Opuscula Math. 39, no. 2 (2019), 227-245
https://doi.org/10.7494/OpMath.2019.39.2.227

Opuscula Mathematica

# On a Robin (p,q)-equation with a logistic reaction

Nikolaos S. Papageorgiou
Calogero Vetro
Francesca Vetro

Abstract. We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a $$p$$-Laplacian and of a $$q$$-Laplacian ($$(p,q)$$-equation) plus an indefinite potential term and a parametric reaction of logistic type (superdiffusive case). We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $$\lambda \gt 0$$ varies. Also, we show that for every admissible parameter $$\lambda \gt 0$$, the problem admits a smallest positive solution.

Keywords: positive solutions, superdiffusive reaction, local minimizers, maximum principle, minimal positive solutions, Robin boundary condition, indefinite potential.

Mathematics Subject Classification: 35J20, 35J60.

Full text (pdf)

1. S. Aizicovici, N.S. Papageorgiou, V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc. 196 (2008) 915, 70 pp.
2. A. Ambrosetti, D. Lupo, On a class of nonlinear Dirichlet problems with multiple solutions, Nonlinear Anal. 8 (1984), 1145-1150.
3. A. Ambrosetti, G. Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal. 3 (1979), 635-645.
4. T. Cardinali, N.S. Papageorgiou, P. Rubbioni, Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type, Ann. Mat. Pura Appl. 193 (2013), 1-21.
5. L. Cherfils, Y. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with $$p$$&$$q$$-Laplacian, Commun. Pure Appl. Anal. 4 (2005) 1, 9-22.
6. W. Dong, A priori estimates and existence of positive solutions for a quasilinear elliptic equation, J. Lond. Math. Soc. 72 (2005), 645-662.
7. W. Dong, J. Chen, Existence and multiplicity results for a degenerate elliptic equation, Acta Math. Sin. (Engl. Ser.) 22 (2008), 665-670.
8. M. Filippakis, D. O'Regan, N.S. Papageorgiou, Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case, Comm. Pure Appl. Anal. 9 (2010) 6, 1507-1527.
9. G. Fragnelli, D. Mugnai, N.S. Papageorgiou, The Brezis-Oswald result for quasilinear Robin problems, Adv. Nonlinear Stud. 16 (2016), 403-422.
10. L. Gasiński, N.S. Papageorgiou, Exercises in Analysis. Part 2, Springer, Cham, 2016.
11. L. Gasiński, N.S. Papageorgiou, Positive solutions for the Robin $$p$$-Laplacian problem with competing nonlinearities, Adv. Calc. Var (2017), doi:10.1515/acv-2016-0039. https://doi.org/10.1515/acv-2016-0039.
12. M.E. Gurtin, R.C. MacCamy, On the diffusion of biological population, Math. Biosci. 33 (1977), 35-49.
13. S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
14. S. Kamin, L. Veron, Flat core properties associated to the $$p$$-Laplace operator, Proc. Amer. Math. Soc. 118 (1993), 1079-1085.
15. G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Diff. Equations 16 (1991), 311-361.
16. S.A. Marano, N.S. Papageorgiou, Constant sign and nodal solutions for a Neumann problem with $$p$$-Laplacian and equidiffusive reaction term, Topol. Methods Nonlinear Anal. 38 (2011), 233-248.
17. S.A. Marano, N.S. Papageorgiou, Positive solutions to a Dirichlet problem with $$p$$-Laplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal. 12 (2013), 815-829.
18. N.S. Papageorgiou, F. Papalini, Constant sign and nodal solutions for logistic-type equations with equidiffusive reaction, Monatsh. Math. 165 (2014), 91-116.
19. N.S. Papageorgiou, F. Papalini, On $$p$$-logistic equations of equidiffusive type, Positivity 21 (2017), 9-21.
20. N.S. Papageorgiou, V.D. Rădulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations 256 (2014), 2449-2479.
21. N.S. Papageorgiou, V.D. Rădulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud. 16 (2016), 737-764.
22. N.S. Papageorgiou, P. Winkert, On a parametric nonlinear Dirichlet problem with subdiffusive and equidiffusive reaction, Adv. Nonlinear Stud. 14 (2014), 565-591.
23. N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dyn. Syst. Ser. A 37 (2017), 2589-2618.
24. N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Positive solutions for superdiffusive mixed problems, Appl. Math. Lett. 77 (2018), 87-93.
25. N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Positive solutions for nonlinear, nonhomogeneous parametric Robin problems, Forum Math. 30 (2018), 553-580.
26. P. Pucci, J. Serrin, The Maximum Principle, Birkhäuser Verlag, Basel, 2007.
27. V.D. Rădulescu, D.D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal. 75 (2012), 1524-1530.
28. M. Struwe, A note on a result of Ambrosetti and Mancini, Ann. Mat. Pura Appl. 131 (1982), 107-115.
29. M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990.
30. S. Takeuchi, Multiplicity result for a degenerate elliptic equation with a logistic reaction, J. Differential Equations 173 (2001), 138-144.
31. S. Takeuchi, Positive solutions of a degenerate elliptic equation with a logistic reaction, Proc. Amer. Math. Soc. 129 (2001), 433-441.
32. V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 29 (1987), 33-66.
• Nikolaos S. Papageorgiou
• National Technical University, Department of Mathematics, Zografou Campus, 15780, Athens, Greece
• Communicated by Giovanni Molica Bisci.
• Revised: 2018-11-07.
• Accepted: 2018-11-07.
• Published online: 2018-12-07.