Topological Methods in Nonlinear Analysis

Topological structure of the solution set of singular equations with sign changing terms under Dirichlet boundary condition

José V. Gonçalves, Marcos R. Marcial, and Olimpio H. Miyagaki

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In this paper we establish existence of connected components of positive solutions of the equation $ -\Delta_{p} u = \lambda f(u)$ in $\Omega$, under Dirichlet boundary conditions, where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$, $\Delta_{p}$ is the $p$-Laplacian, and $f \colon (0,\infty) \rightarrow \mathbb{R} $ is a continuous function which may blow up to $\pm \infty$ at the origin.

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Topol. Methods Nonlinear Anal., Volume 47, Number 1 (2016), 73-89.

First available in Project Euclid: 23 March 2016

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Gonçalves, José V.; Marcial, Marcos R.; Miyagaki, Olimpio H. Topological structure of the solution set of singular equations with sign changing terms under Dirichlet boundary condition. Topol. Methods Nonlinear Anal. 47 (2016), no. 1, 73--89. doi:10.12775/TMNA.2015.091.

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  • A. Anane, Simplicité et isolation de la primiére valeur propre du p-Laplacien avec poids, C.R. Acad. Sci. Paris Sér. I (1987), 725–728.
  • L. Boccardo, F. Murat and J.P. Puel, Résultats d'existence pour certains problémes elliptiques quasilinéaires, Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 (1984), 213–235.
  • H. Brézis, Functional Analysis, Sobolev Spaces and partial differential equations, Springer–Verlag, Berlin, (2011).
  • S. Carl and K. Perera, Generalized solutions of singular $p$-Laplacian problems in $\mathbb{R}^N$, Nonlinear Stud. 18 (2011), 113–124.
  • M.G. Crandall, P.H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193–222.
  • K. Deimling, Nonlinear Functional Analysis, Springer–Verlag, Berlin, 1985.
  • J.I. Diaz, J. Hernandez and F.J. Mancebo, Branches of positive and free boundary solutions for some singular quasilinear elliptic problem, J. Math. Anal. Appl. 352 (2009) 449–474.
  • J.I. Diaz, J.M. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations 12 (1987), 1333–1344.
  • E. DiBenedetto, $C^{1+\alpha}$-local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827–850.
  • M. Ghergu and V. Radulescu, Sublinear singular elliptic problems with two parameters, J. Differential Equations 195 (2003), 520–536.
  • J. Giacomoni, I. Schindler and P. Takac, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 117–158.
  • D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer–Verlag, New York, 1983.
  • J.V. Goncalves, M.C. Rezende and C.A. Santos, Positive solutions for a mixed and singular quasilinear problem, Nonlinear Anal. 74 (2011), 132–140.
  • D.D. Hai, Singular boundary value problems for the $p$-Laplacian, Nonlinear Anal. 73 (2010), 2876–2881.
  • ––––, On a class of singular $p$-Laplacian boundary value problems, J. Math. Anal. Appl. 383 (2011), 619–626.
  • A.C. Lazer and P.J. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc. 111 (1991), 721–730.
  • G.M. Liebermann, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.
  • N.H. Loc and K. Schmitt, Boundary value problems for singular elliptic equations, Rocky Mountain J. Math. 41 (2011), 555–572.
  • A. Mohammed, Positive solutions of the $p$-Laplace equation with singular nonlinearity, J. Math. Anal. Appl. 352 (2009), 234–245.
  • I. Peral, Multiplicity of Solutions for the $p$-Laplacian, Second School on Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, Italy, 1997.
  • J. Shi and M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A 138 (1998), 1389–1401.
  • J. Simon, Regularité de la solution d'une equation non linéaire dans $\R^N$, Lecture Notes in Mathematics, vol. 665, Springer–Verlag, New York, 1978.
  • J.X. Sun and F.M. Song, A property of connected components and its applications, Topology Appl. 125 (2002), 553–560.
  • P. Tolksdorff, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126–150.
  • J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202.
  • G.T. Whyburn, Topological Analysis, Princeton University Press, Princeton NJ, 1955.