DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS
Volume 34, Number 11, November 2014
doi:10.3934/dcds.2014.34.4987
pp. 4987–4987
ERRATUM TO: “ON A FUNCTIONAL SATISFYING A WEAK
PALAIS-SMALE CONDITION”
A. Azzollini
Dipartimento di Matematica ed Informatica, Università degli Studi della Basilicata
Via dell’Ateneo Lucano 10, I-85100 Potenza, Italy
Remark 3 in the paper [1] is incorrect. Indeed Sobolev embedding does not hold,
since it is not true, in general, that there exists a continuous extension operator
T : E(Ω) → E(RN ), being E(Ω) the space defined at page 1834.
Actually, the right statement should have been: if p 6 α 6 p∗ , the set Wr is included
∗
in Ls (RN ) for any s ∈ [α, p∗ ]. To prove this, it is enough to see that Wr ⊂ Lp (RN ).
So, take a function u in Wr and set Λu = {x ∈ Ω |u(x)| > 1} (with Λcu we denote
∗
∗
its complementary). We are going to prove that u ∈ Lp (Λu )∩Lp (Λcu ). By estimate
(6) at page 1835 we deduce that Λu is bounded, so we can take M > 0 sufficiently
large such that Λu ⊂ BM , being BM the M −radius ball centered in 0. By [2,
Proposition 2.1], certainly ∇u ∈ Lp (BM ). On the other hand, since u ∈ Lα (RN ),
we have that u ∈ Lα (BM ). Then, by embedding Lα (BM ) ,→ Lp (BM ), we deduce
∗
that u ∈ W 1,p (BM ). By Sobolev embedding theorems u ∈ Lp (BM ) and, of course,
∗
∗
u ∈ Lp (Λu ). It remains to prove that u ∈ Lp (Λcu ). Actually, since u ∈ L∞ (Λcu )
and we know that u ∈ Lα (Λcu ), by interpolation we conclude.
REFERENCES
[1] A. Azzollini, On a functional satisfying a weak Palais-Smale condition, DCDS series A, 34
(2014), 1829–1840.
[2] M. Badiale, L. Pisani and S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic
equations, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 369–405.
E-mail address: [email protected]
The author is supported by M.I.U.R. - P.R.I.N. “Metodi variazionali e topologici nello studio
di fenomeni non lineari”.
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