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\((p, 2)\)-equations resonant at any variational eigenvalue. (English) Zbl 1444.35042

The authors study a Dirichlet problem driven by the \((p,2)\)-Laplace operator, where \(2<p<+\infty\). A feature of this paper is that the reaction is resonant at \(\pm\infty\) with respect to any variational eigenvalue of the \(p\)-Laplacian. The main abstract results establish two multiplicity properties. The proofs combine variational and topological methods.

MSC:

35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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[1] Benci, V.; D’Avenia, P.; Fortunato, D., Solitons in several space dimensions: Derrick’s problem and infinitely many solutions, Arch Ration Mech Anal, 154, 297-324 (2000) · Zbl 0973.35161 · doi:10.1007/s002050000101
[2] Cherfils, L.; Il’yasov, Y., Commun Pure Appl Anal, 4, 9-22 (2005) · Zbl 1210.35090
[3] Cingolani, S.; Degiovanni, M., Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity, Comm Partial Differ Equ, 30, 1191-1203 (2005) · Zbl 1162.35367 · doi:10.1080/03605300500257594
[4] Gasiski, L.; Papageorgiou, NS., Calc Var Partial Differ Equ, 56, 88, 1-23 (2017) · Zbl 1380.35091
[5] Papageorgiou, NS; Rădulescu, VD., Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl Math Optim, 69, 393-430 (2014) · Zbl 1304.35077 · doi:10.1007/s00245-013-9227-z
[6] Papageorgiou, NS; Rădulescu, VD; Repovs̆, DD., Appl Math Optim, 75, 193-228 (2017) · Zbl 1376.35052 · doi:10.1007/s00245-016-9330-z
[7] Papageorgiou, NS; Rădulescu, VD; Repovs̆, DD., Adv Nonlin Anal · Zbl 1401.35085 · doi:10.1515/anona-2017-0195
[8] Papageorgiou, NS; Vetro, C., Compl Var Elliptic Equ · Zbl 1409.35069 · doi:10.1080/17476933.2017.1409743
[9] Papageorgiou, NS; Vetro, C.; Vetro, F., Discrete Contin Dyn Syst Ser S, 12, 347-374 (2019) · Zbl 1421.35177
[10] Papageorgiou, NS; Winkert, P., Appl Anal, 94, 341-359 (2015) · Zbl 1308.35075 · doi:10.1080/00036811.2014.895332
[11] Sun, M., Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance, J Math Anal Appl, 386, 661-668 (2012) · Zbl 1229.35089 · doi:10.1016/j.jmaa.2011.08.030
[12] Sun, M.; Zhang, M.; Su, J., Critical groups at zero and multiple solutions for a quasilinear elliptic equation, J Math Anal Appl, 428, 696-712 (2015) · Zbl 1317.35085 · doi:10.1016/j.jmaa.2015.03.033
[13] Gasiński, L.; Papageorgiou, NS., Exercises in analysis Part 2. Nonlinear analysis (2016), Cham: Springer, Cham · Zbl 1351.00006
[14] Lieberman, GM., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal, 12, 1203-1219 (1988) · Zbl 0675.35042 · doi:10.1016/0362-546X(88)90053-3
[15] Gasiński, L.; Papageorgiou, NS., Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set-Valued Var Anal, 20, 417-443 (2012) · Zbl 1258.35075 · doi:10.1007/s11228-011-0198-4
[16] Gasiński, L.; Papageorgiou, NS., Nonlinear analysis ser math anal appl 9 (2006), Boca Raton: Chapman and Hall/CRC Press, Boca Raton · Zbl 1086.47001
[17] Motreanu, D.; Motreanu, VV; Papageorgiou, NS., Topological and variational methods with applications to nonlinear boundary value problems (2014), New York: Springer, New York · Zbl 1292.47001
[18] Marano, SA; Papageorgiou, NS., J Math Anal Appl, 443, 123-145 (2016) · Zbl 1348.35082 · doi:10.1016/j.jmaa.2016.05.017
[19] Aizicovici, S.; Papageorgiou, NS; Staicu, V., Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem Amer Math Soc, 196, 915, 1-70 (2008) · Zbl 1165.47041
[20] Su, J., Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal, 48, 881-895 (2002) · Zbl 1018.35037 · doi:10.1016/S0362-546X(00)00221-2
[21] Palais, RS., Homotopy theory of infinite dimensional manifolds, Topology, 5, 1-16 (1966) · Zbl 0138.18302 · doi:10.1016/0040-9383(66)90002-4
[22] Chang, K-C., Infinite-dimensional Morse theory and multiple solution problems (1993), Boston: Birkháuser, Boston · Zbl 0779.58005
[23] Chang, K-C., Methods in nonlinear analysis (2005), Berlin: Springer-Verlag, Berlin · Zbl 1081.47001
[24] Liang, Z.; Su, J., Multiple solutions for semilinear elliptic boundary value problems with double resonance, J Math Anal Appl, 354, 147-158 (2009) · Zbl 1166.35023 · doi:10.1016/j.jmaa.2008.12.053
[25] Perera, K., Nontrivial critical groups in p-Laplacian problems via the Yang index, Topol Methods Nonlinear Anal, 21, 301-309 (2003) · Zbl 1039.47041 · doi:10.12775/TMNA.2003.018
[26] Ladyzhenskaya, OA; Ural’tseva, NN., Linear and quasilinear elliptic equations (1968), New York-London: Academic Press, New York-London · Zbl 0164.13002
[27] Pucci, P.; Serrin, J., The maximum principle (2007), Basel: Birkhäuser Verlag, Basel · Zbl 1134.35001
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