Articles

ANISOTROPIC (p,q)-EQUATIONS WITH COMPETITION PHENOMENA

  • Zhenhai LIU ,
  • Nikolaos S. PAPAGEORGIOU
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  • 1. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, 537000, China;
    2. Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, Nanning, Guangxi, 530006, China;
    3. Department of Mathematics, National Technical University, Zografou Campus, 15780, Athens, Greece

Received date: 2020-06-15

  Revised date: 2021-06-09

  Online published: 2022-02-24

Supported by

The work was supported by NNSF of China (12071413), NSF of Guangxi (2018GXNSFDA138002).

Abstract

We consider a nonlinear Robin problem driven by the anisotropic (p, q)-Laplacian and with a reaction exhibiting the competing effects of a parametric sublinear (concave) term and of a superlinear (convex) term. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter varies. We also prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.

Cite this article

Zhenhai LIU , Nikolaos S. PAPAGEORGIOU . ANISOTROPIC (p,q)-EQUATIONS WITH COMPETITION PHENOMENA[J]. Acta mathematica scientia, Series B, 2022 , 42(1) : 299 -322 . DOI: 10.1007/s10473-022-0117-9

References

[1] Ambrosetti A, Brezis H, Cerami G. Combined effects of concave and convex nonlinearities in some elliptic problems. J Functional Anal, 1994, 22:519-543
[2] Bahrouni A, Radulescu V D, Repovs D. Double phase transonic flow problems with variable growth:nonlinear patterns and stationary waves. Nonlinearity, 2019, 32:2481-2495
[3] 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
[4] Casas E, Fernandez L A. A Green's formula for quasilinear elliptic operators. J Math Anal Appl, 1989, 142:62-73
[5] Chen F F, Yang Y. Existence of solutions for the fractional (p, q)-Laplacian problems involving a critical Sobolev exponent. Acta Mathematica Scientia, 2020, 40B(6):1666-1678
[6] 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
[7] Deng S G. Positive solutions for Robin problem involving the p(x)-Laplacian. J Math Anal Appl, 2009, 360:548-560
[8] Diening L, Harjulehto P, Hästo P, Ruzicka M. Lebesgue and Sobolev Spaces with Variable Exponent. Lecture Notes in Math, Vol 2017. Heidelberg:Springer, 2011
[9] Fan X. Global C1,α regularity for variable exponent elliptic equations in divergence form. J Differential Equ, 2007, 235:397-417
[10] Fan X. Boundary trace embedding theorems for variable exponent Sobolev spaces. J Math Anal Appl, 2008, 339:1395-1412
[11] Fan X X, Zhao D. A class of De Giorgi type and Hölder continuity. Nonlin Anal, 1999, 36:295-318
[12] Garcia Azorero J, Manfredi J, Peral Alonso I. Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Comm Contemp Math, 2000, 2:385-404
[13] Gasinski L, Papageorgiou N S. Nonlinear Analysis. Boca Raton, FL:Chapman & Hall/CRC, 2006
[14] Gasinski L, Papageorgiou N S. Anisotropic nonlinear Neumann problems. Calc Var, 2011, 42:323-354
[15] Hu S, Papageorgiou N S. Handbook of Multivalued Analysis, Volume I:Theory. Dordrecht, The Netherlands:Kluwer Academic Publishers, 1997
[16] Kenmochi N. Pseudomonotone operators and nonlinear elliptic boundary value problems. J Math Soc Japan, 1975, 27:121-149
[17] Leonardi S, Papageorgiou N S. Positive solutions for nonlinear Robin problems with indefinite potential and competing nonlinearities. Positivity, 2020, 24:339-367
[18] Lieberman G. The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations. Comm Partial Diff Equ, 1991, 16:311-361
[19] Marano S, Marino G, Papageorgiou N S. On the Dirichlet problem with (p, q)-Laplacian and parametric concave-convex nonlinearity. J Math Anal Appl, 2019, 475:1093-1107
[20] Marano S, Mosconi S. Some recent results on the Dirichlet problem for (p, q)-Laplace equations. Disc Cont Dyn Syst Ser S, 2018, 11:279-291
[21] Motreanu D, Motreanu V V, Papageorgiou N S. Existence and nonexistence of positive solutions for parametric Neumann problems with p-Laplacian. Tohoku Math J, 2014, 66(1):137-153
[22] Motreanu D, Motreanu V V, Papageorgiou N S. Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. New York:Springer, 2014
[23] Molica Bisci G, Radulescu V, Servadei R. Competition phenomena for elliptic equations involving a general operator in divergence form. Anal Appl, 2017, 15:51-82
[24] Papageorgiou N S, Qin D, Radulescu V D. Anisotropic double phase problems with indefinite potential:Multiplicity of solutions. Anal Math Phys, 2020, 10(4):63
[25] Papageorgiou N S, Radulescu V D, Repovs D. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discr Cont Dyn Syst, 2017, 37:2589-2618
[26] Papageorgiou N S, Radulescu V D, Repovs D. Nonlinear Analysis-Theory and Methods. Swizerland AG:Springer Nature, 2019
[27] Papageorgiou N S, Radulescu V D, Repovs D. Anisotropic equations with indefinite potential and compacting nonlinearities. Nonl Anal, TMA, 2020, 201:Art111861
[28] Papageorgiou N S, Repovs D, Vetro C. Nonlinear nonhomogeneous Robin problems with almost critical and partially concave reaction. J Geometric Anal, doi:10.110z/s12220-019-00278-0
[29] Papageorgiou N S, Vetro C, Vetro F. Landesman-Lazer type (p, q)-equations with Neumann condition. Acta Mathematica Scientia, 2020, 40B(4):991-1000
[30] Papageorgiou N S, Winkert P. Applied Nonlinear Functional Analysis. Berlin:De Grugler, 2018
[31] Papageorgiou N S, Zhang C. Noncoercive resonant (p, 2)-equations with concave terms. Adv Nonlin Anal, 2020, 9:228-249
[32] Radulescu V D. Isotropic and anisotropic double-phase problems:old and new. Opuscula Math, 2019, 39:259-280
[33] Radulescu V D, Repovs D. Partial Differential Equations with Variable Exponents:Variational Methods and Qualitative Analysis. Boca Raton, FL:CRC Press, Taylor and Francis Group, 2015
[34] Takač P, Giacomoni J. A p(x)-Laplacian extension of the Diaz-Saa inequality and some applications. Proc Royal Soc Edinburgh, 2020, 150:205-232
[35] Tan Z, Fang F. Orlicz Sobolev versus Hölder local minimizer and multiplicity results for quasilinear elliptic equations. J Math Anal Appl, 2013, 402:348-370
[36] Zhang Q. A strong maximum principle for differential equations with nonstandard p(x)-growth conditions. J Math Anal Appl, 2005, 312:125-143
[37] Zhikov V V. On variational problems and nonlinear elliptic equations with nonstandard growth conditions. J Math Sci, 2011, 173:463-570
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