Weak solution by the sub-supersolution method for a nonlocal system involving Lebesgue generalized spaces
DOI:
https://doi.org/10.58997/ejde.2022.36Keywords:
Nonlocal problem, p(x)-Laplacian, sub-supersolution; minimal wave speedAbstract
We consider a system of nonlocal elliptic equations $$ \displaylines{ - \mathcal{A}(x, |v|_{L^{r_1(x)}}) \hbox{div}(a_1(|\nabla u|^{p_1(x)})|\nabla u|^{p_1(x)-2}\nabla u)\cr = f_1(x, u,v) |\nabla v|^{\alpha_1(x)}_{L^{q_1(x)}}+g_1(x, u,v) |\nabla v|^{\gamma_1(x)}_{L^{s_1(x)}},\cr - \mathcal{A}(x, |u|_{L^{r_2(x)}}) \hbox{div}(a_2(|\nabla v|^{p_2(x)})|\nabla u|^{p_2(x)-2}\nabla u)\cr = f_2(x, u,v) |\nabla u|^{\alpha_2(x)}_{L^{q_2(x)}}+g_2(x, u,v)
|\nabla u|^{\gamma_2(x)}_{L^{s_2(x)}}, } $$
with Dirichlet boundary condition, where \(\Omega\) is a bounded domain in
\()\mathbb{R}^N\) \((N >1)\) with \(C^2\) boundary. Using sub-supersolution method,
we prove the existence of at least one positive weak solution.
Also, we study a generalized logistic equation and a sublinear system.
For more information see https://ejde.math.txstate.edu/Volumes/2022/36/abstr.html
References
C. O. Alves, A. Moussaoui, L. S. Tavares; An elliptic system with logarithmic nonlinearity, Adv. Nonlinear Anal., 8 (2019), 928-945.
S. Baraket, G. Molica Bisci; Multiplicity results for elliptic Kirchhoff-type problems, Adv. Nonlinear Anal., 6 (2017), no. 1, 85-93.
A. Bose, G. A. Kriegsman; Large amplitude solutions of spatially non-homogeneous nonlocal reaction-diffusion equations, Methods Appl. Anal., 7 (2000), 295-312.
A. Cabadam, F. J .S. A. Corr^ea; Existence of solutions of a nonlocal elliptic system via Galerkin Method, Abstr. Appl. Anal., Art. ID 137379 (2012), 16 pp.
G. F. Carrier; On the nonlinear vibration problem of the elastic string, Q. Appl. Math., 3 (1945), 157-165.
M. Cencelj, D. Repov s, Z. Virk; Multiple perturbations of a singular eigenvalue problem, Nonlinear Anal., 119 (2015), 37-45.
Y. Chen, H. Gao; Existence of positive solutions for nonlocal and nonvariational elliptic system, Bull. Austral. Math. Soc., 72 (2005), no. 2, 271-281.
Y. Chen, S. Levine, M. Rao; Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), no. 4, 1383-1406.
M. Chipot; The diffusion of a population partly driven by its preferences, A.R.M.A. 155 (2000), 237-259.
M. Chipot; Remarks on some class of nonlocal elliptic problems, Recent Advances on Elliptic and Parabolic Issues, World Scienti c (2006), 79-102.
M. Chipot, F. J. S. A. Corr^ea; Boundary layer solutions to functional, Bull. Braz. Math. Soc., New Series 40 (2009), 381-393.
M. Chipot, B. Lovat; Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), no. 7, 4619-4627.
M. Chipot, P. Roy; Existence results for some functional elliptic equations, Differential Integral Equations, 27 (2014), no. 3/4, 289-300.
F. J. S. A. Correa, G. M. Figueiredo, F. P. M. Lopes; On the existence of positive solutions for a nonlocal elliptic problem involving the p-Laplacian and the generalized Lebesgue space
Lp(x)(omega), Differential Integral Equations, 21 (2008), no. 3-4, 305-324.
F. J. S. A. Corr^ea, F. P. M. Lopes; Positive solutions for a class of nonlocal elliptic systems, Comm. Appl. Nonlinear Anal., 14 (2007), no. 2, 67-77.
C. Cowan, A. Razani; Singular solutions of a p-Laplace equation involving the gradient, J. Differential Equations 269 (2020), 3914-3942.
C. Cowan, A. Razani; Singular solutions of a H enon equation involving a nonlinear gradient term, Commun. Pure Appl. Anal., 21 (2022), no. 1, 141-158.
W. Deng, Y. Lie, C. Xie; Blow-up and global existence for a nonlocal degenerate parabolic system, J. Math. Anal. Appl., 277 (2003), no. 1, 199-217.
X. L. Fan; Global C1; regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), no. 2, 397-417.
X. L. Fan; On the sub-super solution method for p(x)-Laplacian equations, J. Math. Anal. Appl., 330 (2007), no. 1, 665-682.
X. L. Fan, Q. H. Zhang; Existence of solution for p(x)-Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), no. 8, 1843-1852.
X. L. Fan, D. Zhao; A class of De Giorgi type and Holder continuity, Nonlinear Anal., 36 (1999), no. 3, 295-318.
X. L. Fan, Y. Z. Zhao, Q. H. Zhang; A strong maximum principle for p(x)-Laplace equations, Chinese J. Contemp. Math., 24 (2003), no. 3, 277-282.
G. M. Figueiredo, A. Razani; The sub-supersolution method for a non-homogeneous elliptic equation involving Lebesgue generalized spaces, Boundary Value Problems, 2021:105 (2021).
A. C. Fowler, I. Frigaard, S. D. Howison; Temperature surges in current limiting circuits devices, SIAM J. Appl. Mah., 52 (1992), 998-1011.
P. Freitas, M. Grinfeld; Stationary solutions of an equation modelling Ohmic heating, Appl. Math. Lett., 7 (1994), 1-6.
G. Kirchhoff; Mechanik, Teubner, Leipzig, 1883.
J. L. Lions; Quelques M ethodes de R esolution de Probl emes aux Limites Non Lineaires, Dunod, Paris, 1969.
J. Liu, Q. Zhang, C. Zha; Existence of positive solutions for p(x)-Laplacian equations with a singular nonlinear term, Electron. J. Differential Equations, 2014 (2014), no. 155, 21 pp.
M. Mihailescu, V. R adulescu, D. Repov s; On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting, J. Math. Pures Appl., (9) 93 (2010), no. 2, 132-148.
G. Molica Bisci, V. R adulescu, R. Servadei; Variational methods for nonlocal fractional problems, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016.
W. E. Olmstead, S. Nemat-Nasser, L. Ni; Shear bands as surfaces of discontinuity, J. Mech. Phy. Solids, 42 (1994), 697-709.
C. V. Pao; Blowing-up of solutions for a nonlocal reaction-diffusion problem in combustion theory, J. Math. Anal. Appl., 166 (1992), 591-600.
P. Pucci, Q. Zhang; Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), no. 5, 1529-1566.
P. Pucci, M. Xiang, B. Zhang; Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations. Adv. Nonlinear Anal., 5 (2016), no. 1, 27-55.
P. H. Rabinowitz; Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.
V. R adulescu, D. Repov s; Partial Differential equations with variable xponents. Variational methods and qualitative analysis, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015.
M. Ruzicka; Electrorheological Fluids: Modelling and Mathematical Theory, Springer-Verlag, Berlin, 2000.
G. C. G. dos Santos, G. M. Figueiredo; Positive Solutions for a class of nonlocal problems involving Lebesgue generalized spaces: Scalar and system cases, J. Elliptic Parabol. Equ., 2 (2016), no. 1-2, 235-266.
G. C. G. dos Santos, G. M. Figueiredo, L. S. Tavares; A sub-supersolution method for a class of nonlocal problems involving the p(x)-Laplacian operator and applications, Acta Appl. Math., 153 (2018), 171-187.
G. C. G. dos Santos, G. M. Figueiredo, L. S. Tavares; Sub-super solution method for nonlocal systems involving the p(x)-Laplacian operator, Electron. J. Differential Equations, 2020 (2020), no. 25, 1-19.
Downloads
Published
License
Copyright (c) 2022 Abdolrahman Razani, Giovany M. Figueiredo
This work is licensed under a Creative Commons Attribution 4.0 International License.