Blow-up solutions for a $p$-Laplacian elliptic equation of logistic type with singular nonlinearity

Claudianor O. Alves, Carlos Alberto Santos, Jiazheng Zhou

DOI: http://dx.doi.org/10.12775/TMNA.2019.022

Abstract


In this paper, we deal with existence, uniqueness and exact rate of boundary behavior of blow-up solutions is for a class of logistic type quasilinear problems in a smooth bounded domain involving the $p$-Laplacian operator, where the nonlinearity can have a singular behavior. In the proof of the existence of solution, we have used the sub and super solution method in conjunction with variational techniques and comparison principles. Related to the rate on boundary and uniqueness, we combine comparison principle with our result of existence of solution.

Keywords


Variational methods; blow-up solution; logistic type; quasilinear equations

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References


Y. Chen and M. Wang, Boundary blow-up solutions of p-Laplacian elliptic equations of logistic type, Proc. Roy. Soc. Edinburg Sect. A 142 (2012), 691–714.

M. Delgado, J. López-Gómez and A. Suárez, Characterizing the existence of large solutions for a class of sublinear problems with nonlinear diffusion, Adv. Differential Equations 7 (2002), 1235–1256.

M. Delgado, J. López-Gómez and A. Suárez, Singular boundary value problems of a porous media logistic equation, Hiroshima Math. J. 34 (2004), 57–80.

J.I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. 1, Elliptic Equations, Pitman Advanced Publishing Program, 1985.

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, World Scientific, 2006.

Y. Du and Z. M. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math. 89 (2003), 277–302.

P. Feng, Remarks on large solutions of a class of semilinear elliptic equations, J. Math. Anal. Appl. 356 (2009), 393–404.

J. Garcı́a-Melián, Large solution for equations involving the p-Laplacian and singular weights, Z. Angew. Math. Phys. 60 (2009), 594–607.

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.

J.V. Gonçalves and C.A.P. Santos, Positive solutions for a class of quasilinear singular equations, Eletron. J. Diferential Equations 56 (2004), 1–15.

J.B. Keller, On solutions of ∆u = f (u), Comm. Pure Appl. Math. 10 (1957), 503–510.

T. Kura, The weak supersolution-subsolution method for second order quasilinear elliptic equations, Hiroshima Math. J. 19 (1989), 1–36.

H. Li, P.Y.H. Pang and M. Wang, Boundary blow-up solutions of p-Laplacian elliptic equations with lower order terms, Z. Angew. Math. Phys. 63 (2012), 295–311.

P. Lindqvist, On the equation div(|∇u|p−2 ∇u) + λ|u|p−2 u = 0, Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164.

J. Matero, Quasilinear elliptic equations with boundary blow-up, J. Anal. Math. 69 (1996), 229–247.

A. Mohammed, Positive solutions of the p-Laplace equation with singular nonlinearity, J. Math. Anal. Appl. 352 (2009), 234–245.

R. Osserman, On the inequality ∆u ≥ f (u), Pacific. J. Math. 7(1957), 1641–1647.

T. Ouyang and Z. Xie, The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation, Nonlinear Anal. 64 (2006), 2129–2142.

Z. Xie and C. Zhao, Blow-up rate and uniqueness of singular radial solutions for a class of quasi-linear elliptic equations, J. Differential Equations 25 (2012), 1776–1788.

L. Wei, The existence of large solutions of semilinear elliptic equations with negative exponent, Nonlinear Anal. 73 (2010), 1739–1746.

L. Wei and M. Wang, Existence of large solutions of a class of quasilinear elliptic equations with singular boundary, Acta Math. Hungar. 129 (2010), 81–95.


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