In this paper, we study the elliptic system of competitive type with nonhomogeneous terms $ \Delta u = u^pv^q+h_1(x) $, $ \Delta v = u^rv^s+h_2(x) $ in $ \Omega $ with two types of boundary conditions: (Ⅰ) $ u = v = +\infty $ and (SF) $ u = +\infty $, $ v = f $ on $ \partial\Omega $, where $ f > 0 $, $ (p-1)(s-1)-qr > 0 $, and $ \Omega \subset \mathbb{R^N} $ is a smooth bounded domain. The nonhomogeneous terms $ h_1(x) $ and $ h_2(x) $ may be unbounded near the boundary and may change sign in $ \Omega $. First, for a single semilinear elliptic equation with a singular weight and nonhomogeneous term, boundary asymptotic behaviour of large positive solutions is established. Using this asymptotic behaviour, we show existence of large positive solutions for this elliptic system with the boundary condition (SF), existence of maximal solution, boundary asymptotic behaviour and uniqueness of large positive solutions for this elliptic system with (Ⅰ).
Citation: Haohao Jia, Feiyao Ma, Weifeng Wo. Large positive solutions to an elliptic system of competitive type with nonhomogeneous terms[J]. AIMS Mathematics, 2021, 6(8): 8191-8204. doi: 10.3934/math.2021474
In this paper, we study the elliptic system of competitive type with nonhomogeneous terms $ \Delta u = u^pv^q+h_1(x) $, $ \Delta v = u^rv^s+h_2(x) $ in $ \Omega $ with two types of boundary conditions: (Ⅰ) $ u = v = +\infty $ and (SF) $ u = +\infty $, $ v = f $ on $ \partial\Omega $, where $ f > 0 $, $ (p-1)(s-1)-qr > 0 $, and $ \Omega \subset \mathbb{R^N} $ is a smooth bounded domain. The nonhomogeneous terms $ h_1(x) $ and $ h_2(x) $ may be unbounded near the boundary and may change sign in $ \Omega $. First, for a single semilinear elliptic equation with a singular weight and nonhomogeneous term, boundary asymptotic behaviour of large positive solutions is established. Using this asymptotic behaviour, we show existence of large positive solutions for this elliptic system with the boundary condition (SF), existence of maximal solution, boundary asymptotic behaviour and uniqueness of large positive solutions for this elliptic system with (Ⅰ).
[1] | M. Chuaqui, C. Cortázar, M. Elgueta, J. García-Melián, Uniqueness and boundary behaviour of large solutions to elliptic problems with singular weights, Commun. Pure Appl. Anal., 3 (2004), 653–662. doi: 10.3934/cpaa.2004.3.653 |
[2] | E. N. Dancer, Y. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal., 34 (2001), 292–314. |
[3] | G. Díaz, R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal. Theor., 20 (1993), 97–125. doi: 10.1016/0362-546X(93)90012-H |
[4] | D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer, 1998. |
[5] | J. López-Gómez, L. Maire, Boundary blow-up rate and uniqueness of the large solution for an elliptic cooperative system of logistic type, Nonlinear Anal. Real, 33 (2017), 298–316. doi: 10.1016/j.nonrwa.2016.07.001 |
[6] | J. López-Gómez, Metasolutions of parabolic equations in population dynamics, CRC Press, 2016. |
[7] | A. Mohammed, G. Porru, Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term, Adv. Nonlinear Anal., 8 (2017), 517–532. doi: 10.1515/anona-2017-0065 |
[8] | A. Mohammed, V. D. R$\rm \breve{a}$dulescu, A. Vitolo, Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness, Adv. Nonlinear Anal., 9 (2018), 39–64. doi: 10.1515/anona-2018-0134 |
[9] | C. Mu, S. Huang, Q. Tian, L. Liu, Large solutions for an elliptic system of competitive type: Existence, uniqueness and asymptotic behavior, Nonlinear Anal. Theor., 71 (2009), 4544–4552. doi: 10.1016/j.na.2009.03.012 |
[10] | J. García-Melián, Large solutions for an elliptic equation with a nonhomogeneous term, J. Math. Anal. Appl., 434 (2016), 872–881. doi: 10.1016/j.jmaa.2015.09.041 |
[11] | J. García-Melián, J. S. De Lis, R. Letelier-Albornoz, The solvability of an elliptic system under a singular boundary condition, P. Roy. Soc. A-Math. phy., 136 (2006), 509–546. |
[12] | J. García-Melián, J. D. Rossi, Boundary blow-up solutions to elliptic systems of competitive type, J. Differ. Equations, 206 (2004), 156–181. doi: 10.1016/j.jde.2003.12.004 |
[13] | J. García-Melián, J. D. Rossi, J. S. De Lis, Elliptic systems with boundary blow-up: Existence, uniqueness and applications to removability of singularities, Commun. Pure Appl. Math., 15 2016,549–562. |
[14] | C. V. Pao, Nonlinear parabolic and elliptic equations, Springer, 1992. |
[15] | S. Qi, P. Zhao, Large solutions for a nonhomogeneous quasilinear elliptic problem, Nonlinear Anal. Real, 41 (2018), 428–442. doi: 10.1016/j.nonrwa.2017.11.001 |
[16] | L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., 59 (1992), 231–250. doi: 10.1007/BF02790229 |
[17] | Y. Wang, Explosive solutions of elliptic equation with singular weight, Appl. Anal., 100 (2021), 1744–1751. doi: 10.1080/00036811.2019.1659959 |
[18] | Z. Wang, F. Ma, Boundary blow-up solutions for elliptic systems with nonhomogeneous term, Journal of Ningbo University (Natural Science & Engineering Edition), 33 (2020), 69–71. |
[19] | Z. Wang, F. Ma, W. Wo, Large solutions of a semilinear elliptic equation with singular weights and nonhomogeneous term, Arch. Math., 115 (2020), 99–110. doi: 10.1007/s00013-019-01432-4 |