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

functionals.

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

identity

−

1

2

∂Ω

|Du|2(x

· ν) ds =

Ω

n − 2

2

uf(u) − nF (u) dx,

where F (u) :=

u

0

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

δ

Ω

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),

∂Ω

[F(x, u, Du) − (Du · Fp(x, 0,Du))](x · ν) ds

=

Ω

[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 ×

Rn,

nF(x, u, p) + (x · Fx(x, u, p)) − auFu(x, u, p) − (a + 1)(p · Fp(x, u, p)) ≥ 0

on Ω × R ×

Rn,

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.