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A Semilinear Dirichlet Problem
Published online by Cambridge University Press: 20 November 2018
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Introduction and notations. Let Ω be a bounded region in Rn. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem
(I)
where Δ is the Laplacian operator, g : Ω × R → R and f : Ω × Rn+1 → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.
We let λ1 < λ2 ≦ … ≦ λm ≦ … denote the sequence of numbers for which the problem
(II)
has nontrivial weak solutions.
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