A TRIBUTE TO PATRIZIA PUCCI 3
(1) can be partly weakened to the existence of 0 r R such that
f(x) ≥ a for r ≤ x ≤ R, f(0) ≤ a, f(e) ≤ a.
In other words, the mountain has to be wide enough when it is not high enough.
Pucci and Serrin also have analyzed the fine structure of the critical set under
the mountain pass geometry, and have given variants of their results for periodic
A famous and widely used formula for the solutions of the semilinear Dirichlet
problem with continuous f
Δu + f(u) = 0 in Ω, u = 0 on ∂Ω (2)
on a bounded star-shaped domaine Ω ⊂ Rn with outer normal ν, is Pohozaev
· ν) ds =
n − 2
uf(u) − nF (u) dx,
where F (u) :=
f(s) ds. It implies that problem (2) has no nontrivial solution
when f verifies the condition
(n − 2)uf(u) − 2nF (u) 0 for u = 0.
In 1986, Pucci and Serrin have found an illuminating and fruitful extension of
Pohozaev identity to general Dirichlet variational problem of the form
F(x, u, Du) dx = 0.
Here F : Ω × R × Rp → R is a suﬃciently smooth function, and the corresponding
Euler-Lagrange equation, for Dirichlet conditions, has the form
div [Fp(x, u, Du)] = Fu(x, u, Du) in Ω, u = 0 on ∂Ω. (3)
The corresponding generalization of Pohozaev identity, now referred as Pucci-
Serrin’s variational identity, is (with a any constant),
[F(x, u, Du) − (Du · Fp(x, 0,Du))](x · ν) ds
[nF(x, u, Du) + (x · Fx(x, u, Du)) − auFu(x, u, Du)
− (a + 1)(Du · Fp(x, u, Du))] dx.
It implies that if, for some a ∈ R,
(p · Fp(x, 0,p)) − F(x, 0,p) ≥ 0 on ∂Ω ×
nF(x, u, p) + (x · Fx(x, u, p)) − auFu(x, u, p) − (a + 1)(p · Fp(x, u, p)) ≥ 0
on Ω × R ×
and that either u = 0 or p = 0 whenever equality holds in the last
inequality, then the Dirichlet problem for (3) has no non-trivial classical solution.
This result, and its variants for systems, not only reveals the variational nature
of Pohozaev’s contribution, but its generality has allowed applications to other
equations or systems of equations, like those involving p-Laplacian operator, mean
curvature, biharmonic operator, problems on unbounded domains. It has inspired
and still inspires a wide literature.