The Dirichlet Problem for Second Order Semilinear Elliptic and Parabolic Equations

In the present paper the Dirichlet problem for semilinear elliptic and parabolic equations in general form is considered. New condition guaranteeing the global solvability of this problem for a wide class of superlinear sources, including e u and |u|p−1u , p > 1 , is formulated. For sublinear case (for example ln(1+ |u|) or |u|p−1u , p < 1) this condition is automatically fulfilled. Our approach gives new a priori estimate of the solution for superlinear, sublinear and linear case as well. 0. Introduction and Main Results I. Elliptic case. Consider semilinear strictly elliptic equation n ∑ i, j=1 ai j(x)uxix j + n ∑ i=1 bi(x)uxi + c(x)g(u) = f (x) in Ω, (0.1) coupled with boundary condition u(x) ∣∣∣ ∂Ω = φ(s). (0.2) Here Ω is a bounded domain in Rn . Concerning the nonlinear term g we assume that |g(ξ )| g(η) for all ξ and η such that |ξ | η . (0.3) For example, functions g(u) = |u|q with q ∈ (0,1) , g(u) = ln(1 + |u|) , g(u) = up for p 1 integer, or g(u) = |u|p−1u for arbitrary p 0 as well as g(u) = eu satisfy condition (0.3). Recall that (0.1) is strictly elliptic if n ∑ i, j=1 ai j(x)ξiξ j ε|ξ |2 for any x ∈Ω, ξ ∈ R, (0.41) Mathematics subject classification (2000): 35J25, 35K20.