July  2017, 16(4): 1293-1314. doi: 10.3934/cpaa.2017063

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

* Corresponding author: Vicenţiu D. Rădulescu

Received  July 2016 Revised  February 2017 Published  April 2017

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 $\lambda \mapsto \tilde{u}_\lambda$.

Citation: Nikolaos S. Papageorgiou, Vicenšiu D. Rădulescu, Dušan D. Repovš. Robin problems with indefinite linear part and competition phenomena. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1293-1314. doi: 10.3934/cpaa.2017063
References:
[1]

S. AizicoviciN. 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.  Google Scholar

[2]

A. AmbrosettiH. 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.  Google Scholar

[3]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

N. S. PapageorgiouV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. AizicoviciN. 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.  Google Scholar

[2]

A. AmbrosettiH. 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.  Google Scholar

[3]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

N. S. PapageorgiouV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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