We consider a nonlinear Robin problem driven by the $(p,q)$-Laplacian plus an indefinite potential term and with a parametric reaction term. Under minimal conditions on the reaction function, which concern only its behavior near zero, we show that, for all $\lambda >0$ small, the problem has a nodal solution $y_{\lambda} \in C^1(\bar{Ω})$ and we have $y_{\lambda} \rightarrow 0$ in $C^1(\bar{Ω})$ as $\lambda \rightarrow 0^+$.
Salvatore LEONARDI
,
Nikolaos S. PAPAGEORGIOU
. ARBITRARILY SMALL NODAL SOLUTIONS FOR PARAMETRIC ROBIN (p,q)-EQUATIONS PLUS AN INDEFINITE POTENTIAL[J]. Acta mathematica scientia, Series B, 2022
, 42(2)
: 561
-574
.
DOI: 10.1007/s10473-022-0210-0
[1] Bahrouni A, Radulescu V D, Repovs D D. Double phase transonic flow problems with variable growth:nonlinear patterns and stationary waves. Nonlinearity, 2019, 32:2481-2495
[2] Benci V, D'Avenia P, Fortunato D, Pisani L. Solitons in several space dimensions:Derrick's problem and infinitely many solutions. Arch Rational Mech Anal, 2000, 154:297-324
[3] Cherfils L, Ilyasov Y. On the stationary solutions of generalized reaction diffusion equations with p&q Laplacian. Comm Pure Appl Anal, 2005, 4:9-22
[4] Diaz J I, Saa J E. Existence et unicit e de solutions positives pour certains equations elliptiques quasilineaires. CRAS Paris, 1987, 305:521-524.
[5] Gasinski L, Papageorgiou N S. Exercises in Analysis. Part 2:Nonlinear Analysis. Cham:Springer, 2016
[6] Gasinski L, Papageorgiou N S. Positive solutions for the Robin p-Laplacian problem with competing nonlinearities. Adv Calc Var, 2019, 12:31-56
[7] Guarnotta U, Marano S A, Papageorgiou N S. Multiple nodal solutions to a Robin problem with signchanging and locally defined reaction. Rend Lincei Mat Appl, 2019, 30:269-294
[8] Hu S, Papageorgiou N S. Handbook of Multivalued Analysis. Vol I:Theory. Dordrecht, The Nederlands:Kluwer Acad Publ, 1997
[9] Leonardi S. Morrey estimates for some classes of elliptic equations with a lower order term. Nonlinear Analysis, 2018, 177(part B):611-627
[10] Leonardi S, Onete F I. Nonlinear Robin problems with indefinite potential. Nonlinear Analysis TMA, 2020, 195
[11] Leonardi S, Papageorgiou N S. Positive solutions for nonlinear Robin problems with indefinite potential and competing nonlinearities. Positivity, 2020, 24:339-367. https://doi.org/10.1007/s11117-019-00681-5
[12] Leonardi S, Papageorgiou N S. On a class of critical Robin problems. Forum Math, 2020, 32(1). https://doi.org/10.1515/forum-2019-0160
[13] Leonardi S, Papageorgiou N S. Existence and multiplicity of positive solutions for parametric nonlinear nonhomogeneous singular Robin problems. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matemáticas, 2020, 114:100. https://doi.org/10.1007/s13398-020-00830-6
[14] Lieberman G M. Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal, 1988, 12:1203-1219
[15] Mugnai D, Papageorgiou N S. Resonant nonlinear Neumann problems with indefinite weight. Ann Scu Norm Sup Pisa, 2012, 11(5):729-788
[16] Papageorgiou N S, Radulescu V D. Coercive and noncoercive nonlinear Neumann problems with indefinite potentials. Forum Math, 2016, 28:545-571
[17] Papageorgiou N S, Radulescu V D. Nonlinear nonhomogeneous Robin problem with superlinear reaction term. Adv Nonlin Stud, 2016, 16:737-764
[18] Papageorgiou N S, Radulescu V D. Infinitely many nodal solutions for nonlinear non homogeneous Robin problems. Adv Nonlinear Studies, 2016, 16:287-300
[19] Papageorgiou N S, Radulescu V D, Repovs D. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discr Cont Dyn Systems, 2017, 37:2589-2618
[20] Papageorgiou N S, Radulescu V D, Repovs D. Nodal solutions fon nonlinear non homogeneous Robin problems. Rend Lincei Mat Applicable, 2018, 29:731-738
[21] Papageorgiou N S, Radulescu V D, Repovs D. Nonlinear Analysis-Theory and Methods. Switzerland:Springer, 2019
[22] Papageorgiou N S, Radulescu V D, Repovs D D. Positive solutions for nonlinear parametric singular Dirichlet problems. Bull Math Sci, 2019, 9(3):1950011, 21 pp
[23] Papageorgiou N S, Vetro C, Vetro F. Multiple nodal solutions for semilinear Robin problems with indefinite linear part and concave terms. Topol Methods Nonlin Anal, 2017, 60:269-286
[24] Pucci P, Serrin J. The Maximum Principle. Basel:Birkhäuser, 2007
