# 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:

show all references

##### References:
 [1] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 [2] Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083 [3] Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 [4] Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810 [5] Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021009 [6] Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 [7] Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $\Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109 [8] Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475 [9] Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026 [10] Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029 [11] Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 [12] Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912 [13] Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005 [14] Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817 [15] Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 [16] Palash Sarkar, Subhadip Singha. Verifying solutions to LWE with implications for concrete security. Advances in Mathematics of Communications, 2021, 15 (2) : 257-266. doi: 10.3934/amc.2020057 [17] Yu-Hsien Liao. Solutions and characterizations under multicriteria management systems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021041 [18] Alexander Tolstonogov. BV solutions of a convex sweeping process with a composed perturbation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021012 [19] Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2899-2920. doi: 10.3934/dcdsb.2020210 [20] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

2019 Impact Factor: 1.105