Article - preprint
Let Ω be a bounded region in R^n. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem:
Δu(x) + g(x, u(x)) + f(x, u(x), ∇u(x)) = 0 ; x є Ω
u(x) = 0 ; x є ∂Ω
where Δ is the Laplacian operator, g : Ω x R → R and f : Ω x Rn+1 → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.
© 1979 Canadian Mathematical Society
Castro, Alfonso. “A semilinear Dirichlet problem” Canadian J. of Math., Vol. XXXI, No. 2(1979), pp. 337-340.