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Existence theorems for a semilinear elliptic boundary value problem

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Let Ω be a bounded domain in $ℝ^n$, n ≥ 3, with a smooth boundary ∂Ω; let L be a linear, second order, elliptic operator; let f and g be two real-valued functions defined on Ω × ℝ such that f(x,z) ≤ g(x,z) for almost every x ∈ Ω and every z ∈ ℝ. In this paper we prove that, under suitable assumptions, the problem
{ f(x,u) ≤ Lu ≤ g(x,u)   in Ω,
   u = 0     on ∂Ω,
has at least one strong solution $u ∈ W^{2,p}(Ω) ∩ W^{1,p}_0(Ω). Next, we present some remarkable special cases.
Twórcy
  • Dipartimento di Matematica Università di Catania Viale A. Doria 6 95125 Catania, Italy
Bibliografia
  • [1] R. A. Adams, Sobolev Spaces, Academic Press, 1975.
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  • [3] H. Amann, Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problems, in: Lecture Notes in Math. 543, Springer, 1976, 1-55.
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  • [9] J. K. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567-597.
  • [10] S. A. Marano, Existence theorems for a multivalued boundary value problem, Bull. Austral. Math. Soc. 45 (1992), 249-260.
  • [11] J. Mawhin, J. R. Ward, Jr., and M. Willem, Variational methods and semilinear elliptic equations, Arch. Rational Mech. Anal. 95 (1986), 269-277.
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  • [13] O. Naselli Ricceri and B. Ricceri, An existence theorem for inclusions of the type Ψ(u)(t) ∈ F(t,Φ(u)(t)) and application to a multivalued boundary value problem, Appl. Anal. 38 (1990), 259-270.
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  • [17] M. Zuluaga, Existence of solutions for some elliptic problems with critical Sobolev exponents, Rev. Mat. Iberoamericana 5 (1989), 183-193.
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