Hendrik J. Kuiper

Existence and comparison theorems for nonlinear diffusion systems

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v1(x), v2(x),…, vm(x)) from Ω ⊂ Rn into Rm which satisfy ψi(x, t) ⩽ vi(x) ⩽ θi(x, t) for all x ϵ Ω and 1 ⩽ i ⩽ m, where ψiand θi are extended real-valued functions on \gW × [0, T). We find conditions which will ensure that a solution U(x, t) ≡ (u1(x, t), u2(x, t),…, um(x, t)) which satisfies U(x, 0) ϵ K(0) will also satisfy U(x, t) ϵ K(t) for all 0 ⩽ t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.

On the number of radially symmetric solutions to dirichlet problems with jumping nonlinearities of superlinear order

This paper is concerned with the multiplicity of radially symmetric solutions u(x) to the Dirichlet problem ∆u+ f(u) = h(x) + cφ(x) on the unit ball Ω ⊂ RN with boundary condition u = 0 on ∂Ω. Here φ(x) is a positive function and f(u) is a function that is superlinear (but of subcritical growth) for large positive u, while for large negative u we have that f ′(u) < μ, where μ is the smallest positive eigenvalue for ∆ψ + μψ = 0 in Ω with ψ = 0 on ∂Ω. It is shown that, given any integer k ≥ 0, the value c may be chosen so large that there are 2k+1 solutions with k or less interior nodes. Existence of positive solutions is excluded for large enough values of c.