The goal of this correction is to correct a mistake that appears in the proof of both lemma 3.9 and 3.10 in the article Mihăilescu & Rădulescu (2006).

The mistake in the proof of lemma 3.9 and 3.10 is mainly owing to the fact that we wrongly computed the expression of function *G*(*x*,*t*) when *t*>*u*_{1}(*x*). The correct computation reads as follows:
1.1
We remember the result of Mihăilescu & Rădulescu (2006, lemma 3.9) and provide the correct proof.

### Lemma 3.9

*There exists ρ*∈(0,∥*u*_{1}∥) *and a* > 0 *such that J(u) ≥ a, for all u∈E with* ∥*u*∥=*ρ*.

### Proof.

Let *u*∈*E* be fixed, such that ∥*u*∥<1. It is clear that
Define
If *x*∈*Ω*\*Ω*_{u} then and we have
If *x*∈*Ω*_{u}∩{*x*;*u*_{1}(*x*)<*u*(*x*)<1} then
Define
Thus, provided that ∥*u*∥<1 by condition (A5), the above estimates and relation (1.5) we get
1.2
Since , it follows that *p*^{+}<*p*^{☆}(*x*) for all *x*∈** Ω**. Then, there exists

*q*∈(

*p*

^{+},

*Np*

^{−}/(

*N*−

*p*

^{−})) such that

*E*is continuously embedded in

*L*

^{q}(

*Ω*). Thus, there exists a positive constant

*C*>0 such that Using the definition of

*G*and the above estimate, we obtain 1.3 where

*D*and

*D*

_{1}are two positive constants. Combining inequalities (1.2) and (1.3), we find that for a small enough we have and taking into account that

*q*>

*p*

^{+}, we infer that the conclusion of this lemma holds true. ■

Regarding the proof of Mihăilescu & Rădulescu (2006, lemma 3.10) it can be carried out with the same arguments as in the original proof, but the expression of *G*(*x*,*t*) with *t*>*u*_{1}(*x*) should be replaced in the proof with the one given by relation (1.1).

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