Abstract
We consider semilinear parametric Robin problems driven by the Laplacian plus an indefinite and unbounded potential. In the reaction we have two competing nonlinearities. However, the competition is different from the usual one in “concave-convex” problems. Using a combination of different tools we prove a multiplicity theorem producing seven nontrivial smooth solutions all with sign information (four of constant sign and three nodal).
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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. In: Memoirs. AMS, vol. 196 (2008)
Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic equation. J. Funct. Anal. 122, 519–543 (1994)
Castro, A., Cossio, J., Vélez, C.: Existence of seven solutions for an asymptotically linear Dirichlet problem without symmetries. Ann. Mat. Pura Appl. 192, 607–619 (2013)
Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhauser, Boston (1993)
Cossio, J., Heron, S., Vélez, C.: Multiple solutions for nonlinear Dirichlet problems via bifurcation and additional results. J. Math. Anal. Appl. 399, 166–179 (2013)
D’Agui, G., Marano, S.A., Papageorgiou, N.S.: Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction. J. Math. Anal. Appl. 433, 1821–1845 (2016)
Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton (2006)
Gasinski, L., Papageorgiou, N.S.: Exercises in Analysis. Springer, Cham (2016)
Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, Volume I: Theory. Kluwer Academic Publishers, Dordrecht (1997)
Li, G., Yang, C.: The existence of a nontrivial solution to a nonlinear boundary value problem of \(p\)-Laplacian type without the Ambrosetti-Rabinowitz condition. Nonlinear Anal. 72, 4602–4613 (2010)
Marano, S.A., Papageorgiou, N.S.: Positive solutions to a Dirichlet problem with \(p\)-Laplacian and concave-convex nonlinearity depending on a parameter. Commun. Pure Appl. Anal. 12, 815–829 (2013)
Motreanu, D., Motreanu, 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)
Palais, R.: Homotopy theory of infinite dimensional manifolds. Topology 5, 115–132 (1966)
Papageorgiou, N.S., Papalini, F.: Seven solutions for superlinear Dirichlet problems with sign information for sublinear equations with indefinite and unbounded potential and no symmetries. Isr. J. Math. 201, 761–794 (2014)
Papageorgiou, N.S., Radulescu, V.D.: Neumann problems with indefinite and unbounded potential and concave terms. Proc. Am. Math. Soc. 143, 4803–4816 (2015)
Papageorgiou, N.S., Radulescu, V.D.: Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential. Trans. Am. Math. Soc. 367, 8723–8756 (2015)
Papageorgiou, N.S., Radulescu, V.D.: Bifurcation of positive solutions for nonlinear nonhomogeneous Neumann and Robin problems with competing nonlinearities. Discrete Contin. Dyn. Syst. 35, 5003–5036 (2015)
Papageorgiou, N.S., Radulescu, V.D.: Nonlinear nonhomogeneous Robin problems with superlinear reaction term. Adv. Nonlinear Stud. 16, 737–764 (2016)
Papageorgiou, N.S., Radulescu, V.D.: Robin problems with indefinite unbounded potential and reaction of arbitrary growth. Rev. Mat. Complut. 19, 91–126 (2016)
Papageorgiou, N.S., Radulescu, V.D., Repovs, D.: Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete Contin. Dyn. Syst. 37, 2589–2618 (2017)
Papageorgiou, N.S., Radulescu, V.D., Repovs, D.: Robin problems with an indefinite linear part and competition phenomena. Commun. Pure Appl. Anal. 16, 1293–1314 (2017)
Polác̆ik, P.: On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains. Discrete Contin. Dyn. Syst. 34, 2657–2667 (2014)
Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)
Sacks, P., Warma, M.: Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and \(L^{1}\)-data. Discrete Contin. Dyn. Syst. 34, 761–787 (2014)
Su, J.: Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues. Nonlinear Anal. 48, 881–895 (2002)
Wang, X.: Neumann problems of semilinear elliptic equations involving critical Sobolev exponents. J. Differ. Equ. 93, 283–310 (1991)
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Hu, S., Papageorgiou, N.S. Semilinear Robin Problems with Indefinite Potential and Competition Phenomena. Acta Appl Math 168, 187–216 (2020). https://doi.org/10.1007/s10440-019-00284-y
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DOI: https://doi.org/10.1007/s10440-019-00284-y
Keywords
- Competition phenomena
- Constant sign solutions
- Nodal solutions
- Critical groups
- Hopf boundary point theorem
- Flow invariance