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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
W. Dambrosio, Time-map techniques for some boundary value problems, Rocky Mountain J. Math., 28 (1998), 885-926.
doi: 10.1216/rmjm/1181071745. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems, Sci. Math. Jpn., 61 (2005), 493-516. |
[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. |
[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. |
[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. |
[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. |
show all references
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
W. Dambrosio, Time-map techniques for some boundary value problems, Rocky Mountain J. Math., 28 (1998), 885-926.
doi: 10.1216/rmjm/1181071745. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems, Sci. Math. Jpn., 61 (2005), 493-516. |
[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. |
[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. |
[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. |
[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. |
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