-
Previous Article
Non-topological solutions in a generalized Chern-Simons model on torus
- CPAA Home
- This Issue
-
Next Article
Stability of the composite wave for the inflow problem on the micropolar fluid model
Robin problems with indefinite linear part and competition phenomena
1. | Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece |
2. | Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia |
3. | Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13,200585 Craiova, Romania |
4. | Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, Kardeljeva ploščad 16, SI-1000 Ljubljana, Slovenia |
We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $\lambda > 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_\lambda$ and determine the monotonicity and continuity properties of the map $\lambda \mapsto \tilde{u}_\lambda$.
References:
[1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu,
Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Memoirs Amer. Math. Soc., 196 (2008), pp. 70.
doi: 10.1090/memo/0915. |
[2] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[3] |
A. Ambrosetti and P. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
|
[4] |
T. Bartsch and M. Willem,
On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), 3555-3561.
doi: 10.2307/2161107. |
[5] |
M. Filippakis and N. S. Papageorgiou,
Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations, 245 (2008), 1883-1992.
doi: 10.1016/j.jde.2008.07.004. |
[6] |
L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. |
[7] |
S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume Ⅰ: Theory, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
doi: 10.1007/978-1-4615-4665-8\_17. |
[8] |
S. Hu and N. S. Papageorgiou,
Positive solutions for Robin problems with general potential and logistic reaction, Comm. Pure Appl. Anal., 15 (2016), 2489-2507.
doi: 10.3934/cpaa.2016046. |
[9] |
S. Li, S. Wu and H. S. Zhou,
Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations, 185 (2002), 200-224.
doi: 10.1006/jdeq.2001.4167. |
[10] |
C. Li and C. Yang,
The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of a p-Laplacian type without the Ambrosetti-Rabinowitz condition, Nonlinear Anal., 72 (2010), 4602-4613.
doi: 10.1016/j.na.2010.02.037. |
[11] |
D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9323-5. |
[12] |
N. S. Papageorgiou and V. D. Rădulescu,
Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.
doi: 10.1016/j.jde.2014.01.010. |
[13] |
N. S. Papageorgiou and V. D. Rădulescu,
Neumann problems with indefinite and unbounded potential and concave terms, Proc. Amer. Math. Soc., 143 (2015), 4803-4816.
doi: 10.1090/proc/12600. |
[14] |
N. S. Papageorgiou and V. D. Rădulescu,
Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential, Trans. Amer. Math. Soc., 366 (2014), 8723-8756.
doi: 10.1090/S0002-9947-2014-06518-5. |
[15] |
N. S. Papageorgiou and V. D. Rădulescu,
Bifurcation of positive solutions for nonlinear nonhomogeneous Neumann and Robin problems with competing nonlinearities, Discrete Contin. Dyn. Syst., 35 (2015), 5003-5036.
doi: 10.3934/dcds.2015.35.5003. |
[16] |
N. S. Papageorgiou and V. D. Rădulescu,
Robin problems with indefinite, unbounded potential and reaction of arbitrary growth, Rev. Mat. Complut., 29 (2016), 91-126.
doi: 10.1007/s13163-015-0181-y. |
[17] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš,
Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dyn. Syst., 37 (2017), 2589-2618.
doi: 10.3934/dcds.2017111. |
[18] |
N. S. Papageorgiou and G. Smyrlis,
Positive solutions for parametric p-Laplacian equations, Comm. Pure Appl. Math., 15 (2016), 1545-1570.
doi: 10.3934/cpaa.2016002. |
[19] |
V. D. Rădulescu and D. Repovš,
Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530.
doi: 10.1016/j.na.2011.01.037. |
[20] |
X. Wang,
Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310.
doi: 10.1016/0022-0396(91)90014-Z. |
show all references
References:
[1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu,
Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Memoirs Amer. Math. Soc., 196 (2008), pp. 70.
doi: 10.1090/memo/0915. |
[2] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[3] |
A. Ambrosetti and P. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
|
[4] |
T. Bartsch and M. Willem,
On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), 3555-3561.
doi: 10.2307/2161107. |
[5] |
M. Filippakis and N. S. Papageorgiou,
Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations, 245 (2008), 1883-1992.
doi: 10.1016/j.jde.2008.07.004. |
[6] |
L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. |
[7] |
S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume Ⅰ: Theory, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
doi: 10.1007/978-1-4615-4665-8\_17. |
[8] |
S. Hu and N. S. Papageorgiou,
Positive solutions for Robin problems with general potential and logistic reaction, Comm. Pure Appl. Anal., 15 (2016), 2489-2507.
doi: 10.3934/cpaa.2016046. |
[9] |
S. Li, S. Wu and H. S. Zhou,
Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations, 185 (2002), 200-224.
doi: 10.1006/jdeq.2001.4167. |
[10] |
C. Li and C. Yang,
The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of a p-Laplacian type without the Ambrosetti-Rabinowitz condition, Nonlinear Anal., 72 (2010), 4602-4613.
doi: 10.1016/j.na.2010.02.037. |
[11] |
D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9323-5. |
[12] |
N. S. Papageorgiou and V. D. Rădulescu,
Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.
doi: 10.1016/j.jde.2014.01.010. |
[13] |
N. S. Papageorgiou and V. D. Rădulescu,
Neumann problems with indefinite and unbounded potential and concave terms, Proc. Amer. Math. Soc., 143 (2015), 4803-4816.
doi: 10.1090/proc/12600. |
[14] |
N. S. Papageorgiou and V. D. Rădulescu,
Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential, Trans. Amer. Math. Soc., 366 (2014), 8723-8756.
doi: 10.1090/S0002-9947-2014-06518-5. |
[15] |
N. S. Papageorgiou and V. D. Rădulescu,
Bifurcation of positive solutions for nonlinear nonhomogeneous Neumann and Robin problems with competing nonlinearities, Discrete Contin. Dyn. Syst., 35 (2015), 5003-5036.
doi: 10.3934/dcds.2015.35.5003. |
[16] |
N. S. Papageorgiou and V. D. Rădulescu,
Robin problems with indefinite, unbounded potential and reaction of arbitrary growth, Rev. Mat. Complut., 29 (2016), 91-126.
doi: 10.1007/s13163-015-0181-y. |
[17] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš,
Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dyn. Syst., 37 (2017), 2589-2618.
doi: 10.3934/dcds.2017111. |
[18] |
N. S. Papageorgiou and G. Smyrlis,
Positive solutions for parametric p-Laplacian equations, Comm. Pure Appl. Math., 15 (2016), 1545-1570.
doi: 10.3934/cpaa.2016002. |
[19] |
V. D. Rădulescu and D. Repovš,
Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530.
doi: 10.1016/j.na.2011.01.037. |
[20] |
X. Wang,
Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310.
doi: 10.1016/0022-0396(91)90014-Z. |
[1] |
Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6133-6166. doi: 10.3934/dcds.2016068 |
[2] |
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2589-2618. doi: 10.3934/dcds.2017111 |
[3] |
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5003-5036. doi: 10.3934/dcds.2015.35.5003 |
[4] |
Shouchuan Hu, Nikolaos S. Papageorgiou. Positive solutions for Robin problems with general potential and logistic reaction. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2489-2507. doi: 10.3934/cpaa.2016046 |
[5] |
Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623 |
[6] |
Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 |
[7] |
Santiago Cano-Casanova. Bifurcation to positive solutions in BVPs of logistic type with nonlinear indefinite mixed boundary conditions. Conference Publications, 2013, 2013 (special) : 95-104. doi: 10.3934/proc.2013.2013.95 |
[8] |
M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure and Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411 |
[9] |
Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436 |
[10] |
Rushun Tian, Zhi-Qiang Wang. Bifurcation results on positive solutions of an indefinite nonlinear elliptic system. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 335-344. doi: 10.3934/dcds.2013.33.335 |
[11] |
Vladimir Lubyshev. Precise range of the existence of positive solutions of a nonlinear, indefinite in sign Neumann problem. Communications on Pure and Applied Analysis, 2009, 8 (3) : 999-1018. doi: 10.3934/cpaa.2009.8.999 |
[12] |
Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675 |
[13] |
Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615 |
[14] |
John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83 |
[15] |
Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084 |
[16] |
Gennaro Infante. Positive solutions of differential equations with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 432-438. doi: 10.3934/proc.2003.2003.432 |
[17] |
Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. A curve of positive solutions for an indefinite sublinear Dirichlet problem. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 817-845. doi: 10.3934/dcds.2020063 |
[18] |
Guglielmo Feltrin. Positive subharmonic solutions to superlinear ODEs with indefinite weight. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 257-277. doi: 10.3934/dcdss.2018014 |
[19] |
Alberto Boscaggin, Maurizio Garrione. Positive solutions to indefinite Neumann problems when the weight has positive average. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5231-5244. doi: 10.3934/dcds.2016028 |
[20] |
Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]