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.  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.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  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.   Google Scholar

[17]

J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems,, Sci. Math. Jpn., 61 (2005), 493.   Google Scholar

[18]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in Population Dynamics. A dream of Volterra,, in Handbook of Differential Equations, (2005), 211.  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.   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.  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.  doi: 10.1016/S0022-0396(02)00073-6.  Google Scholar

show all references

References:
[1]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign,, J. Funct. Anal., 141 (1996), 159.  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.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  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.   Google Scholar

[17]

J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems,, Sci. Math. Jpn., 61 (2005), 493.   Google Scholar

[18]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in Population Dynamics. A dream of Volterra,, in Handbook of Differential Equations, (2005), 211.  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.   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.  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.  doi: 10.1016/S0022-0396(02)00073-6.  Google Scholar

[1]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[2]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[3]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[4]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[5]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[6]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[7]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[8]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

[9]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[10]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[11]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[12]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[13]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[14]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[15]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[16]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[17]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[18]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[19]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[20]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (26)

[Back to Top]