Abstract
We consider parametric Dirichlet problems driven by the sum of a p-Laplacian (\(p>2\)) and a Laplacian ((p, 2)-equation) and with a reaction term which exhibits competing nonlinearities. We prove two multiplicity theorems. In the first the competing terms are not decoupled, the dependence on the parameter is not necessarily linear and the reaction term has a general polynomial growth, possibly supercritical. We produce three nontrivial solutions for small values of the parameter. We provide sign information for all solutions (two of constant sign and the third nodal). Then we decouple the competing nonlinearities and allow for resonance to occur at \(\pm \,\infty \). We produce six nontrivial smooth solutions for small values of the parameter. We provide sign information for five of these solutions.
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Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Am. Math. Soc. 196(915), 70 (2008)
Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Nodal solutions for \((p,2)\)-equations. Trans. Am. Math. Soc. 367, 7343–7372 (2015)
Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Benci, V., D’Avenia, P., Fortunato, D., Pisani, L.: Solitons in several space dimensions: Derrick’s problem and infinitely many solutions. Arch. Ration. Mech. Anal. 154, 297–324 (2000)
Chang, K.-C.: Methods in Nonlinear Analysis. Springer, Berlin (2005)
Cherfils, L., Il’yasov, Y.: On the stationary solutions of generalized reaction diffusion equations with \(p\)&\(q\)-Laplacian. Commun. Pure Appl. Anal. 4, 9–22 (2005)
Cingolani, S., Degiovanni, M.: Nontrivial solutions for \(p\)-Laplace equations with right hand side having \(p\)-linear growth at infinity. Commun. Partial Differ. Equ. 30, 1191–1203 (2005)
Diaz, J.I., Saa, J.E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C. R. Acad. Sci. Paris Sér. I Math. 305, 521–524 (1987)
Fadell, E.R., Rabinowitz, P.H.: Generalized cohomological index theories for Lie group actions with an applications to bifurcation questions for Hamiltonian systems. Invent. Math. 45, 139–174 (1978)
Filippakis, M.E., Papageorgiou, N.S.: Multiple constant sign and nodal solutions for nonlinear elliptic equations with the \(p\)-Laplacian. J. Differ. Equ. 245(7), 1883–1922 (2008)
Gasiński, L., Klimczak, L., Papageorgiou, N.S.: Nonlinear Dirichlet problems with no growth restriction on the reaction. Z. Anal. Anwend. 36(2), 209–238 (2017)
Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis. Mathematical Analysis and Applications, 9. Chapman & Hall, Boca Raton, FL (2006)
Gasiński, L., Papageorgiou, N.S.: Nodal and multiple constant sign solutions for resonant \(p\)-Laplacian equations with a nonsmooth potential. Nonlinear Anal. 71(11), 5747–5772 (2009)
Gasiński, L., Papageorgiou, N.S.: Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential. Set-Valued Var. Anal. 20, 417–443 (2012)
Gasiński, L., Papageorgiou, N.S.: Multiplicity of positive solutions for eigenvalue problems of \((p,2)\)-equations. Bound. Value Probl. 2012(152), 1–17 (2012)
Gasiński, L., Papageorgiou, N.S.: On generalized logistic equations with a non-homogeneous differential operator. Dyn. Syst. Int. J. 29, 190–207 (2014)
Gasiński, L., Papageorgiou, N.S.: A pair of positive solutions for \((p, q)\)-equations with combined nonlinearities. Commun. Pure Appl. Anal. 13(1), 203–215 (2014)
Gasiński, L., Papageorgiou, N.S.: Dirichlet \((p,q)\)-equations at resonance. Discrete Contin. Dyn. Syst. Ser. A 34(5), 2037–2060 (2014)
Gasiński, L., Papageorgiou, N.S.: Nonlinear elliptic equations with a jumping reaction. J. Math. Anal. Appl. 443(2), 1033–1070 (2016)
Gasiński, L., Papageorgiou, N.S.: Multiplicity theorems for \((p,2)\)-equations. J. Nonlinear Convex Anal. 18(7), 1297–1323 (2017)
Guedda, M., Véron, L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13, 879–902 (1989)
Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Vol. I. Theory, Mathematics and its Applications, Vol. 419. Kluwer Academic Publishers, Dordrecht (1997)
Ladyzhenskaya, O.A., Uraltseva, N.: Linear and Quasilinear Elliptic Equations. Mathematics in Science and Engineering, vol. 46. Academic Press, New York (1968)
Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)
Liang, Z., Su, J.: Multiple solutions for semilinear elliptic boundary value problems with double resonance. J. Math. Anal. Appl. 354(1), 147–158 (2009)
Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)
Mugnai, D., Papageorgiou, N.S.: Wang’s multiplicity result for superlinear \((p, q)\)-equations without the Ambrosetti–Rabinowitz condition. Trans. Am. Math. Soc. 366, 4919–4937 (2014)
Papageorgiou, N.S., Rădulescu, V.D.: Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance. Appl. Math. Optim. 69, 393–430 (2014)
Papageorgiou, N.S., Rădulescu, V.D.: Resonant \((p,2)\)-equations with asymmetric reaction. Anal. Appl. 13(5), 481–506 (2015)
Papageorgiou, N.S., Rădulescu, V.D.: Noncoercive resonant \((p,2)\)-equations. Appl. Math. Optim. 76(3), 621–639 (2017)
Papageorgiou, N.S., Smyrlis, G.: On nonlinear nonhomogeneous resonant Dirichlet equations. Pac. J. Math. 264, 421–453 (2013)
Papageorgiou, N.S., Winkert, P.: Resonant \((p,2)\)-equations with concave terms. Appl. Anal. 94(2), 342–360 (2015)
Pucci, P., Serrin, J.: The Maximum Principle. Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel (2007)
Sun, M.: Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance. J. Math. Anal. Appl. 386, 661–668 (2012)
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)
Troianiello, G.M.: Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York (1987)
Yang, D., Bai, C.: Nonlinear elliptic problem of \(2\)-\(q\)-Laplacian type with asymmetric nonlinearities. Electron. J. Differ. Equ. 170, 1–13 (2014)
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The research was supported by the National Science Center of Poland under Project No. 2015/19/B/ST1/01169.
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Gasiński, L., Papageorgiou, N.S. Multiple Solutions for \({\varvec{(p,2)}}\)-Equations with Resonance and Concave Terms. Results Math 74, 79 (2019). https://doi.org/10.1007/s00025-019-0996-9
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DOI: https://doi.org/10.1007/s00025-019-0996-9