Opuscula Math. 43, no. 4 (2023), 603-613
https://doi.org/10.7494/OpMath.2023.43.4.603
Opuscula Mathematica
Solutions for a nonhomogeneous p&q-Laplacian problem via variational methods and sub-supersolution technique
Leandro S. Tavares
J. Vanterler C. Sousa
Abstract. In this paper it is obtained, through variational methods and sub-supersolution arguments, existence and multiplicity of solutions for a nonhomogeneous problem which arise in several branches of science such as chemical reactions, biophysics and plasma physics. Under a general hypothesis it is proved an existence result and multiple solutions are obtained by considering an additional natural condition.
Keywords: \(p\&q\)-Laplacian operator, nonhomogeneous operator, sub-supersolutions, existence, multiplicity.
Mathematics Subject Classification: 35A15, 35J60.
- A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
- V. Ambrosio, Fractional \((p,q)\)-Schrödinger equations with critical and supercritical growth, Appl. Math. Optim. 86 (2022), Article no. 31. https://doi.org/10.1007/s00245-022-09893-w
- V. Ambrosio, T. Isernia, Multiplicity of positive solutions for a fractional \(p\&q\)-Laplacian problem in \(\mathbb{R}^N\), J. Math. Anal. Appl. 501 (2021), 31 pp. https://doi.org/10.1016/j.jmaa.2020.124487
- V. Ambrosio, T. Isernia, A multiplicity result for a \((p,q)\)-Schrödinger-Kirchhoff type equation, 201 (2022), 943-984. https://doi.org/10.1007/s10231-021-01145-y
- V. Ambrosio, V. Rădulescu, Fractional double-phase patterns: concentration and multiplicity of solutions, J. Math. Pures Appl. 9 (2020), 101-145. https://doi.org/10.1016/j.matpur.2020.08.011
- V. Ambrosio, D. Repovš, Multiplicity and concentration results for a \((p,q)\)-Laplacian problem in \(\mathbb{R}^N\), Z. Angew. Math. Phys. 72 (2021), 33 pp. https://doi.org/10.1007/s00033-020-01466-7
- S.C.Q. Arruda, R.G. Nascimento, Existence and multiplicity of positive solutions for singular \(p\&q\)-Laplacian problems via sub-supersolution method, Electron. J. Differential Equations 2021 (2021), 11 pp.
- S.C.Q. Arruda, R.G. Nascimento, Existence and multiplicity of positive solutions for a singular system via sub-supersolution method and mountain pass theorem, Electron. J. Qual. Theory Differ. Equ. 2021 (2021), 20 pp. https://doi.org/10.14232/ejqtde.2021.1.26
- L. Cherfils, Y. Ilyasov, On the stationary solutions of generalized reaction diffusion equations with \(p\&q\)-Laplacian, Commun. Pure Appl. Anal. 4 (2005), 9-22. https://doi.org/10.3934/cpaa.2005.4.9
- F.J.S.A. Corrêa, A.S.S. Corrêa, G.M. Figueiredo, Existence of positive solution for a singular system involving general quasilinear operators, Differ. Equ. Appl. 6 (2014), 481-494. https://doi.org/10.7153/dea-06-28
- F.J.S.A. Corrêa, A.S.S. Corrêa, G.M. Figueiredo, Positive solution for a class of \(p\&q\)-singular elliptic equation, Nonlinear Anal. Real World Appl 16 (2014), 163-169. https://doi.org/10.1016/j.nonrwa.2013.09.015
- G.M. Figueiredo, Existence of positive solutions for a class of \(p\&q\) elliptic problems with critical growth on \(\mathbb{R}^N\), J. Math. Anal. Appl. 378 (2011), 507-518. https://doi.org/10.1016/j.jmaa.2011.02.017
- Y.T. Guo, G.J. Ye, Existence and uniqueness of weak solutions to variable-order fractional Laplacian equations with variable exponents, J. Funct. Spaces 2021 (2021), Article ID 6686213. https://doi.org/10.1155/2021/6686213
- C. He, G. Li, The regularity of weak solutions to nonlinear scalar field elliptic equations containing \(p\&q\)-Laplacians, Ann. Acad. Sci. Fenn. Math. 33 (2008), 337-371.
- T. Isernia, D. Repovš, Nodal solutions for double phase Kirchhoff problems with vanishing potentials, Asymptot. Anal. 124 (2021), 371-396. https://doi.org/10.3233/ASY-201648
- M.A. Ragusa, Parabolic Herz spaces and their applications, Appl. Math. Lett. 25 (2012), 1270-1273. https://doi.org/10.1016/j.aml.2011.11.022
- K.C.V. de Sousa, L.S. Tavares, Multiple solutions for a class of problems involving the \(p(x)\)-Laplacian operator, Appl. Anal. 101 (2021), 5415-5423. https://doi.org/10.1080/00036811.2021.1892081
- G.C.G. dos Santos, G. Figueiredo, J.R.S. Silva, Multiplicity of positive solutions for an anisotropic problem via sub-supersolution method and mountain pass theorem, J. Convex Anal. Analysis 27 (2020), 1363-1374.
- M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 2nd ed., Springer, Berlin, 1996.
- J.B. Zuo, R. Guefaifia, F. Kamache, S. Boulaaras, Multiplicity of solutions for perturbed nonlinear fractional \(p\)-Laplacian boundary value systems related with two control parameters, Filomat 35 (2021), 2827-2848. https://doi.org/10.2298/FIL2108827Z
- Leandro S. Tavares (corresponding author)
https://orcid.org/0000-0003-0953-8037- Universidade Federal do Cariri, Centro de Ciências e Tecnologia, Juazeiro do Norte/CE, Brazil
- J. Vanterler C. Sousa
- https://orcid.org/0000-0002-5379-2975
- Universidade Federal do ABC, Centro de Matemática, Computação e Cognição, Santo André/SP, Brazil
- Communicated by Giovany Figueiredo.
- Received: 2022-12-26.
- Revised: 2023-04-01.
- Accepted: 2023-04-13.
- Published online: 2023-06-13.
