prototype model under non-homogeneous boundary conditions, measured by $M\leq \infty$. Rather strikingly, there are ranges of values of the parameters involved in its setting for which the model admits an arbitrarily large number of positive solutions, as a result of their fast oscillatory behavior, for sufficiently large $M$. Further, using the amplitude of the superlinear term as the main bifurcation parameter, we can ascertain the global bifurcation diagram of the positive solutions. This seems to be the first work where these multiplicity results have been documented.
Citation: Julián López-Góme, Andrea Tellini, F. Zanolin. High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 1-73. doi: 10.3934/cpaa.2014.13.1
##### References:
 [1] S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215. doi: 10.1006/jfan.1996.0125.  Google Scholar [2] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Eqns., 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440.  Google Scholar [3] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Top. meth. Nonl. Anal., 4 (1994), 59-78.  Google Scholar [4] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, Nonl. Diff. Eqns. Appns., 2 (1995), 553-572. doi: 10.1007/BF01210623.  Google Scholar [5] M. Bertsch and R. Rostamian, The principle of linearized stability for a class of degenerate diffusion equations, J. Diff. Eqns., 57 (1985), 373-405. doi: 10.1016/0022-0396(85)90062-2.  Google Scholar [6] S. Cano-Casanova, Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Anal. TMA, 49 (2002), 361-430. doi: 10.1016/S0362-546X(01)00116-X.  Google Scholar [7] W. Dambrosio, Time-map techniques for some boundary value problems, Rocky Mountain J. Math., 28 (1998), 885-926. doi: 10.1216/rmjm/1181071745.  Google Scholar [8] J. Fraile, P. Koch-Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Diff. Eqns., 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071.  Google Scholar [9] J. García-Melián, Multiplicity of positive solutions to boundary blow up elliptic problems with sign-changing weights, J. Funct. Anal., 261 (2011), 1775-1798. doi: 10.1016/j.jfa.2011.05.018.  Google Scholar [10] J. García-Melián, R. Gómez-Reñasco, J. López-Gómez and J. C. Sabina de Lis, Point-wise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Rat. Mech. Anal., 145 (1998), 261-289. doi: 10.1007/s002050050130.  Google Scholar [11] R. García-Melián and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction diffusion equations, J. Diff. Eqns., 167 (2000), 36-72. doi: 10.1006/jdeq.2000.3772.  Google Scholar [12] R. García-Melián and J. López-Gómez, The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations, Diff. Int. Eqns., 14 (2001), 751-768.  Google Scholar [13] G. A. Harris, The influence of boundary data on the number of solutions of boundary value problems with jumping nonlinearities, Trans. Amer. Math. Soc., 321 (1990), 417-464. doi: 10.2307/2001568.  Google Scholar [14] J. López-Gómez, On the existence of positive solutions for some indefinite superlinear elliptic problems, Comm. Part. Diff. Eqns., 22 (1997), 1787-1804. doi: 10.1080/03605309708821320.  Google Scholar [15] J. López-Gómez, Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems, Trans. Amer. Math. Soc., 352 (1999), 1825-1858. doi: 10.1090/S0002-9947-99-02352-1.  Google Scholar [16] J. López-Gómez, Large solutions, metasolutions, and asymptotic behavior of the regular positive solutions of a class of sublinear parabolic problems, El. J. Diff. Eqns. Conf., 5 (2000), 135-171.  Google Scholar [17] J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems, Sci. Math. Jpn., 61 (2005), 493-516.  Google Scholar [18] J. López-Gómez, Metasolutions: Malthus versus Verhulst in Population Dynamics. A dream of Volterra, in Handbook of Differential Equations Stationary Partial Differential Equations", edited by M. Chipot and P. Quittner, Elsevier Science B. V., North Holland, Chapter 4, pp. 211-309, Amsterdam 2005. doi: 10.1016/S1874-5733(05)80012-9.  Google Scholar [19] J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications, Diff. Int. Eqns., 7 (1994), 383-398.  Google Scholar [20] J. López-Gómez and J. C. Sabina de Lis, First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs, J. Diff. Eqns., 148 (1998), 47-64. doi: 10.1006/jdeq.1998.3456.  Google Scholar [21] J. Mawhin, D. Papini and F. Zanolin, Boundary blow-up for differential equations with indefinite weight, J. Diff. Eqns., 188 (2003), 33-51. doi: 10.1016/S0022-0396(02)00073-6.  Google Scholar

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##### References:
 [1] S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215. doi: 10.1006/jfan.1996.0125.  Google Scholar [2] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Eqns., 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440.  Google Scholar [3] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Top. meth. Nonl. Anal., 4 (1994), 59-78.  Google Scholar [4] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, Nonl. Diff. Eqns. Appns., 2 (1995), 553-572. doi: 10.1007/BF01210623.  Google Scholar [5] M. Bertsch and R. Rostamian, The principle of linearized stability for a class of degenerate diffusion equations, J. Diff. Eqns., 57 (1985), 373-405. doi: 10.1016/0022-0396(85)90062-2.  Google Scholar [6] S. Cano-Casanova, Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Anal. TMA, 49 (2002), 361-430. doi: 10.1016/S0362-546X(01)00116-X.  Google Scholar [7] W. Dambrosio, Time-map techniques for some boundary value problems, Rocky Mountain J. Math., 28 (1998), 885-926. doi: 10.1216/rmjm/1181071745.  Google Scholar [8] J. Fraile, P. Koch-Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Diff. Eqns., 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071.  Google Scholar [9] J. García-Melián, Multiplicity of positive solutions to boundary blow up elliptic problems with sign-changing weights, J. Funct. Anal., 261 (2011), 1775-1798. doi: 10.1016/j.jfa.2011.05.018.  Google Scholar [10] J. García-Melián, R. Gómez-Reñasco, J. López-Gómez and J. C. Sabina de Lis, Point-wise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Rat. Mech. Anal., 145 (1998), 261-289. doi: 10.1007/s002050050130.  Google Scholar [11] R. García-Melián and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction diffusion equations, J. Diff. Eqns., 167 (2000), 36-72. doi: 10.1006/jdeq.2000.3772.  Google Scholar [12] R. García-Melián and J. López-Gómez, The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations, Diff. Int. Eqns., 14 (2001), 751-768.  Google Scholar [13] G. A. Harris, The influence of boundary data on the number of solutions of boundary value problems with jumping nonlinearities, Trans. Amer. Math. Soc., 321 (1990), 417-464. doi: 10.2307/2001568.  Google Scholar [14] J. López-Gómez, On the existence of positive solutions for some indefinite superlinear elliptic problems, Comm. Part. Diff. Eqns., 22 (1997), 1787-1804. doi: 10.1080/03605309708821320.  Google Scholar [15] J. López-Gómez, Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems, Trans. Amer. Math. Soc., 352 (1999), 1825-1858. doi: 10.1090/S0002-9947-99-02352-1.  Google Scholar [16] J. López-Gómez, Large solutions, metasolutions, and asymptotic behavior of the regular positive solutions of a class of sublinear parabolic problems, El. J. Diff. Eqns. Conf., 5 (2000), 135-171.  Google Scholar [17] J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems, Sci. Math. Jpn., 61 (2005), 493-516.  Google Scholar [18] J. López-Gómez, Metasolutions: Malthus versus Verhulst in Population Dynamics. A dream of Volterra, in Handbook of Differential Equations Stationary Partial Differential Equations", edited by M. Chipot and P. Quittner, Elsevier Science B. V., North Holland, Chapter 4, pp. 211-309, Amsterdam 2005. doi: 10.1016/S1874-5733(05)80012-9.  Google Scholar [19] J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications, Diff. Int. Eqns., 7 (1994), 383-398.  Google Scholar [20] J. López-Gómez and J. C. Sabina de Lis, First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs, J. Diff. Eqns., 148 (1998), 47-64. doi: 10.1006/jdeq.1998.3456.  Google Scholar [21] J. Mawhin, D. Papini and F. Zanolin, Boundary blow-up for differential equations with indefinite weight, J. Diff. Eqns., 188 (2003), 33-51. doi: 10.1016/S0022-0396(02)00073-6.  Google Scholar
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