Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity

Liuyang Shao; Yingmin Wang

doi:10.1515/math-2021-0025

Open Access Published by De Gruyter Open Access May 10, 2021

Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity

Liuyang Shao and Yingmin Wang

From the journal Open Mathematics

https://doi.org/10.1515/math-2021-0025

Abstract

In this study, we consider the following quasilinear Choquard equation with singularity

− Δ u + V ( x ) u − u Δ u 2 + λ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u = K ( x ) u − γ , x ∈ R N , u > 0 , x ∈ R N ,

where I α is a Riesz potential, 0 < α < N , and N + α N < p < N + α N − 2 , with λ > 0 . Under suitable assumption on V and K , we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as λ → 0 .

Keywords: quasilinear Schrödinger equation; singularity; Choquard type; variational methods

MSC 2010: 35B09; 35J20

1 Introduction and main results

In this study, we investigate the following quasilinear Choquard equation with singularity

(1.1) − Δ u + V ( x ) u − u Δ u 2 + λ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u = K ( x ) u − γ , x ∈ R N , u > 0 , x ∈ R N ,

where I α is a Riesz potential, 0 < α < N , and N + α N < p < N + α N − 2 , with λ > 0 . Quasilinear Schrödinger equations of the form

(1.2) i ∂ t z = − Δ z + V ( x ) z − f ( ∣ z ∣ 2 ) z − Δ h ( ∣ z ∣ 2 ) h ′ ( ∣ z ∣ 2 ) z

have been derived as models of several physical phenomena. Here, V = V ( x ) , x ∈ R N , is a given potential, and K , V are real functions. For instance, in the case h ( s ) = s , we obtain

(1.3) i ∂ t z = − Δ z + V ( x ) z − f ( ∣ z ∣ 2 ) z − ( Δ ∣ z ∣ 2 ) z ,

which has been called the superfluid film equation in plasma physics by Kurihara in [1] (cf. [2,3]). In the case h ( s ) = ( 1 + s ) 1 2 , equation (1.2) models the self-channeling of a high-power ultra-short laser in mater, see [4,5] and the references in [6]. Equation (1.2) also appears in plasma physics and fluid mechanics [7,8], in the theory of Heisenberg ferromagnets and magnons [9], in dissipative quantum mechanics, and in condensed matter theory [10,11]. For more details, we refer the readers to [12,13] and the references therein.

In recent years, the study on the quasilinear Schrödinger equation (1.2) is always a topic of great interest. Mathematicians have established several methods to treat equation (1.3), for example, the dual approach, the perturbation method, and the Nehari method, see for instance [14,15, 16,17,18, 19,20,21, 22,23,24, 25,26,27, 28,29], and the references therein. However, the system (1.1) with Choquard type nonlinearity has only been studied in [30,31].

It is remarkable that there are few papers investigating quasilinear equation with singularity. To the best of our knowledge, it only appears in [32], J. Marcos do Ó and A. Moameni established the singular quasilinear Schrödinger equation

− Δ u − 1 2 Δ ( u 2 ) u = λ u 3 − u − u − α , u > 0 , x ∈ Ω ,

where Ω is a ball in R N ( N ≥ 2 ) centered at the origin, 0 < α < 1 . Furthermore, they obtained the existence of radially symmetric positive solutions by taking advantage of Nehari manifold and some techniques about implicit function theorem when λ belongs to a certain neighborhood of the first eigenvalue λ 1 of the eigenvalue problem

− Δ u − 1 2 Δ ( u 2 ) u = λ u 3

In [33], they studied the following Choquard-type quasilinear Schrödinger equation:

− Δ u + V ( x ) u − Δ ( u 2 ) u = ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u , x ∈ R N

where N ≥ 3 , 0 < α < N , 2 ( N + α ) N < p < 2 ( N + α ) N − 2 , V : R N → R is radial potential, and I α is a Riesz potential. They consider the existence of ground state solutions.

To the best of our knowledge, there seems to be little progress on the existence of a positive solution for quasilinear Choquard equation with singularity. By the motivation of the above work, in our study, we establish the existence of a positive solution for problem (1.1) with singularity. First, the nonlinearity of problem (1.1) is nonlocal, and it is much more difficult to obtain the existence of positive solutions. Second, we investigate the relationships between quasilinear Choquard equation involving and without convolution, which makes our studies more interesting. At last, we obtain the asymptotic behavior of solutions as λ → 0 .

Before stating our main result, we suppose that the functions V ( x ) and K ( x ) satisfy the following assumptions:

V ∈ C ( R N ) satisfies inf x ∈ R N V ( x ) > V 0 > 0 , where V 0 is a constant.
meas { x ∈ R N : − ∞ < V ( x ) ≤ μ } < + ∞ for all μ ∈ R .
K ∈ L 2 2 ∗ 2 2 ∗ − 1 + r ( R N ) is a nonnegative function.

Now, we state our main results as follows.

Theorem 1.1

Suppose that γ ∈ ( 0 , 1 ) , 0 < α < N , N + α N < p < N + α N − 2 and ( V 1 ) , ( V 2 ) , ( K 1 ) hold, then equation (1.1) admits a unique solution in E .

Theorem 1.2

Suppose that γ ∈ ( 0 , 1 ) , 0 < α < N , N + α N < p < N + α N − 2 ( V 1 ) , ( V 2 ) , ( K 1 ) are satisfied for any sequence { λ n } > 0 with λ n → 0 as n → ∞ . w λ n are the corresponding solutions of problem (1.1) obtained in Theorem 1.1 with λ = λ n , then w λ n → w 0 in E where w 0 is the unique positive solution to problem

− Δ u + V ( x ) u − Δ ( u 2 ) u = K ( x ) u − γ .

Notation. In this study, we make use of the following notations: C will denote various positive constants; the strong (resp. weak) convergence is denoted by → (resp. ⇀ ); o ( 1 ) denotes o ( 1 ) → 0 as n → ∞ , and B ρ ( 0 ) denotes a ball centered at the origin with radius ρ > 0 .

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented, and in Section 3, we give the proof of our main results.

2 Variational setting and preliminaries

To prove our conclusion, we give some basic notations and preliminaries. First, we can rewrite (1.1) as

− Δ u + V ( x ) u − Δ ( u 2 ) u + λ I α ∗ ∣ u ∣ p ∣ u ∣ p − 2 u − K ( x ) ∣ u ∣ − γ = 0 in R N .

It may also be noted that we can not apply directly the variational method to study (1.1), since the natural associated functional I given by

I ( u ) = 1 2 ∫ R N ( 1 + 2 u 2 ) ∣ ∇ u ∣ 2 d x + 1 2 ∫ R N V ( x ) u 2 d x + λ 2 p ∫ R N I α ∗ ∣ u ∣ p ∣ u ∣ p − 1 u d x − 1 1 − γ ∫ R N K ( x ) u 1 − γ d x

is not well defined in general. We make the changing of variables w = f − 1 ( u ) , where f is defined by: f ′ ( t ) = 1 1 + 2 f 2 ( t ) on [ 0 , ∞ ) and f ( t ) = f ( − t ) on ( − ∞ , 0 ] .

If we make the change of variable u = f ( w ) , we may rewrite equation I ( u ) in the form

(2.1) J λ ( w ) = 1 2 ∫ R N ( ∣ ∇ w ∣ 2 + V ( x ) f 2 ( w ) ) d x + λ 2 p ∫ R N I α ∗ ∣ f ( w ) ∣ p ∣ f ( w ) ∣ p − 1 f ( w ) d x − 1 1 − γ ∫ R N K ( x ) ∣ f ( w ) ∣ 1 − γ d x .

It can be easily proved that the functional J λ ( w ) is of class C 1 (see [34]) in E . Moreover, the critical points of J λ are weak solutions of the equation

∫ R N ∣ ∇ w ∇ φ ∣ + V ( x ) f ( w ) f ′ ( w ) φ + λ ∫ R N ( I α ∗ ∣ f ( w ) ∣ p ∣ f ( w ) ∣ p − 1 f ′ ( w ) φ − ∫ R N K ( x ) f − γ ( w ) f ′ ( w ) φ = 0

Be aware that if w is a critical point of J λ , then, u = f ( w ) is a weak solution of the problem (1.1). For any w ∈ E , let

E = w ∈ H 1 ( R N ) ∣ ∫ R N V ( x ) f 2 ( w ) d x < ∞

be a Hilbert space endowed with inner product and norm

∥ w ∥ E ≔ ⟨ w , w ⟩ = ∫ R N ( ∣ ∇ w ∣ 2 + V ( x ) w 2 ) d x 1 2 .

We denote by ∥ • ∥ p the usual L p -norm in the sequel for convenience, where 1 ≤ p ≤ + ∞ . In this step, we see that (1.1) is variational and its weak solutions are the critical points of the functional given by

J λ ( w ) = 1 2 ∫ R N ∣ ∇ w ∣ 2 + V ( x ) f 2 ( w ) + λ 2 p ∫ R N I α ∗ ∣ f ( w ) ∣ p ∣ f ( w ) ∣ p − 1 f ( w ) d x − 1 ( 1 − γ ) ∫ R N K ( x ) ∣ f ( w ) ∣ 1 − γ d x .

Lemma 2.1

(See [35]) The function f satisfies the following properties:

f is uniquely defined C ∞ function and invertible;
∣ f ′ ( s ) ∣ ≤ 1 and ∣ f ( s ) ∣ ≤ ∣ s ∣ for all s ∈ R ;
f ( s ) s → 1 as s → 0 ;
f ( s ) s → 2 1 4 as s → ∞ ;
f ( s ) 2 ≤ s f ′ ( s ) ≤ f ( s ) for all s ≥ 0 ;
∣ f ( s ) ∣ ≤ 2 1 4 ∣ s ∣ 1 2 for all s ∈ R ;
the function f 2 ( s ) is strictly convex;
there exists a positive constant C such that
∣ f ( s ) ∣ ≥ C ∣ s ∣ , ∣ s ∣ ≤ 1 , C ∣ s ∣ 1 2 , ∣ s ∣ ≥ 1 ;
for each λ > 1 , we have f 2 ( λ s ) ≤ λ f 2 ( s ) for all t ∈ R ;
the function f − q ( s ) f ′ ( s ) is strictly decreasing for s > 0 and 0 < q < 1 ;
the function f q ( s ) f ′ ( s ) s − 1 is strictly increasing for q ≥ 3 and s > 0 .

3 Proof of Theorem 1.1

To prove Theorem 1.1, we need the following results.

Lemma 3.1

Suppose that ( V 1 ) , ( V 2 ) , ( K 1 ) are satisfied, then (1.1) has the global minimizer in E . In other words, there exists w 0 ∈ E such that J λ ( w 0 ) = m λ = inf E J λ < 0 .

Proof

By the Sobolev inequality, Hölder inequality and Lemma 2.1 ( A 6 ) yield

(3.1) ∫ R N K ( x ) ∣ f ( w ) ∣ 1 − γ ≤ C ∥ K ∥ 2 2 ∗ 2 2 ∗ − 1 + γ ∥ w ∥ 1 − γ 2 .

For any w ∈ E , using (2.1) and (3.1), for λ > 0 and 0 < γ < 1

(3.2) J λ ( w ) = 1 2 ∫ R N ( ∣ ∇ w ∣ 2 + V ( x ) f 2 ( w ) ) + λ 2 p ∫ R N ( I α ∗ ∣ f ( w ) ∣ p ) ∣ f ( w ) ∣ p − 1 1 − γ ∫ R N K ( x ) ∣ f ( w ) ∣ 1 − γ ≥ 1 2 ∫ R N ( ∣ ∇ w ∣ 2 + V ( x ) f 2 ( w ) ) − 1 1 − γ ∫ R N K ( x ) ∣ f ( w ) ∣ 1 − γ ≥ 1 2 ∫ R N ( ∣ ∇ w ∣ 2 + V ( x ) f 2 ( w ) ) − C 1 − γ ∥ K ∥ 2 2 ∗ 2 2 ∗ − 1 + γ ∥ w ∥ 1 − γ 2 .

Since γ ∈ ( 0 , 1 ) , J λ is coercive and bounded from below on E for each λ > 0 . Thus m λ ≔ inf E J λ is well defined. For t > 0 and given w ∈ E \ { 0 } , by Lemma 2.1 ( A 3 ) , we have

(3.3) J λ ( t w ) = t 2 ∫ R N ∣ ∇ w ∣ 2 + V ( x ) f 2 ( t w ) + λ 2 p ( I α ∗ ∣ f ( t w ) ∣ p ) ∣ f ( t w ) ∣ p − 1 1 − γ ∫ R N K ( x ) ∣ f ( t w ) ∣ 1 − γ ≤ t 2 2 ∫ R N ∣ ∇ w ∣ 2 + V ( x ) w 2 + λ t 2 p 2 p ∫ R N ( I α ∗ ∣ w ∣ p ) ∣ w ∣ p − 1 1 − γ f 1 − γ ( t ) ∫ R N K ( x ) ∣ w ∣ 1 − γ .

Let

g ( t ) = t 2 2 ∫ R N ∣ ∇ w ∣ 2 + V ( x ) w 2 + λ t 2 p 2 p ∫ R N ( I α ∗ ∣ w ∣ p ) ∣ w ∣ p − 1 1 − γ f 1 − γ ( t ) ∫ R N K ( x ) ∣ w ∣ 1 − γ ,

lim t → 0 + g ( t ) t ( 1 − γ ) = lim t → 0 + t ( 1 + γ ) ∫ R N ∣ ∇ w ∣ 2 + V ( x ) w 2 + λ t ( 2 p − 1 + γ ) 2 p ∫ R N ( I α ∗ ∣ w ∣ p ) ∣ w ∣ p − 1 1 − γ lim t → 0 + f 1 − γ ( t ) t ( 1 − γ ) ∫ R N K ( x ) ∣ w ∣ 1 − γ .

Therefore, by (3.3), we obtain J λ ( t w ) < 0 , for all w ≢ 0 and t > 0 , and there exists a minimizing sequence { w n } ⊂ E such that lim n → ∞ J λ ( w n ) = m λ < 0 . Since J λ ( ∣ w n ∣ ) = J λ ( w n ) , we could suppose that w n ≥ 0 . The coerciveness of J λ on E shows that { w n } is bounded in E . Going if necessary to a subsequence, we can assume that w n ⇀ w 0 in E , w n → w 0 in L p ( R N ) , p ∈ [ 2 , 2 ∗ ) and w n → w 0 , a.e. in R N , since 0 < γ < 1 , K ∈ L 2 2 ∗ 2 2 ∗ − 1 + γ ( R N ) is nonnegative, by Hölder’s inequality. Similar to (3.1), we have

(3.4) lim n → ∞ ∫ R N K ( x ) f 1 − γ ( w n ) = ∫ R N K ( x ) f 1 − γ ( w 0 ) .

Then, by the weakly lower semi-continuity of the norm, Lemma 2.4 in [36] and (3.4), we obtain

J λ ( w 0 ) = 1 2 ∫ R N ∣ ∇ w 0 ∣ 2 + V ( x ) ∣ f ( w 0 ) ∣ 2 + λ 2 p ∫ R N ( I α ∗ ∣ f ( w 0 ) ∣ n ) ∣ f ( w 0 ) ∣ p − 1 1 − γ ∫ R N K ( x ) f 1 − γ ( w 0 ) ≤ lim n → ∞ inf 1 2 ∫ R N ∣ ∇ w n ∣ 2 + V ( x ) ∣ f ( w n ) ∣ 2 + λ 2 p ∫ R N ( I α ∗ ∣ f ( w n ) ∣ p ) ∣ f ( w n ) ∣ p − 1 1 − γ ∫ R N K ( x ) f 1 − γ ( w n ) = lim n → ∞ inf J λ ( w n ) = m λ .

In addition, J λ ( w 0 ) = m λ < 0 . We complete the proof.□

Proof of Theorem 1.1

We divide the proof into three parts.

We will prove that for any 0 ≤ φ ∈ E
∫ R N ∣ ∇ w 0 ∇ φ ∣ + V ( x ) f ( w 0 ) f ′ ( w 0 ) φ + λ ∫ R N ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − 1 f ′ ( w 0 ) φ − ∫ R N K ( x ) f − γ ( w 0 ) f ′ ( w 0 ) φ ≥ 0 .
On the basis of Lemma 3.1, w 0 is bounded in E and w 0 ≥ 0 with w 0 ≢ 0 . For 0 ≤ φ ∈ E and δ > 0 , one has
(3.5) 0 ≤ J λ ( w 0 + δ φ ) − J λ ( w 0 ) = 1 2 ∥ w 0 + δ φ ∥ 2 − 1 2 ∥ w 0 ∥ 2 + 1 2 ∫ R N V ( x ) ∣ f ( w 0 + δ φ ) ∣ 2 − 1 2 ∫ R N V ( x ) ∣ f ( w 0 ) ∣ 2 + λ ∫ R N ( I α ∗ ∣ f ( w 0 + δ φ ) ∣ p ) ∣ f ( w 0 + δ φ ) ∣ p − ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − 1 1 − γ ∫ R N K ( x ) [ f 1 − γ ( w 0 + δ φ ) − f 1 − γ ( w 0 ) ] ,
since γ ∈ ( 0 , 1 ) and K ( x ) is nonnegative.

Dividing (3.5) by δ > 0 and passing to the liminf as δ → 0 + , then we can get from Fatou’s Lemma that
(3.6) 1 1 − γ lim δ → 0 + inf ∫ R N f 1 − γ ( w 0 + δ φ ) − f 1 − γ ( w 0 ) δ ≤ ∫ R N ∇ w 0 ∇ φ + V ( x ) f ( w 0 ) f ′ ( w 0 ) φ + λ ∫ R N ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − 1 f ′ ( w 0 ) φ .
Since
∫ R N K ( x ) f 1 − γ ( w 0 + δ φ ) − f 1 − γ ( w 0 ) δ ≤ ( 1 − γ ) ∫ R N K ( x ) f − γ ( w 0 + δ φ ) f ′ ( w 0 + δ φ ) φ ,
by the Beppolevi Monotone convergence Theorem and Lemma 2.1( A 10 ), we have
lim δ → 0 + inf 1 1 − γ ∫ R N K ( x ) f 1 − γ ( w 0 + δ φ ) − f 1 − γ ( w 0 ) δ = ( 1 − γ ) ∫ R N K ( x ) f − γ ( w 0 + δ θ φ ) f ′ ( w 0 + θ δ φ ) φ ,
where 0 < θ < 1 , which together with (3.6) implies that
(3.7) ∫ R N ∇ w 0 ∇ φ + V ( x ) f ( w 0 ) f ′ ( w 0 ) φ + λ ∫ R N ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − 1 f ′ ( w 0 ) φ − ∫ R N K ( x ) f − γ ( w 0 ) f ′ ( w 0 ) φ ≥ 0
We show that w 0 > 0 in R N and w 0 is a solution of problem (1.1). Given ε > 0 , define g : [ − ε , ε ] → R by g ( t ) = J λ ( w 0 + t w 0 ) . Then, g attains its minimum at t = 0 by Lemma 3.1, which implies that
(3.8) g ′ ( 0 ) = ∫ R N ∣ ∇ w 0 ∣ 2 + V ( x ) ∣ f ( w 0 ) ∣ 2 + λ ∫ R N ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − ∫ R N K ( x ) ∣ f ( w 0 ) ∣ 1 − γ = 0
For any v ∈ E and ε > 0 , set φ ε = ( w 0 + ε v ) + and Ω ε = { x ∈ R N : φ ε ≤ 0 } . Then, using (3.7) and (3.8) with φ = φ ε lead to
0 ≤ ∫ R N ∇ w 0 ∇ φ + V ( x ) f ( w 0 ) f ′ ( w 0 ) φ + λ ∫ R N ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − 1 f ′ ( w 0 ) φ − ∫ R N K ( x ) f − γ ( w 0 ) f ′ ( w 0 ) φ = ∫ R N − ∫ Ω ε ∇ w 0 ∇ ( w 0 + ε v ) + V ( x ) f ( w 0 ) f ′ ( w 0 ) ( w 0 + ε v ) + λ ∫ R N − ∫ Ω ε ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − 1 f ′ ( w 0 ) ( w 0 + ε v ) − ∫ R N − ∫ Ω ε K ( x ) f − γ ( w 0 ) f ′ ( w 0 ) ( w 0 + ε v ) = ε ∇ w 0 ∇ v + V ( x ) f ( w 0 ) f ′ ( w 0 ) v + λ ∫ R N ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − 1 f ′ ( w 0 ) v − ∫ R N K ( x ) f − γ ( w 0 ) f ′ ( w 0 ) v − ∫ Ω ε ∇ w 0 ∇ ( w 0 + ε v ) + V ( x ) f ( w 0 ) f ′ ( w 0 ) ( w 0 + ε v ) − λ ∫ Ω ε ( I α ∗ ∣ f ( w ) ∣ p ) ∣ f ( w ) ∣ p − 1 f ′ ( w ) ( w + ε v ) + ∫ R N − ∫ Ω ε K ( x ) f − γ ( w 0 ) f ′ ( w 0 ) ( w 0 + ε v ) .
Taking ε → 0 + to the above inequality and based on the fact that Ω ε → 0 as ε → 0 + , we get
∫ R N ∇ w 0 ∇ φ + V ( x ) f ( w 0 ) f ′ ( w 0 ) φ + λ ∫ R N ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − 1 f ′ ( w 0 ) φ − ∫ R N K ( x ) f − γ ( w 0 ) f ′ ( w 0 ) φ ≥ 0 ∀ φ ∈ E
The above inequality also holds for − v ; hence, we have
(3.9) ∫ R N ∇ w 0 ∇ φ + V ( x ) f ( w 0 ) f ′ ( w 0 ) φ + λ ∫ R N ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − 1 f ′ ( w 0 ) φ − ∫ R N K ( x ) f − γ ( w 0 ) f ′ ( w 0 ) φ = 0
Analogous to the proof of [32, Theorem 1], we obtain w 0 ∈ C loc 2 ( R N ) . Since w 0 ≥ 0 , the strong maximum principle implies w 0 > 0 , and w 0 ∈ E is a solution of problem (1.1).
We show that the solution w 0 is unique. Assume that w ¯ ∈ E is also a solution, then for any φ ∈ E
(3.10) ∫ R N ∇ w ¯ ∇ φ + V ( x ) f ( w ¯ ) f ′ ( w ¯ ) φ + λ ∫ R N ( I α ∗ ∣ f ( w ¯ ) ∣ p ) ∣ f ( w ¯ ) ∣ p − 1 f ′ ( w ¯ ) φ − ∫ R N K ( x ) f − γ ( w ¯ ) f ′ ( w ¯ ) φ = 0
Subtracting (3.9) and (3.10), since K ( x ) > 0 , it follows from Lemma 2.4 (see [36]) and λ > 0 that
∥ w 0 − w ¯ ∥ 2 = ∫ R N K ( x ) [ f − γ ( w 0 ) f ′ ( w 0 ) − f − γ ( w ¯ ) f ′ ( w ¯ ) ] ( w 0 − w ¯ ) − ∫ R N V ( x ) f ( w 0 ) f ′ ( w 0 ) ( w 0 − w ¯ ) − λ ∫ R N ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − 1 f ′ ( w 0 ) − ( I α ∗ ∣ f ( w 0 ) ∣ p ) ∣ f ( w 0 ) ∣ p − 1 f ′ ( w ¯ ) ( w 0 − w ¯ ) ≤ 0
Hence, w 0 = w ¯ and w 0 is the unique solution of problem (1.1), which completes the proof.□

Proof of Theorem 1.2

From the proof of Lemma 3.1 and Theorem 1.1, we can get that λ = 0 is allowed. Therefore, under the conditional assumptions of Theorem 1.2, equation (1.1) has a unique positive solution w 0 ∈ E , i.e., for any w ∈ E , we obtain

∫ R N ∇ w 0 ∇ φ + V ( x ) f ( w 0 ) f ′ ( w 0 ) φ = ∫ R N K ( x ) ∣ f ( w 0 ) ∣ − γ f ′ ( w 0 ) φ .

For any sequence { λ n } > 0 with λ n → 0 , as n → ∞ , according to Theorem 1.1, we can obtain a positive solution sequence { w λ n } ⊂ E corresponding solution of problem (1.1) with λ = λ n for n ∈ N . Thus, we obtain

(3.11) ∫ R N ∇ w λ n ∇ φ + V ( x ) ∣ f ( w λ n ) ∣ f ′ ( w λ n ) φ + λ n ∫ R N ( I α ∗ ∣ f ( w λ n ) ∣ p ) ∣ f ( w λ n ) ∣ p − 2 f ( w ) f ′ ( w λ n ) φ = ∫ R N K ( x ) ∣ f ( w λ n ) ∣ − γ f ′ ( w λ n ) φ

for any w λ n ∈ E . From Lemma 2.1 and the proof of Theorem 1.1, we get J λ n = m λ n < 0 and then { w λ n } is bounded in E since J λ n is coercive according to (3.3). As a result, there exist a subsequence of { w λ n } (still denoted by { w λ n } ) and a nonnegative function w 0 ∈ E such that w λ n ⇀ w 0 in E , w λ n → w 0 in L p ( R N ) , p ∈ [ 2 , 2 ∗ ) and w λ n → w 0 a.e in R N . Let us define w n = w λ n in (3.11) and passing to the liminf as n → ∞ , we can obtain from Lemma 2.4 (see [36]), (3.3) and the weakly lower semi-continuity of the norm that for any φ ∈ C 0 ∞ ( R N ) , the support of φ is contained in B R 0 ( 0 ) for some R 0 > 0 since w n ⇀ w 0 in H 1 ( R N ) , we have

(3.12) ∫ R N ∇ w n ∇ φ − ∇ w 0 ∇ φ → 0 .

By w n → w 0 in L loc 2 ( R N ) ,

(3.13) ∫ R N V ( x ) [ f ( w n ) f ′ ( w n ) φ − f ( w 0 ) f ′ ( w 0 ) ] φ ≤ μ ∫ B R 0 ( 0 ) ∣ f ( w n ) f ′ ( w n ) φ − f ( w 0 ) f ′ ( w 0 ) φ ∣ ≤ μ ∫ B R 0 ( 0 ) ∣ f ( w n ) f ′ ( w n ) − f ( w 0 ) f ′ ( w 0 ) ∣ 2 1 2 ∫ B R 0 ( 0 ) ∣ φ ∣ 2 1 2 → 0 .

Passing to the liminf as n → ∞ in (3.11), by (3.12), (3.13) and the weakly lower semi-continuity of the definition, we have

(3.14) ∫ R N ∇ w 0 ∇ φ + V ( x ) f ( w 0 ) f ′ ( w 0 ) φ ≤ ∫ R N K ( x ) ∣ f ( w 0 ) ∣ − γ f ′ ( w 0 ) φ .

Furthermore, passing to the liminf as n → ∞ in (3.11), by (3.12), (3.13) and the Fatou’s Lemma, we obtain

(3.15) ∫ R N ∇ w 0 ∇ φ + V ( x ) f ( w 0 ) f ′ ( w 0 ) φ ≥ ∫ R N K ( x ) ∣ f ( w 0 ) ∣ − γ f ′ ( w 0 ) φ .

Using (3.14) and (3.15), we have

∫ R N ∇ w 0 ∇ φ + V ( x ) f ( w 0 ) f ′ ( w 0 ) φ = ∫ R N K ( x ) ∣ f ( w 0 ) ∣ − γ f ′ ( w 0 ) φ .

Analogous to step (2) in the proof of Theorem 1.1, we can get that 0 < w 0 ∈ E is also a solution. Therefore, w λ n → w 0 in E and w 0 is the unique positive solution to the equation (1.2). We complete the proof.□

Acknowledgements

The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.

Funding information: This work was partially supported by the Fundamental Research Funds for the National Natural Science Foundation of China 11671403 and Guizhou University of Finance and Economics of 2019XYB15.
Conflict of interest: Authors state no conflict of interest.

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Received: 2020-08-02

Revised: 2021-01-20

Accepted: 2021-01-24

Published Online: 2021-05-10

This work is licensed under the Creative Commons Attribution 4.0 International License.

Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity

Abstract

1 Introduction and main results

Theorem 1.1

Theorem 1.2

2 Variational setting and preliminaries

Lemma 2.1

3 Proof of Theorem 1.1

Lemma 3.1

Proof

Proof of Theorem 1.1

Proof of Theorem 1.2

Acknowledgements

References

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