Erratum to “Positive Solutions for a Class of Quasilinear Schrödinger Equations with Nonlocal Term” [Journal of Applied Mathematics and Physics (2022) 347-359] ()
1. Introduction
where
,
,
,
, the function
, g is a
even function with
for all
,
,
,
.
2. Preliminary Results
Next, we introduce some minimization with corresponding energy functional and define
where
and
We also define
where
and
Proof: For any
, let
, from the definition of g, we get
and
so
. It follow that
, hence
, moreover, for any
, let
, then
. We assume
, since
,
, then
. By Hardy-Little-Sobolev-inequality, we have
n
The proof of Lemma 2.4
Proof: (1) For any
, we have
, where
, similarly as the proof of Lemma2.3, by Hardy-Little-Sobolev-inequality, we have
With the proof of continuity, note that F consist of three terms. By Lemma 1.1, we need to check the convolution term only. Using Hardy-Little-Sobolev-inequality
and
where
. We know
if
. So
is bounded in
. By Sobolev embedding theorem and Lemma 3.4 [22]
For (2) we consider the second and the third terms of the functional F, we see for
, using Hölder inequality, we get
Using the definition of g and Lemma 1.1, we know
By the dominated convergence theorem
For the third term, we have
and
where
,
. Since
if
,
embedding into
is compact and
is bounded in
. Using Lemma 3.4 [22], we know
By Lemma 1.1
from Sobolev embedding theorem, we get
is a continuous linear functional on
.n
3. Main Conclusion
Remark 3.1. From assumption of V, we know
embedding into
is compact. In the process of the proof of theorem 3.1, it is important for us to construct auxiliary function, then by implicit function theorem to prove it and lemma 3.4 [22] play a great role in this paper. Moreover, when
is a open question for Equation (1.1), someone could do it if they are interested.
Proof of Theorem 3.1: Step 1: By the assumptions of (V1) or (V2),
is achieved at some
with
.
Let
be a minimizing sequence for
. Set
, then
is a minimizing sequence for
. We can assume
. It shows that
, so there exist
such that
By Hölder inequality and Hardy-Little-Sobolev-inequality,
where
,
,
,
,
, then
Taking
small enough such that
. It implies that
is bounded in
. By the compact embedding result from
into
for
. We may assume that
in
,
in
for
and
a.e
. Hence
, since
,
and
. Similarly as the proof of Lemma 2.4 (1), we have
Hence
Step 2: Set
for
, then
.
In fact, for any
, by Lemma 1.1 and Hölder’s inequality, we have
then
.
(1)
and
Since
in
,
embedding into
is compact and
is bounded in
, where
. By
and Lemma 3.4 [22], we have
then
for
.
Step 3: For any
, there exist
such that
is a weak solution of Equation (1.1) with
. In fact, by lemma2.4,
Take limit
, we get
, by arbitrariness of v, one has
. It follows that
, for every
. Set
be such that
, for every
, let
Then
, it means
, i.e.
Put
, we have
n