Erratum to “Positive Solutions for a Class of Quasilinear Schrödinger Equations with Nonlocal Term” [Journal of Applied Mathematics and Physics (2022) 347-359]

Abstract

The original online version of this article (Liao, P., Ping, R. and Chen, S. (2022) Positive Solutions for a Class of Quasilinear Schr?dinger Equations with Nonlocal Term. Journal of Applied Mathematics and Physics, 10, 347-359. https://doi.org/10.4236/jamp.2022.102027) needs some further amendments and clarification.

Keywords

Erratum

Share and Cite:

Liao, P. , Ping, R. and Chen, S. (2022) Erratum to “Positive Solutions for a Class of Quasilinear Schrödinger Equations with Nonlocal Term” [Journal of Applied Mathematics and Physics (2022) 347-359]. Journal of Applied Mathematics and Physics, 10, 1297-1303. doi: 10.4236/jamp.2022.104091.

1. Introduction

where N 3 , 0 < μ < N , 1 q N N 2 , 2 N μ N p < 2 N μ N 2 , the function V C ( N , + ) , g is a C 1 even function with g ( t ) 0 for all t > 0 , g ( 0 ) = 0 , lim t + g ( t ) = a , 0 < a < 1 .

2. Preliminary Results

Next, we introduce some minimization with corresponding energy functional and define

m b = inf u M b E ( u ) ,

where

M b = { u H 1 ( N ) : u L q + 1 = b } , b > 0 ,

and

E ( u ) = 1 2 N [ g 2 ( u ) | u | 2 + V ( x ) u 2 ] λ 2 p N [ ( | x | | u | p ) | u | p ] .

We also define

ω b = inf v W b F ( v ) ,

where

W b = { v H V 1 ( N ) : G 1 ( v ) L q + 1 = b } , b > 0 ,

and

F ( v ) = 1 2 N ( | v | 2 + V ( x ) G 1 ( v ) 2 ) λ 2 p N [ ( | x | | G 1 ( v ) | p ) | G 1 ( v ) | p ] .

Proof: For any v W b , let u = G 1 ( v ) , from the definition of g, we get

N | u | 2 = N | v | 2 g 2 ( G 1 ( v ) ) 1 a 2 N | v | 2 < + ,

and

N u 2 N V ( x ) G 1 ( v ) 2 < + ,

so u M b . It follow that F ( v ) = E ( G 1 ( v ) ) = E ( u ) m b , hence ω b m b , moreover, for any u M b , let v = G ( u ) , then u = G 1 ( v ) . We assume E ( u ) < + , since u H 1 ( N ) , 2 < 2 N p 2 N μ < 2 , then u L 2 N p 2 N μ ( N ) . By Hardy-Little-Sobolev-inequality, we have

1 2 N [ g 2 ( u ) | u | 2 + V ( x ) u 2 ] = E ( u ) + λ 2 p N ( | x | | u | p ) | u | p E ( u ) + λ C 2 p ( N | u | 2 N p 2 N μ ) 2 N μ N < + . n

The proof of Lemma 2.4

Proof: (1) For any v H V 1 ( N ) , we have N | G 1 ( v ) | 2 N p 2 N μ C N | v | 2 N p 2 N μ < + , where 2 2 N p 2 N μ < 2 * , similarly as the proof of Lemma2.3, by Hardy-Little-Sobolev-inequality, we have

F ( v ) = 1 2 N ( | v | 2 + V ( x ) G 1 ( v ) 2 ) λ 2 p N ( | x | | G 1 ( v ) | p ) | G 1 ( v ) | p 1 2 a 2 N ( | v | 2 + V ( x ) v 2 ) + λ C 2 p ( N | G 1 ( v ) | 2 N p 2 N μ ) 2 N μ N 1 2 a 2 N ( | v | 2 + V ( x ) v 2 ) + λ C 2 p ( N | v | 2 N p 2 N μ ) 2 N μ N < + .

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

λ 2 p | N ( | x | | G 1 ( v n ) | p ) | G 1 ( v n ) | p N ( | x | | G 1 ( v ) | p ) | G 1 ( v ) | p | λ 2 p ( | N | x | ( | G 1 ( v n ) | p | G 1 ( v ) | p ) | G 1 ( v n ) | p | + | N ( | x | | G 1 ( v ) | p ) ( | G 1 ( v n ) | p | G 1 ( v ) | p ) | ) C | ( N | G 1 ( v n ) | p r ) 1 r ( N ( | G 1 ( v n ) | p | G 1 ( v ) | p ) r ) 1 r | + C | ( N | G 1 ( v ) | p r ) 1 r ( N ( | G 1 ( v n ) | p | G 1 ( v ) | p ) r ) 1 r | ,

and

| | G 1 ( v n ) | p | G 1 ( v ) | p | r C ( | v n | p r + | v | p r ) ,

where r = 2 N 2 N μ . We know v n v H V 1 ( N ) 0 if n + . So { v n } is bounded in H V 1 ( N ) . By Sobolev embedding theorem and Lemma 3.4 [22]

| N ( | x | | G 1 ( v n ) | p ) | G 1 ( v n ) | p N ( | x | | G 1 ( v ) | p ) | G 1 ( v ) | p | 0, n + .

For (2) we consider the second and the third terms of the functional F, we see for ϕ H V 1 ( N ) , using Hölder inequality, we get

| 1 2 t N V ( x ) ( G 1 ( v + t ϕ ) 2 G 1 ( v ) 2 ) N V ( x ) G 1 ( v ) g ( G 1 ( v ) ) ϕ | = | 0 1 d s N V ( x ) ( G 1 ( v + t s ϕ ) g ( G 1 ( v + t s ϕ ) ) G 1 ( v ) g ( G 1 ( v ) ) ) ϕ | 0 1 d s ( N V ( x ) | G 1 ( v + t s ϕ ) g ( G 1 ( v + t s ϕ ) ) G 1 ( v ) g ( G 1 ( v ) ) | 2 ) 1 2 0 1 d s ( N V ( x ) ϕ 2 ) 1 2 .

Using the definition of g and Lemma 1.1, we know

| G 1 ( v + t s ϕ ) g ( G 1 ( v + t s ϕ ) ) G 1 ( v ) g ( G 1 ( v ) ) | 2 | G 1 ( v + t s ϕ ) + G 1 ( v ) | 2 C ( | G 1 ( v + t s ϕ ) | 2 + | G 1 ( v ) | 2 ) C ( | v + t s ϕ | 2 + | v | 2 ) C ( | v | 2 + | ϕ | 2 ) .

By the dominated convergence theorem

| 0 1 d s N V ( x ) ( G 1 ( v + t s ϕ ) g ( G 1 ( v + t s ϕ ) ) G 1 ( v ) g ( G 1 ( v ) ) ) ϕ | 0, t 0.

For the third term, we have

λ | N [ | x | μ | G 1 ( v n ) | p ] | G 1 ( v n ) | p 2 G 1 ( v n ) g ( G 1 ( v n ) ) ϕ N [ | x | μ | G 1 ( v ) | p ] | G 1 ( v ) | p 2 G 1 ( v ) g ( G 1 ( v ) ) ϕ | λ | N [ | x | μ ( | G 1 ( v n ) | p | G 1 ( v ) | p ) ] | G 1 ( v n ) | p 2 G 1 ( v n ) g ( G 1 ( v n ) ) ϕ | + λ | N ( | x | μ | G 1 ( v ) | p ) ( | G 1 ( v n ) | p 2 G 1 ( v n ) g ( G 1 ( v n ) ) | G 1 ( v ) | p 2 G 1 ( v ) g ( G 1 ( v ) ) ) ϕ | C N ( ( | G 1 ( v n ) | p | G 1 ( v ) | p ) r ) 1 r N ( | | G 1 ( v n ) | p 2 G 1 ( v n ) g ( G 1 ( v n ) ) ϕ | r ) 1 r + C ( N | G 1 ( v ) | p r ) 1 r ( N ( | G 1 ( v n ) | p 2 G 1 ( v n ) g ( G 1 ( v n ) ) | G 1 ( v ) | p 2 G 1 ( v ) g ( G 1 ( v ) ) ) p p 1 r ) p 1 p r ( N | ϕ | p r ) 1 p r .

and

| | G 1 ( v n ) | p | G 1 ( v ) | p | r C ( | v n | p r + | v | p r ) ,

| | G 1 ( v n ) | p 2 G 1 ( v n ) g ( G 1 ( v n ) ) | G 1 ( v ) | p 2 G 1 ( v ) g ( G 1 ( v ) ) | p p 1 r C ( | v n | p r + | v | p r ) ,

where r = 2 N 2 N μ , 2 p r < 2 * . Since v n v H V 1 ( N ) 0 if n + , H V 1 ( N ) embedding into L r ( N ) is compact and { v n } is bounded in H V 1 ( N ) . Using Lemma 3.4 [22], we know

| N [ | x | μ | G 1 ( v n ) | p ] | G 1 ( v n ) | p 2 G 1 ( v n ) g ( G 1 ( v n ) ) ϕ N [ | x | μ | G 1 ( v ) | p ] | G 1 ( v ) | p 2 G 1 ( v ) g ( G 1 ( v ) ) ϕ | 0, n + .

By Lemma 1.1

| F ( v ) , ϕ | = | N v ϕ + N V ( x ) G 1 ( v ) g ( G 1 ( v ) ) ϕ N [ | x | μ | G 1 ( v ) | p ] | G 1 ( v ) | p 2 G 1 ( v ) g ( G 1 ( v ) ) ϕ | C v H V 1 ( N ) ϕ H V 1 ( N ) + C v L p r ( N ) p 1 ϕ L p r ( N ) .

from Sobolev embedding theorem, we get F ( v ) is a continuous linear functional on H V 1 ( N ) .n

3. Main Conclusion

Remark 3.1. From assumption of V, we know H V 1 ( N ) embedding into L p ( N ) 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 q 2 * 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), ω b is achieved at some 0 v b W b with v b 0 .

Let { v n } W b be a minimizing sequence for ω b . Set u n = G 1 ( v n ) , then { u n } M b is a minimizing sequence for m b . We can assume u n 0 . It shows that E ( u n ) m b , so there exist C > 0 such that

C E ( u n ) = 1 2 N [ g 2 ( u n ) | u n | 2 + V ( x ) u n 2 ] λ 2 p N [ ( | x | | u n | p ) | u n | p ] a 2 N [ | u | 2 + V ( x ) u n 2 ] λ 2 p N ( | x | | u n | p ) | u n | p .

By Hölder inequality and Hardy-Little-Sobolev-inequality,

N [ ( | x | | u n | p ) | u n | p ] ( N | u n | p r ) 2 r ( ( N | u n | 2 ) θ p r 2 ( N | u n | q + 1 ) ( 1 θ ) p r q + 1 ) 2 r = ( N | u n | 2 ) θ p b 2 ( 1 θ ) p q + 1 b 2 p θ p + θ p q q 1 q + 1 ( ε N | u n | 2 + C ( ε ) b ) .

where θ = 2 ( q + 1 ) 2 p r ( q 1 ) p r , 0 < θ p < 1 , r = 2 N 2 N μ , ε > 0 , C ( ε ) = ( ε θ p ) θ p 1 θ p ( 1 θ p ) , then

C E ( u n ) a 2 N [ | u n | 2 + V ( x ) u n 2 ] λ b 2 p θ p + θ p q q 1 q + 1 2 p ( ε N V ( x ) u n 2 + C ( ε ) b ) ( a 2 λ ε b 2 p θ p + θ p q q 1 q + 1 2 p ) ( N | u n | 2 + V ( x ) | u n | 2 ) λ 2 p C ( ε ) b 2 p θ p + θ p q q + 1 .

Taking ε > 0 small enough such that a 2 λ ε b 2 p θ p + θ p q q 1 q + 1 2 p > 0 . It implies that u n ( x ) is bounded in H V 1 ( N ) . By the compact embedding result from H V 1 ( N ) into L r ( N ) for 2 r < 2 . We may assume that u n u b in H V 1 ( N ) , u n u b in L r ( N ) for 2 r < 2 and u n ( x ) u b ( x ) a.e x N . Hence u b M b , since u n 0 , u b 0 and u b 0 . Similarly as the proof of Lemma 2.4 (1), we have

N ( | x | | u n | p ) | u n | p N ( | x | | u b | p ) | u b | p , n + .

Hence

m b = lim n E ( u n ) lim inf n { 1 2 N [ g 2 ( u n ) | u n | 2 + V ( x ) u n 2 ] λ 2 p N ( | x | | u n | p ) | u n | p } E ( u b ) .

Step 2: Set h q + 1 ( v ) = 1 q + 1 N | G 1 ( v ( x ) ) | q + 1 for 2 q + 1 < 2 * , then h q + 1 ( v ) C 1 ( H V 1 ( N ) , ) .

In fact, for any φ H V 1 ( N ) , by Lemma 1.1 and Hölder’s inequality, we have

| h q + 1 ( v ) , φ | = | N | G 1 ( v ) | q 1 G 1 ( v ) g ( G 1 ( v ) ) φ | C ( N | v | q + 1 ) q q + 1 ( N | φ | q + 1 ) 1 q + 1 C φ H V 1 ( N ) .

then h q + 1 ( v ) ( H V 1 ( N ) ) * .

| h q + 1 ( v n ) h q + 1 ( v ) , φ | = | N ( | G 1 ( v n ) | q 1 G 1 ( v n ) g ( G 1 ( v n ) ) | G 1 ( v ) | q 1 G 1 ( v ) g ( G 1 ( v ) ) ) φ | ( N | | G 1 ( v n ) | q 1 G 1 ( v n ) g ( G 1 ( v n ) ) | G 1 ( v ) | q 1 G 1 ( v ) g ( G 1 ( v ) ) | q + 1 q ) q q + 1 ( N | φ | q + 1 ) 1 q + 1 . (1)

and

| | G 1 ( v n ) | q 1 G 1 ( v n ) g ( G 1 ( v n ) ) | G 1 ( v ) | q 1 G 1 ( v ) g ( G 1 ( v ) ) | q + 1 q C ( | v n | q + 1 + | v | q + 1 ) ,

Since v n v in H V 1 ( N ) , H V 1 ( N ) embedding into L r ( N ) is compact and { v n } is bounded in H V 1 ( N ) , where 2 r < 2 * . By 2 q + 1 < 2 * and Lemma 3.4 [22], we have

| h q + 1 ( v n ) h q + 1 ( v ) , φ | 0, n + .

then h q + 1 ( v ) C 1 ( H V 1 ( N ) , ) for 2 q + 1 < 2 * .

Step 3: For any b 0 , there exist β ( b ) such that 0 < u b = G 1 ( v b ) M b is a weak solution of Equation (1.1) with λ = λ ( b ) . In fact, by lemma2.4,

F ( v ) , φ = N v φ + N V ( x ) G 1 ( v ) g ( G 1 ( v ) ) φ λ N [ | x | μ | G 1 ( v ) | p ] | G 1 ( v ) | p 2 G 1 ( v ) g ( G 1 ( v ) ) φ .

Take limit t 0 , we get F ( v b ) , v 0 , by arbitrariness of v, one has F ( v b ) , v 0 . It follows that F ( v b ) , v = 0 , for every v N ( h q + 1 ( v b ) ) . Set v H V 1 ( N ) be such that h q + 1 ( v b ) , v = 1 , for every φ H V 1 ( N ) , let

ψ = φ h q + 1 ( v b ) , φ v .

Then ψ N ( h q + 1 ( v b ) ) , it means F ( v b ) , ψ = 0 , i.e.

F ( v b ) , φ = F ( v b ) , v h q + 1 ( v b ) , φ .

Put β = β ( b ) = F ( v b ) , v , we have

F ( v b ) , φ = β h q + 1 ( v b ) , φ ,

n

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.