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Elliptic Partial Differential Equation Involving a Singularity and a Radon Measure


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1 Department of Mathematics, National Institute of Technology, Rourkela, India
     

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The aim of this paper is to prove the existence of solution for a partial differential equation involving a singularity with a general nonnegative, Radon measure μ as its nonhomogenous term which is given as

−Δu = f(x)h(u) + μ in Ω,

u = 0 on ∂Ω,

u > 0 on Ω,

where Ω is a bounded domain of RN, f is a nonnegative function over Ω.

Keywords

Elliptic PDE, Sobolev Space, Schauder Fixed Point Theorem
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  • D. Arcoya, J. Carmona, T. Leonori, P. J. Martinez-Aparicio, L. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations, Journal of Differential Equations, 246 (2009), 4006-4042.
  • P. Benilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre and J. L. Vazquez, An L1 theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1995), 241-273.
  • M. Bhakta and M. Marcus, Reduced limit for semilinear boundary value problems with measure data, J. Differential Equations, 256 (2014), 2691-2710.
  • L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var., 37 (2010), 363-380.
  • H. Brezis and X. Cabre, Some simple nonlinear PDE’s without solutions, Bollettino dell’Unione Matematica Italiana, Serie 8, 1-B (1998), 223-262.
  • A. Canino, Minimax methods for singular elliptic equations with an application to a jumping problem, Journal of Differential Equations, 221(1) (2006), 210-223.
  • A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations, Journal of Convex Analysis, 11 (2004), 147-162.
  • A. Canino, M. Grandinetti and B. Sciunzi, Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities, Journal of Differential Equations, 255 (2013), 4437-4447.
  • L. M. De Cave and F. Oliva, Elliptic equations with general singular lower order term and measure data, Nonlinear Analysis, 128 (2015), 391-411.
  • L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (2010).
  • G. B. Folland, Real analysis (Modern techniques and their applications), Second edition, A Wiley-Interscience publication, 2nd Edition.
  • J. A. Gatica, V. Oliker and P. Waltman, Singular nonlinear boundary-value problems for second-order ordinary differential equations, Journal of Differential Equations, 79(1) (1989), 62-78.
  • D. Giachetti, P. J. Martinez-Aparicio and F. Murat, A semilinear elliptic equation with a mild singularity at u = 0: Existence and homogenization, Journal de Mathematiques Pures et Appliquees, 107 (2017), 41-77.
  • D. Giachetti, P. J. Martinez-Aparicio and F. Murat, Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0, Annali della Scuola Normale Superiore di Pisa, 2017.
  • A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111(3) (1991), 721-730.
  • J. Leray and J. L. Lions, Quelques resultats de Visik sur les problemes elliptiques non lineaires par les methodes de Minty-Browder, Bulletin de la S.M.F., 93 (1965), 97-107.
  • M. Marcus and L. Veron, Nonlinear second order elliptic equations involving measures, de Gruyter Series in Nonlinear Analysis and Applications, 21 (2013).
  • M. Montenegro and A. C. Ponce, The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc., 136(7) (2008), 2429-2438.
  • F. Oliva and F. Petitta, Finite and infinite energy solutions of singular elliptic problems: Existence and Uniqueness, Journal of Differential Equations, 264 (2018), 311-340.
  • F. Oliva and F. Petitta, On singular elliptic equations with measure sources, ESAIM: Control, Optimisation and Calculus of Variations, 22 (2016), 289-308.
  • G. Stampacchia, Le probleme de Dirichlet pour lesequations elliptiques du second ordre a coefficientes discontinus, Annales de l'institut Fourier, 15(1) (1965), 189-257.
  • Y. Sun and D. Zhang, The role of the power 3 for elliptic equations with negative exponents, Calculus of Variations, 49 (2014), 909-922.
  • S. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Analysis, Theory, Methods & Applications, 3(6) (1979), 897-904.

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  • Elliptic Partial Differential Equation Involving a Singularity and a Radon Measure

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Authors

Akasmika Panda
Department of Mathematics, National Institute of Technology, Rourkela, India
Sekhar Ghosh
Department of Mathematics, National Institute of Technology, Rourkela, India
Debajyoti Choudhuri
Department of Mathematics, National Institute of Technology, Rourkela, India

Abstract


The aim of this paper is to prove the existence of solution for a partial differential equation involving a singularity with a general nonnegative, Radon measure μ as its nonhomogenous term which is given as

−Δu = f(x)h(u) + μ in Ω,

u = 0 on ∂Ω,

u > 0 on Ω,

where Ω is a bounded domain of RN, f is a nonnegative function over Ω.

Keywords


Elliptic PDE, Sobolev Space, Schauder Fixed Point Theorem

References