A version of Pohožaev’s non-existence result via maximum principles^{∗}

Wolfgang Reichel, Mathematisches Institut Universität zu Köln Weyertal 86 - 90 D - 50931 Köln

                wreichel@MI.Uni-Koeln.de

In 1965, S.I. Pohožaev discovered an important identity: if Ω ⊂ IR^{N} is a bounded, smooth domain and if u ∈ C^{2}(Ω) ∩ C^{1}(\overline{Ω}) satisfies

Δu + f(u) = 0 in Ω, u = 0 on ∂Ω (1)

then

∫_{∂Ω} \frac{1}{2}(∇u)^{2}x · ν dσ = ∫_{∂Ω} NF (u) + \frac{2 − N}{2} f(u)u dx, (2)

where ν is the exterior normal and F (s) = ∫_{0}^{s} f(t) dt. The proof is by integration by parts. An important application is non-existence of positive solutions of (1), if Ω is strictly star-shaped and

(3) F (s)/s^{\frac{2N}{N−2}} is nondecreasing in s ∈ (0, ∞).

We prove non-existence via maximum principles under the assumption that

(4) f(s)/s^{\frac{N+2}{N−2}} is nondecreasing in s ∈ (0, ∞).

Similar non-existence conditions for non-autonomous problems Δu + g(x, u) = 0 can also be formulated. Condition (4) implies (3), but our approach needs less regularity of Ω and u, and it sheds new light on the supercriticality and starshapedness assumptions.

∗joint work with Henghui Zou, University of Alabama, Birmingham, U.S.A.