We consider a nonlinear parametric Neumann problem driven by the sum of a p-Laplacian and of a q-Laplacian and exhibiting in the reaction the competing effects of a singular term and of a resonant term. Using variational methods together with suitable truncation and comparison techniques, we show that for small values of the parameter the problem has at least two positive smooth 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), 1–70 (2008)
Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Cherfils, L., Il’yasov, Y.: On the stationary solutions of generalized reaction diffusion equations with \(p\)&\(q\)-Laplacian. Commun. Pure Appl. Anal. 4(1), 9–22 (2005)
Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis, Series Mathematical Analysis and Applications, vol. 9. CRC Press, Boca Raton (2006)
Gasiński, L., Papageorgiou, N.S.: Nonlinear elliptic equations with singular terms and combined nonlinearities. Ann. Henri Poincaré 13(3), 481–512 (2012)
Giacomoni, J., Schindler, I., Takáč, P.: Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6(1), 117–158 (2007)
Hirano, N., Saccon, C., Shioji, N.: Brezis–Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem. J. Differ. Equ. 245(8), 1997–2037 (2008)
Kyritsi, S., Papageorgiou, N.S.: Pairs of positive solutions for singular \(p\)-Laplacian equations with a \(p\)-superlinear potential. Nonlinear Anal. 73(5), 1136–1142 (2010)
Lair, A.V., Shaker, A.W.: Entire solution of a singular semilinear elliptic problem. J. Math. Anal. Appl. 200(2), 498–505 (1996)
Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural\(^\prime \)tseva for elliptic equations. Commun. Partial Differ. Equ. 16(2–3), 311–361 (1991)
Marano, S.A., Mosconi, S.J.N.: Some recent results on the Dirichlet problem for (\(p, q\))-Laplace equations. Discrete Contin. Dyn. Syst. Ser. S 11(2), 279–291 (2018)
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(2), 815–829 (2013)
Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)
Papageorgiou, N.S., Rădulescu, V.D.: Combined effects of singular and sublinear nonlinearities in some elliptic problems. Nonlinear Anal. 109, 236–244 (2014)
Papageorgiou, N.S., Rădulescu, V.D.: Multiple solutions with precise sign for nonlinear parametric Robin problems. J. Differ. Equ. 256(7), 2449–2479 (2014)
Papageorgiou, N.S., Rădulescu, V.D.: Nonlinear nonhomogeneous Robin problems with superlinear reaction term. Adv. Nonlinear Stud. 16(4), 737–764 (2016)
Papageorgiou, N.S., Rădulescu, V.D., Repovs̆, D.D.: Pairs of positive solutions for resonant singular equations with the \(p\)-Laplacian. Electron. J. Differ. Equ. 2017(249), 1–22 (2017)
Papageorgiou, N.S., Smyrlis, G.: A bifurcation-type theorem for singular nonlinear elliptic equations. Methods Appl. Anal. 22(2), 147–170 (2015)
Papageorgiou, N.S., Vetro, C.: Superlinear \((p(z), q(z))\)-equations. Complex Var. Elliptic Equ. 64(1), 8–25 (2019)
Perera, K., Zhang, Z.: Multiple positive solutions of singular \(p\)-Laplacian problems by variational methods. Bound. Value Probl. 2005(3), 377–382 (2005)
Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)
Sun, Y., Wu, S., Long, Y.: Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. J. Differ. Equ. 176(2), 511–531 (2001)
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Papageorgiou, N.S., Vetro, C. & Vetro, F. Singular Neumann (p, q)-equations. Positivity 24, 1017–1040 (2020). https://doi.org/10.1007/s11117-019-00717-w
- Singular term
- Resonant nonlinearity
- Nonlinear regularity
- Truncation and comparison
- Nonlinear strong maximum principle
- (p, q)-equation
Mathematics Subject Classification