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

• Autorzy: Onete Florin-Iulian , Papageorgiou Nikolaos S. , Vetro Calogero

Identyfikatory

DOI

10.7494/OpMath.2020.40.1.131

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

superlinear reaction; constant sign and nodal solutions extremal solutions; nonlinear regularity; nonlinear maximum principle; critical groups

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2020
• Tom: Vol. 40, no. 1
• Strony: 131--149
• Opis fizyczny: Bibliogr. 31 poz.

Twórcy

autor

Onete Florin-Iulian

• Liceul Tehnologic Petre Banita 207170 Calarasi, Dolj, Romania

autor

Papageorgiou Nikolaos S.

• National Technical University Department of Mathematics Zografou Campus, 15780, Athens, Greece

autor

Vetro Calogero

• University of Palermo Department of Mathematics and Computer Science Via Archirafi 34, 90123, Palermo, Italy

Bibliografia

• [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. Radulescu, D.D. Repovs, 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 (jp,2)-Laplacian operator, Acta Appl. Math. 165 (2020) 1, 65-79.
• [6] V. Bobkov, M. Tanaka, Remarks on minimizers for (jp,q)-Laplace equations with two parameters, Commun. Pure Appl. Anal. 17 (2018) 3, 1219-1253.
• [7] L. Cherfils, Y. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with p&zq-Lapladan, 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. Gasihski, 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. Radulescu, Qualitative phenomena for some classes of quasilin-ear elliptic equations with multiple resonance, Appl. Math. Optim. 69 (2014), 393-430.
• [16] N.S. Papageorgiou, V.D. Radulescu, Coercive and noncoercive nonlinear Neumann problems with indefinite potential, Forum Math. 28 (2016) 3, 545-571.
• [17] N.S. Papageorgiou, V.D. Radulescu, Nonlinear nonhorn.oge.ne.ous Robin problems with superlinear reaction term, Adv. Nonlinear Stud. 16 (2016), 737-764.
• [18] N.S. Papageorgiou, V.D. Radulescu, D.D. Repovs, 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. Radulescu, D.D. Repovs, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys. 69 (2018), Art. 108, 21 pp.
• [20] N.S. Papageorgiou, V.D. Radulescu, D.D. Repovs, Double phase problems and a discontinuity property of the spectrum, Proc. Amer. Math. Soc. 147 (2019), 2899-2910.
• [21] N.S. Papageorgiou, V.D. Radulescu, D.D. Repovs, 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
• [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, Birkhauser Verlag, Basel, 2007.
• [28] V.D. Radulescu, 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. USSRIzv. 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.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).