Global multiplicity for parametric anisotropic Neumann (p,q)-equations
DOI:
https://doi.org/10.12775/TMNA.2022.010Keywords
Anisotropic operator, superlinear reaction, positive and nodal solutions, critical groupsAbstract
We consider a Neumann boundary value problem driven by the anisotropic $(p,q)$-Laplacian plus a parametric potential term. The reaction is ``superlinear". We prove a global (with respect to the parameter) multiplicity result for positive solutions. Also, we show the existence of a minimal positive solution and finally, we produce a nodal solution.References
S.G. Deng and Q. Wang, Nonexistence, existence and multiplicity of positive solutions to the p(x)-Laplacian nonlinear Neumann boundary value problems, Nonlinear Anal. 73 (2010), 2170–2183.
L. Diening, P. Harjulehto, P. Hästo and M. Ruzička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math, vol. 2017, Springer, Heidelberg, 2011.
X. Fan, Global C 1+α regularity for variable exponent elliptic equations in divergence form, J. Differential Equations 235 (2007), 397–417.
X. Fan and S.G. Deng, Multiplicity of positive solutions for a class of inhomogeneous Neumann problems involving the p(x)-Laplacian, Nonlinear Differ. Equ. Appl. 14 (2009), 255–271.
X. Fan and X. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in RN , Nonlinear Anal. 59 (2004), no. 1–2, 173–188.
M. Galewski, On a Dirichlet problem with p(x)-Laplacian, J. Math. Anal. Appl. 337 (2008), no. 1, 281–291.
L. Gasinski and N.S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. 42 (2011), 323–354.
S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory, Kluwer Academic Publishers, Dodrecht, The Netherlands, 1997.
K. Kefi, On the Robin problem with indefinite weight in Sobolev spaces with variable exponents, Z. Anal. Anwend. 37 (2018), no. 1, 25–38.
G. Mingione and V.D. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl. 501 (2021), no. 1, paper no. 125197, 41 pp.
N.S. Papageorgiou, D. Qin and V.D. Rădulescu, Anisotropic double-phase problems with indefinite potential, Anal. Math. Phys. (2020), 10:63.
N.S. Papageorgiou and V.D. Rădulescu, Coercive and noncoercive nonlinear Neumann problems with indefinite potential, Forum Math. 28 (2016), 545–571.
N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dyn. Syst. 37 (2017), 2589–2618.
N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš, Nonlinear Analysis–Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019.
N.S. Papageorgiou, V.D. Rădulescu and Y. Zhang, Anisotropic singular double phase Dirichlet problems, Discrete Contin. Dyn. Syst. Ser. S 14 (2021), 4465–4502.
N.S. Papageorgiou and Y. Zhang, Constant sign and nodal solutions for superlinear (p, q)-equations with indefinite potential and a concave boundary term, Adv. Nonlinear Anal. 10 (2020), 76–101.
V.D. Rădulescu and D.D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015.
P. Takač and J. Giacomoni, A p(x)-Laplacian extension of the Diaz–Saa inequality and some applications, Proc. Roy. Soc. Edinburgh 150 (2020), 205–232.
Z. Tan and F. Fang, Orlicz–Sobolev versus Hölder local minimizers and multiplicity results for quasilinear ellipltic equations, J. Math. Anal. Appl. 402 (2013), 348–370.
X.H. Tang and S.T. Chen, Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations 56 (2017), 110.
X.H. Tang and B.T. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations 261 (2016), 2384–2402.
P. Winkert and R. Zacher, A priori bounds of solutions to elliptic equations with nonstandard growth, Discrete Contin. Dyn. Syst. Ser. S 5 (2012), 865–878.
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Copyright (c) 2023 Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš
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