On the inequality Δ≧f(u)

Abstract

We are interested in solutions of the non-linear differential inequality (1) Δ≧f(u) where u(x1,…, xn) is to be defined in some region of Euclidean w-space and is the Laplacian of u. Wittich [5] considered the corresponding equation (la) Δu = f(u) in two dimensions and found conditions on f(u) which guarantee that (la) has no solution valid in the whole plane. Haviland [1] found a slightly weaker result in 3 dimensions, and Walter [4] generalized Wittich’s theorem to w-dimensions. The method is essentially the same in all three papers, resulting on the one hand in the requirement that the function f(u) be convex, and on the other hand in a rather involved argument for the w-dimensional case. The proofs do extend immediately to the inequality (1). © 1957 by Pacific Journal of Mathematics.