Opuscula Math. 40, no. 1 (2020), 131-149
https://doi.org/10.7494/OpMath.2020.40.1.131

 
Opuscula Mathematica

A multiplicity theorem for parametric superlinear (p,q)-equations

Florin-Iulian Onete
Nikolaos S. Papageorgiou
Calogero Vetro

Abstract. We consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information.

Keywords: superlinear reaction, constant sign and nodal solutions, extremal solutions, nonlinear regularity, nonlinear maximum principle, critical groups.

Mathematics Subject Classification: 35J20, 35J60.

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  1. P. Baroni, M. Colombo, G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations 57 (2018), Art. 62, 48 pp.
  2. A. Bahrouni, V.D. Rădulescu, D.D. Repovš, Double phase transonic flow problem with variable growth: nonlinear patterns and stationary waves, Nonlinearity (2019), to appear.
  3. V. Benci, P. D'Avenia, D. Fortunato, L. Pisani, Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Arch. Ration. Mech. Anal. 154 (2000) 4, 297-324.
  4. N. Benouhiba, Z. Belyacine, On the solutions of the \((p,q)\)-Laplacian problem at resonance, Nonlinear Anal. 77 (2013), 74-81.
  5. T. Bhattacharya, B. Emamizadeh, A. Farjudian, Existence of continuous eigenvalues for a class of parametric problems involving the \((p,2)\)-Laplacian operator, Acta Appl. Math. 165 (2020) 1, 65-79.
  6. V. Bobkov, M. Tanaka, Remarks on minimizers for \((p,q)\)-Laplace equations with two parameters, Commun. Pure Appl. Anal. 17 (2018) 3, 1219-125.
  7. L. Cherfils, Y. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with \(p\& q\)-Laplacian, Commun. Pure Appl. Anal. 4 (2005), 9-22.
  8. M. Colombo, G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015) 2, 443-496.
  9. M. Colombo, G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015) 1, 219-273.
  10. L. Gasiński, N.S. Papageorgiou, Positive solutions for the Robin \(p\)-Laplacian problem with competing nonlinearities, Adv. Calc. Var. 12 (2019) 1, 31-56.
  11. S. Hu, N.S. Papageorgiou, Handbook of multivalued analysis, Vol. I: Theory, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
  12. G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), 311-361.
  13. P. Marcellini, Regularity and existence of solutions of elliptic equations with \(p,q\)-growth conditions, J. Differential Equations 90 (1991) 1, 1-30.
  14. D. Mugnai, N.S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), 729-788.
  15. N.S. Papageorgiou, V.D. Rădulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim. 69 (2014), 393-430.
  16. N.S. Papageorgiou, V.D. Rădulescu, Coercive and noncoercive nonlinear Neumann problems with indefinite potential, Forum Math. 28 (2016) 3, 545-571.
  17. N.S. Papageorgiou, V.D. Rădulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud. 16 (2016), 737-764.
  18. N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite and unbounded potential, Discrete Contin. Dyn. Syst. Ser. A 37 (2017), 2589-2618.
  19. N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys. 69 (2018), Art. 108, 21 pp.
  20. N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Double phase problems and a discontinuity property of the spectrum, Proc. Amer. Math. Soc. 147 (2019), 2899-2910.
  21. N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Nonlinear Analysis - Theory and Methods, Springer, Switzerland, 2019.
  22. N.S. Papageorgiou, C. Vetro, F. Vetro, On a Robin \((p,q)\)-equation with a logistic reaction, Opuscula Math. 39 (2019) 2, 227-245.
  23. N.S. Papageorgiou, C. Vetro, F. Vetro, Solutions with sign information for nonlinear Robin problems with no growth restriction on reaction, Appl. Anal., doi.org/10.1080/00036811.2019.1597059. https://doi.org/10.1080/00036811.2019.1597059
  24. N.S. Papageorgiou, P. Winkert, Applied Nonlinear Functional Analysis. An Introduction, De Gruyter, Berlin, 2018.
  25. N.S. Papageorgiou, C. Zhang, Noncoercive resonant \((p,2)\)-equations with concave terms, Adv. Nonlinear Anal. 9 (2020) 1, 228-249.
  26. N.S. Papageorgiou, C. Zhang, Double phase problem with critical and locally defined reaction terms, Asympt. Anal. 116 (2020) 2, 73-92.
  27. P. Pucci, J. Serrin, The Maximum Principle, Birkhäuser Verlag, Basel, 2007.
  28. V.D. Rădulescu, Isotropic and anisotropic double-phase problems: old and new, Opuscula Math. 39 (2019) 2, 259-279.
  29. M. Tanaka, Generalized eigenvalue problems for \((p,q)\)-Laplacian with indefinite weight, J. Math. Anal. Appl. 419 (2014) 2, 1181-1192.
  30. V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 29 (1987), 33-36.
  31. V.V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci. 173 (2011) 5, 463-570.
  • Florin-Iulian Onete
  • Liceul Tehnologic Petre Baniţǎ, 207170 Cǎlǎraşi, Dolj, Romania
  • Nikolaos S. Papageorgiou
  • National Technical University, Department of Mathematics, Zografou Campus, 15780, Athens, Greece
  • Calogero Vetro (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-5836-6847
  • University of Palermo, Department of Mathematics and Computer Science, Via Archirafi 34, 90123, Palermo, Italy
  • Communicated by Marius Ghergu.
  • Received: 2019-05-07.
  • Accepted: 2019-05-17.
  • Published online: 2020-02-17.
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Cite this article as:
Florin-Iulian Onete, Nikolaos S. Papageorgiou, Calogero Vetro, A multiplicity theorem for parametric superlinear (p,q)-equations, Opuscula Math. 40, no. 1 (2020), 131-149, https://doi.org/10.7494/OpMath.2020.40.1.131

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