Abstract
Based on a comparison principle, we discuss the existence, uniqueness and asymptotic behaviour of various boundary blow-up solutions, for a class of quasilinear elliptic equations, which are then used to obtain a rather complete understanding of some quasilinear elliptic problems on a bounded domain or over the entireR N.
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[AT] S. Alama and G. Tarantello,On the solvability of a semilinear elliptic equations via an associated eigenvalue problem, Math. Z.221 (1996), 467–493.
[AG] A. Ambrosetti and J. L. Gámez,Branches of positive solutions for some semilinear Schrödinger equations, Math. Z.224 (1997), 347–362.
[BM1] C. Bandle and M. Marcus,Large solutions of semilinear elliptic equations: existence, uniqueness, and asymptotic behaviour, J. Analyse Math.58 (1992), 9–24.
[BM2] C. Bandle and M. Marcus,Asymptotic behaviour of solutions and derivatives for semilinear elliptic problems with blow-up on the boundary, Ann Inst. H. Poincaré12 (1995), 155–171.
[BL] C. Bandle and H. Leutwiler,On a quasilinear elliptic equation and a Riemannian metric invariant under Möbius transformation, Aequationes Math.42 (1991), 166–181.
[CDG] A. Cañada, P. Drábek and J. L. Gamez,Existence of positive solutions for some problems with nonlinear diffusion, Trans. Amer. Math. Soc.349 (1997), 4231–4249.
[dP] M. A. del Pino,Positive solutions of a semilinear elliptic equation on compact manifold, Nonlinear Anal.22 (1994), 1423–1430.
[DL] G. Diaz and R. Letelier,Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal.20 (1993), 97–125.
[D] J. Diaz,Nonlinear Partial Differential Equations and Free Boundary Problems, Vol. 1,Elliptic Equations, Pitman Research Notes in Math., Vol. 106, Boston, 1985.
[Dr] P. Drábek,The least eigenvalues of nonhomogeneous degenerate quasilinear eigenvalue problems, Math. Bohem.120 (2) (1995), 169–195.
[DH] Y. Du and Q. Huang,Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal.31 (1) (1999), 1–18.
[DM] Y. Du and L. Ma,Logistic type equations on R N by a squeezing method involving boundary blow-up solutions, J. London Math. Soc.64 (2001), 107–124.
[Gu1] Z. M. Guo,Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems, Nonlinear Anal.18 (1992), 957–971.
[Gu2] Z. M. Guo,Uniqueness and flat core of positive solutions for quasilinear elliptic eigenvalue problems in general smooth domains, Math. Nachr.243 (2002), 43–74.
[GW1] Z. M. Guo and J. R. L. Webb,Uniqueness of positive solutions for quasilinear, elliptic equations when a parameter is large, Proc. Royal Soc. Edinburgh124A (1994), 189–198.
[GW2] Z. M. Guo and J. R. L. Webb,Large and small solutions of a class of quasilinear elliptic eigenvalue problems, J. Differential Equations180 (2002), 1–50.
[Hs] P. Hess,Periodic-Parabolic Boundary Value Problems and Positivity, Longman Scientific and Technical, Harlow, UK, 1991.
[Ka1] B. Kawohl,Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math.1150, Springer-Verlag, Berlin, 1985.
[Ka2] B. Kawohl,On a family of torsional creep problems, J. Reine Angew. Math.410 (1990), 1–22.
[Ke] J. B. Keller,On solutions of Δu=f(u), Comm. Pure Appl. Math.10 (1957), 503–510.
[Lin] P. Lindqvist,On the equation div(⋎Δu⋎p−2Δu)+λ⋎u⋎p−2 u=0. Proc. Amer. Math. Soc.109 (1990), 157–164.
[LN] O. A. Ladyzhenskaya and N.N. Ural'tseva,Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
[Ma] J. Matero,Quasilinear elliptic equations with boundary blow-up, J. Analyse Math.69 (1996), 229–246.
[ML] J. Garcia-Melian and J. Sabina de Lis,Stationary profiles of degenerate problems when a parameter is large, Differential Integral Equations13 (2000), 1201–1232.
[MV1] M. Marcus and L. Véron,Uniqueness of solutions with blowup at the boundary for a class of nonlinear elliptic equations, C.R. Acad. Sci. Paris, Série I317 (1993), 559–563.
[MV2] M. Marcus and L. Véron,Uniqueness, and asymptotic behaviour of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire,14 (1997), 237–274.
[Ou] T. Ouyang,On the positive solutions of semilinear equations Δu+λu−hu p=0on the compact manifolds, Trans. Amer. Math. Soc.,331 (1992), 503–527.
[Os] R. Osserman,On the inequality Δu≥f(u), Pacific J. Math.7 (1957), 1641–1647.
[PR] M. C. Pélissier and M. L. Reynaud, Etude d'un modèle mathématique découlement de glacier, C. R. Acad. Sci. Paris Sér. I Math.279 (1974), 531–534.
[SW] R. E. Showalter and N. J. Walkington,Diffusion of fluid in a fissured medium with microstructure,, SIAM J. Math. Anal.22 (1991), 1702–1722.
[Ta] S. Takeuchi,Positive solutions of a degenerate elliptic equation with logistic reaction, Proc. Amer. Math. Soc.129 (2), (2000), 433–441.
[To1] P. Tolksdorf,On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations,8 (1983), 773–817.
[To2] P. Tolksdorf,Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations,51 (1984), 126–150.
[Va] J. L. Vazquez,A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim.12 (1984), 191–202.
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Du, Y., Guo, Z. Boundary blow-up solutions and their applications in quasilinear elliptic equations. J. Anal. Math. 89, 277–302 (2003). https://doi.org/10.1007/BF02893084
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DOI: https://doi.org/10.1007/BF02893084