We consider a nonlinear Dirichlet problem driven by a (p,q)-Laplace differential operator (1 < q < p). The reaction is (p-1)-linear near ±∞ and the problem is noncoercive. Using variational tools and truncation and comparison techniques together with critical groups, we produce five nontrivial smooth solutions all with sign information and ordered. In the particular case when q=2, we produce a second nodal solution for a total of six nontrivial smooth solutions all with sign information.
Nikolaos S. PAPAGEORGIOU
,
Nikolaos S. PAPAGEORGIOU
,
Calogero VETRO
. ON NONCOERCIVE (p,q)-EQUATIONS[J]. Acta mathematica scientia, Series B, 2021
, 41(5)
: 1788
-1808
.
DOI: 10.1007/s10473-021-0524-3
[1] Aizicovici S, Papageorgiou N S, Staicu V. Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem Amer Math Soc, 2008, 196:1-70
[2] Cingolani S, Degiovanni M. Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity. Comm Partial Differential Equations, 2005, 30:1191-1203
[3] Díaz J I, Saa J E. Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C R Acad Sci Paris Sér I Math, 1987, 305:521-524
[4] Filippakis M E, Papageorgiou N S. Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian. J Differential Equations, 2008, 245:1883-1922
[5] Gasiński L, Papageorgiou N S. Nonlinear Analysis, Ser Math Anal Appl 9. Boca Raton:Chapman & Hall/CRC, 2006
[6] Gasiński L, Papageorgiou N S. Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential. Set-Valued Var Anal, 2012, 20:417-443
[7] Gasiński L, Papageorgiou N S. Multiple solutions for asymptotically (p-1)-homogeneous p-Laplacian equations. J Funct Anal, 2012, 262:2403-2435
[8] Gasiński L, Papageorgiou N S. Exercises in Analysis. Part 2. Nonlinear Analysis, Problem Books in Mathematics. Cham:Springer, 2016
[9] Gasiński L, Papageorgiou N S. Positive solutions for the Robin p-Laplacian problem with competing nonlinearities. Adv Calc Var, 2019, 12:31-56
[10] He T, Lei Y, Zhang M, Sun H. Nodal solutions for resonant and superlinear (p, 2)-equations. Math Nachr, 2018, 291:2565-2577
[11] Hu S, Papageorgiou N S. Handbook of Multivalued Analysis, Vol. I:Theory, Mathematics and its Applications, 419. Dordrecht:Kluwer Academic Publishers, The Netherlands, 1997
[12] Ladyzhenskaya O, Ural'tseva N. Linear and Quasilinear Elliptic Equations. New York:Academic Press, 1968
[13] Leonardi S, Papageorgiou N S. On a class of critical Robin problems. Forum Math, 2020, 32:95-109
[14] Lieberman G M. The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations. Comm Partial Differential Equations, 1991, 16:311-361
[15] Marano S A, Papageorgiou N S. Constant sign and nodal solutions of coercive (p, q)-Laplacian problems. Nonlinear Anal, 2013, 77:118-129
[16] Medeiros E, Perera K. Multiplicity of solutions for a quasilinear elliptic problem via the cohomological index. Nonlinear Anal, 2009, 71:3654-3660
[17] Papageorgiou N S, Rădulescu V D. Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance. Appl Math Optim, 2014, 69:393-430
[18] Papageorgiou N S, Rădulescu V D, Repovš D D. On a class of parametric (p, 2)-equations. Appl Math Optim, 2017, 75:193-228
[19] Papageorgiou N S, Rădulescu V D, Repovš D D. Noncoercive resonant (p, 2)-equations. Appl Math Optim, 2017, 76:621-639
[20] Papageorgiou N S, Rădulescu V D, Repovš D D. Nonlinear Analysis-Theory and Methods. Switzerland:Springer, 2019
[21] Papageorgiou N S, Vetro C, Vetro F. Multiple solutions with sign information for a class of coercive (p, 2)-equations. Bull Malays Math Sci Soc, 2020, 43:2343-2371
[22] Papageorgiou N S, Vetro C, Vetro F. Multiple solutions with sign information for a (p, 2)-equation with combined nonlinearities. Nonlinear Anal, 2020, 192:111716
[23] Pucci P, Serrin J. The Maximum Principle. Basel:Birkhäuser Verlag, 2007
