Abstract
We study a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which has z-dependent zeros of constant sign. For all big values of the parameter \(\lambda >0\), we prove two multiplicity theorems producing two positive solutions, two negative solutions, and two nodal solutions. In the first we do not impose any asymptotic conditions on the reaction \(f(z,\cdot )\) at zero. In the second we do not impose any asymptotic conditions on the reaction \(f(z,\cdot )\) at \(\pm \infty \). Then we produce a total of twelve nontrivial smooth solutions all with sign information. Our proofs use variational methods together with flow invariance arguments and suitable truncation techniques.
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Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Constant sign and nodal solutions for nonlinear elliptic equations with combined nonlinearities. Methods Appl. Anal. 22, 221–248 (2015)
Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Multiplicity of solutions for a class of nonlinear nonhomogeneous elliptic equations. J. Nonlinear Convex Anal. 15, 7–34 (2014)
Barletta, G., Papageorgiou, N.S.: A multiplicity theorem for \(p\)-superlinear Neumann problems with a nonhomogeneous differential operator. Adv. Nonlinear Stud. 14, 895–913 (2014)
Bartsch, T., Liu, Z.L.: Location and Morse indices of critical points in Banach spaces with an application to nonlinear eigenvalue problems. Adv. Differ. Equ. 9, 645–676 (2004)
Bartsch, T., Liu, Z.L.: On a superlinear elliptic \(p\)-Laplacian equation. J. Differ. Equ. 198, 149–175 (2004)
Bartsch, T., Liu, Z.L.: Mmultiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the \(p\)-Laplacian. Commun. Contemp. Math. 6, 245–258 (2004)
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)
Brock, F., Iturriaga, L., Ubilla, P.: A multiplicity result for the \(p\)-Laplacian involving a parameter. Ann. Henri Poincaré 9, 1371–1386 (2008)
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)
Filippakis, M., O’Regan, D., Papageorgiou, N.S.: Multiple and nodal solutions for nonlinear equations with a nonhomogeneous differential operator and concave–convex terms. Tohoku Math. J. 66, 583–608 (2014)
Fragnelli, G., Mugnai, D., Papageorgiou, N.S.: Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential. Discrete Cont. Dyn. Syst. 36, 6133–6166 (2016)
Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman and Hall, CRC Press, Boca Raton (2006)
Gasiński, L., Papageorgiou, N.S.: Multiple solutions with sign information for a class of parametric superlinear (\(p\), 2)-equations. Appl. Math. Optim. (2019). https://doi.org/10.1007/s00245-019-09595-w
Gasinski, L., Papageorgiou, N.S.: Bifurcation-type results for nonlinear parametric elliptic equations. Proc. R. Soc. Edinb. 142A, 595–623 (2012)
Gasiński, L., Papageorgiou, N.S.: On a nonlinear parametric Robin problem with a locally defined reaction. Nonlinear Anal. 185, 374–387 (2019)
Gasiński, L., Papageorgiou, N.S.: Multiplicity theorems for (\(p\), 2)-equations. J. Nonlinear Convex Anal. 18, 1297–1323 (2017)
Gasiński, L., O’Regan, D., Papageorgiou, N.S.: A variational approach to nonlinear logistic equations. Commun. Contem. Math. 17, 1450021 (2015)
He, T., Guo, P., Huang, Y., Lei, Y.: Multiple nodal solutions for nonlinear nonhomogeneous elliptic problems with a superlinear reaction. Nonlinear Anal. 42, 207–219 (2018)
He, T., Huang, Y., Liang, K., Lei, Y.: Nodal solutions for noncoercive nonlinear Neumann problems with indefinite potential. Appl. Math. Lett. 71, 67–73 (2017)
He, T., Lei, Y., Zhang, M., Sun, H.: Nodal solutions for resonant and superlinear (\(p\), 2)-equations. Math. Nachr. 291, 2565–2577 (2018)
He, T., Yao, Z., Sun, Z.: Multiple and nodal solutions for parametric Neumann problems with nonhomogeneous differential operator and critical growth. J. Math. Anal. Appl. 449, 1133–1151 (2017)
Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Commun. Partial Differ. Equ. 16, 311–361 (1991)
Liu, Z.L., Sun, J.X.: Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. J. Differ. Equ. 172, 257–299 (2001)
Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10, 1–27 (2011)
Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)
Motreanu, D., Tanaka, M.: Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter. Ann. Mat. Pura Appl. 193, 1255–1282 (2014)
Papageorgiou, N.S.: On parametric evolution inclusions of the subdifferential type with applications to optimal control problems. Trans. Am. Math. Soc. 347, 203–231 (1995)
Papageorgiou, N.S., Papalini, F.: Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries. Isr. J. Math. 201, 761–796 (2014)
Papageorgiou, N.S., Rădulescu, V.D.: Multiplicity theorems for resonant and superlinear nonhomogeneous elliptic equations. Topol. Methods Nonlinear Anal. 48, 283–320 (2016)
Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: Double-phase problems with reaction of arbitrary growth. Z. Angew. Math. Phys. 69, 108 (2018)
Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: On a class of parametric (\(p\), 2)-equations. Appl. Math. Optim. 75, 193–228 (2017)
Papageorgiou, N.S., Scapellato, A.: Constant sign and nodal solutions for parametric (\(p\),2)-equations. Adv. Nonlinear Anal. 9, 449–478 (2019)
Papageorgiou, N.S., Vetro, C., Vetro, F.: Multiple solutions with sign information for a (\(p\), 2)-equation with combined nonlinearities. Nonlinear Anal. 192, 111716 (2020)
Papageorgiou, N.S., Winkert, P.: Positive solutions for nonlinear nonhomogeneous Dirichlet problems with concave–convex nonlinearities. Positivity 20, 945–979 (2016)
Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)
Sokołowski, J.: Optimal control in coefficients for weak variational problems in Hilbert space. Appl. Math. Optim. 7, 283–293 (1981)
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The authors wish to thank two very knowledgeable referees very much for their valuable comments and helpful suggestions which improved the paper considerably.
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Communicated by Y. Giga.
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Supported by Guangdong Basic and Applied Basic Research Foundation (Nos. 2020A1515010459, 2020A1515110958).
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He, T., Guo, P. & Liu, L. Multiple constant sign and nodal solutions for nonlinear nonhomogeneous elliptic equations depending on a parameter. Calc. Var. 60, 82 (2021). https://doi.org/10.1007/s00526-021-01977-9
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DOI: https://doi.org/10.1007/s00526-021-01977-9