In this paper, we will study the following equation
− ε 2 Δ w + V ( x ) w = ε − θ ( Y 1 ( w ) + Y 2 ( w ) ) , (1.1)
where ε > 0 , N > 2 , θ ∈ ( 0 , N ) , Y 1 ( w ) = W 1 ( x ) [ I θ ∗ ( W 1 | w | p ) ] | w | p − 2 w ,
Y 2 ( w ) = W 2 ( x ) [ I θ ∗ ( W 2 | w | q ) ] | w | q − 2 w , 2 ≤ p < q < N + θ N − 2 , V , W i , i = 1 , 2 are con
tinuous bounded positive functions and the Riesz potential I θ is defined as follows:
I θ : = Γ ( N − θ 2 ) 2 θ π N / 2 Γ ( θ 2 ) | x | θ − N , x ∈ ℝ N \ { 0 } . (1.2)
When ε = 1 , Equation (1.1) is related to the local nonlinear perturbation of the famous Choquard equation
− Δ u + u = ( I 2 ∗ u 2 ) u in ℝ N . (1.3)
This equation for N = 3 was first proposed by Pekar [
J. N. Correia and C. P. Oliveira [
− Δ u + ω u = ( K μ ∗ | u | q + 1 ) | u | q + ϵ ( K μ ∗ | u | 2 μ ∗ ) | u | 2 μ ∗ − 1 in ℝ N .
where ω = 1 , 2 ≤ q + 1 < 2 μ ∗ = 2 N − μ N − 2 , ϵ is a positive parameter. They proved
existence of positive solutions for a class of problems involving the Choquard term in exterior domain and the nonlinearity with critical growth by using variational method combined with Brouwer theory of degree and Deformation lemma. S. Yao, J. Sun and T. Wu [
− Δ u + λ V ( x ) u = ( I α ∗ K | u | p ) K | u | p − 2 u − | u | q − 2 u in ℝ N .
When N ≥ 3 , λ > 0 , K ( x ) ≥ 0 , 1 + α N < p < N + α N − 2 and 2 < q < 2 ∗ = 2 N N − 2 .
They proved different relationship between p and q when the competing effect of the nonlocal term with the perturbation happens.
For the semiclassical states of Choquard equation, we can refer to the following references. Y. Su a and Z. Liu [
− ε 2 Δ u + V ( x ) u = ε − α g ( u ) in ℝ N , (1.4)
where N ≥ 5 , α ∈ ( 0 , N ) , g ( u ) = ( I α ∗ F ( u ) ) F ′ ( u ) , F ( u ) = λ 2 α # | u | 2 α # + 1 2 α ∗ | u | 2 α ∗ , 2 α # = N + α N , 2 α ∗ : = N + α N − 2 . Working in a variational setting, they showed the
existence, multiplicity and concentration of positive solutions for such equations when the potential satisfies some suitable conditions. Y. Meng and X. He [
Y. Ding and J. Wei [
− ε 2 Δ w + V ( x ) w = W ( x ) | w | p − 2 w in ℝ N ,
where ε > 0 , p ∈ ( 2 , 2 N N − 2 ) and V , W are continuous bounded positive
functions, they proved existence and concentration phenomena of semiclassical positive groundstate solutions, and multiplicity of solutions including at least one pair of sign-changing ones by pseudo-index theory and Nehari method. Later, M. Liu and Z. Tang [
Motivated by the above conclusions, this article mainly discusses the existence, convergence, concentration, and asymptotic property of positive groundstate solution of Equation (1.1). We also establish the multiplicity of semiclassical solutions for Equation (1.1) by pseudo-index theory which was imposed by V. Benci. The equation studied in this paper has two convolution terms and two nonlinear potentials, which bring new challenge in our arguements. Our method of proof is inspired by [
Before stating the main results, we need to make some assumptions.
( A 1 ) V , W i ∈ C 0, μ ( ℝ N ) are bounded with some μ ∈ ( 0,1 ) , V ( x ) achieves a global minimum on ℝ N with min ℝ N V ( x ) > 0 , and W i ( x ) achieves a global maximum on ℝ N with inf ℝ N W i ( x ) > 0 inf ℝ N W i ( x ) > 0 , i = 1 , 2 .
For i = 1 , 2 , we denote by
τ : = min ℝ N V , V : = { x ∈ ℝ N : V ( x ) = τ } , τ ∞ : = lim inf | x | → ∞ V ( x ) ;
k i : = max ℝ N W i , W i : = { x ∈ ℝ N : W i ( x ) = k i } , k i ∞ : = lim sup | x | → ∞ W i ( x ) .
( A 2 ): W 1 ∩ W 2 ≠ ∅ .
We set
x i v ∈ V , k i v : = max V W i ( x ) = W i ( x i v ) , i = 1,2 ;
x w ∈ W 1 ∩ W 2 , τ w : = min W 1 ∩ W 2 V ( x ) = V ( x w ) .
For vector b = ( b 1 , b 2 ) ∈ ℝ 2 , we define
m ( a , b ) = ( ( τ ∞ a ) θ + 2 p 2 ( p − 1 ) − N 2 ( b 1 k 1 ∞ ) 2 p − 1 if k 2 ∞ b 2 ≤ ( a τ ∞ ) 2 + θ 4 q − p p − 1 ( k 1 ∞ b 1 ) q − 1 p − 1 , ( τ ∞ a ) θ + 2 q 2 ( q − 1 ) − N 2 ( b 2 k 2 ∞ ) 2 q − 1 otherwise
and denote k = ( k 1 , k 2 ) , k ∞ = ( k 1 ∞ , k 2 ∞ ) , k v = ( k 1 v , k 2 v ) . Similarly, we set W 0 : = ( W 10 , W 20 ) , W ( x ) = ( W 1 ( x ) , W 2 ( x ) ) . For b i = ( b 1 i , b 2 i ) ∈ ℝ 2 , i = 1,2 , we use b 1 ≤ b 2 to mean min { b 1 2 − b 1 1 , b 2 2 − b 2 1 } ≥ 0 and use b 1 < b 2 to show min { b 1 2 − b 1 1 , b 2 2 − b 2 1 } ≥ 0 and max { b 1 2 − b 1 1 , b 2 2 − b 2 1 } > 0 .
( A 3 ): (i) τ < τ ∞ , and there exists R v > 0 such that W i ( x ) ≤ k i v , i = 1 , 2 for | x | ≥ R v ;
(ii) k > k ∞ , and there exists R w > 0 such that V ( x ) ≥ τ w for | x | ≥ R w .
If ( A 3 )-(i) holds, we let
S v : = { x ∈ V : W i ( x ) = k i v , i = 1,2 } ∪ { x ∉ V : W 1 ( x ) > k 1 v or W 2 ( x ) > k 2 v } .
If ( A 3 )-(ii) holds, we let
S w : = { x ∈ W 1 ∩ W 2 : V ( x ) = τ w } ∪ { x ∉ W 1 ∩ W 2 : V ( x ) < τ w } .
In the following, in the case ( A 3 )-(i), S stands for S v and S stands for S w in the case ( A 3 )-(ii). Clearly, S is bounded. Moreover, S = V ∩ ( W 1 ∩ W 2 ) , if V ∩ ( W 1 ∩ W 2 ) ≠ ∅ .
The next theorems contain the main results of this paper.
Theorem 1.1. Assume that ( A 1 ) holds and
τ < τ ∞ , k v ≥ k ∞ . (1.5)
Then there exists m v ≥ m ( τ , k v ) such that for the maximal integer m ∈ ℕ with m < m v , Equation (1.1) possesses at least m pairs of solutions for small ε > 0 . Moreover, Equation (1.1) has a positive and a negative groundstate solution.
Theorem 1.2. Assume that ( A 1 )-( A 2 ) holds and
τ w ≤ τ ∞ , k > k ∞ . (1.6)
Then there exists m w ≥ m ( τ w , k ) such that for the maximal integer m ∈ ℕ with m < m w , all the conclusions of Theorem 1.1 remain true.
Theorem 1.3. Assume that ( A 1 )-( A 3 ) hold. Then for sufficiently small ε > 0 , Equation (1.1) has a positive groundstate solution w ε . If V , W i ∈ C 1 ( ℝ N ) and ∇ V , ∇ W i , i = 1 , 2 are bounded additionally, then w ε satisfies that
1) There exists a maximum point x ε of w ε with l i m ε → 0 dist ( x ε , S ) = 0 ;
2) There exist C > 0 and sufficiently large R > 0 such that
w ε ( x ) ≤ C ε N − 1 2 | x − x ε | 1 − N 2 exp ( − τ 4 ε | x − x ε | ) , ∀ | x | ≥ R ;
3) Letting v ε ( x ) : = w ε ( ε x + x ε ) , then for any sequence x ε → x 0 ( ε → 0 ), there holds v ε → v in H 1 ( ℝ N ) as ε → 0 , where v is a least energy solution of
− Δ v + V ( x 0 ) v = W 1 2 ( x 0 ) ( I θ ∗ v p ) v p − 1 + W 2 2 ( x 0 ) ( I θ ∗ v q ) v q − 1 , v > 0. (1.7)
If V ∩ ( W 1 ∩ W 2 ) ≠ ∅ particularly, then l i m ε → 0 dist ( x ε , V ∩ ( W 1 ∩ W 2 ) ) = 0 and up to a sequence, v ε → v in H 1 ( ℝ N ) as ε → 0 with v being a least energy solution of
− Δ v + τ v = k 1 2 ( I θ ∗ v p ) v p − 1 + k 2 2 ( I θ ∗ v q ) v q − 1 , v > 0. (1.8)
To prove the above results, we need the following basic conclusions.
Lemma 1.4. ( [
q ∈ [ 2,2 * ] , 2 * : = 2 N N − 2 , and H 1 ( ℝ N ) ↪ L loc q ( ℝ N ) is compact for q ∈ [ 2,2 * ) .
Moreover, H r 1 ( ℝ N ) : = { u ∈ H 1 ( ℝ N ) : u ( x ) = u ( | x | ) } is compactly embedded into L q ( ℝ N ) for q ∈ ( 2,2 * ) .
Lemma 1.5. ( [
sup y ∈ ℝ N ∫ B r ( y ) | ω n | q d x → 0 as n → ∞ ,
then ω n → 0 in L μ ( ℝ N ) for any μ ∈ ( 2,2 * ) .
For simplicity, we set
‖ w ‖ 1 : = ‖ w ‖ H 1 ( ℝ N ) , | w | q : = ‖ w ‖ L q ( ℝ N ) , w + : = max { 0, ω } , w − : = min { 0, w } , ℝ + : = ( 0, ∞ ) ,
and use ∫ ℝ N f ( x ) to denote ∫ ℝ N f ( x ) d x in some cases. Moreover, we use different forms of C to mean various positive constants and o ( 1 ) to represent the quantities which tend to 0 as n → ∞ or j → ∞ in the following.
This paper is organized as follows. Section 2 is an introduction to some conclusions about the Riesz potential, which plays a very important role in the subsequent proof process. In Section 3, we provide some preliminary results for the limit equation and the auxiliary equation which are the foundation for the proof of the main theorems. Section 4 contributes to the proofs of main results. We prove the multiplicity of semiclassical solutions by Benci pseudo-index theory and show the existence of the groundstate solutions and concentration of the positive groundstate solution in Section 4.
The Riesz potential with order θ ∈ ( 0, N ) of a function f ∈ L loc 1 ( ℝ N ) is defined by
( I θ ∗ f ) ( x ) : = ∫ ℝ N Γ ( N − θ 2 ) 2 θ π N / 2 Γ ( θ 2 ) f ( y ) | x − y | N − θ d y . (2.1)
The integral in Equation (2.1) converges in the classical Lebesgue sense for a.e. x ∈ ℝ N if and only if f ∈ L 1 ( ℝ N , ( 1 + | x | ) θ − N ) . Moreover, if
f ∉ L 1 ( ℝ N , ( 1 + | x | ) θ − N ) , then (1) diverges everywhere in ℝ N . The Riesz potential I θ is well-defined as an operator in L q ( ℝ N ) if and only if q ∈ [ 1, N θ ) . In addition, if q ∈ ( 1, N θ ) and r : = N q N − θ q , then I θ : L q ( ℝ N ) → L r ( ℝ N ) is a
bounded linear operator, which can be disclosed by the Hardy-Littlewood-Sobolev inequality.
Lemma 2.1. ( [
I θ ∗ f ∈ L N q / ( N − θ q ) ( ℝ N ) and | I θ ∗ f | N q / ( N − θ q ) ≤ C N , θ , q | f | q .
Applying Lemma 2.1 to the function f = | u | p ∈ L 2 N / ( N + θ ) ( ℝ N ) , we obtain the following result.
Lemma 2.2. ( [
∫ ℝ N ( I θ ∗ | ω | p ) | ω | p d x ≤ C N , θ | ω | 2 N p / ( N + θ ) 2 p .
In particular, if N > 2 , p ∈ [ N + θ N , N + θ N − 2 ] and ω ∈ H 1 ( ℝ N ) , then
∫ ℝ N ( I θ ∗ | ω | p ) | ω | p d x ≤ C N , θ , p ( ∫ ℝ N ( | ∇ u | 2 + | u | 2 ) d x ) p .
Actually, p ∈ [ N + θ N , N + θ N − 2 ] if and only if 2 N p N + θ ∈ [ 2,2 * ] . The Brézis-Lieb
type lemma we use next also applies to the Riesz potential.
Lemma 2.3. ( [
H 1 ( ℝ N ) as n → ∞ , then
1) B ( v n ) − B ( v n − v ) → B ( v ) as n → ∞ ;
2) B ′ ( v n ) − B ′ ( v n − v ) → B ′ ( v ) in H − 1 ( ℝ N ) as n → ∞ ,
where B ( v ) : = ∫ ℝ N ( I θ ∗ | v | p ) | v | p d x .
Lemma 2.4. ( [
H 1 ( ℝ N ) as n → ∞ , then for any u ∈ H 1 ( ℝ N ) , 〈 B ′ ( v n ) , u 〉 → 〈 B ′ ( v ) , u 〉 as n → ∞ , where B ( v ) is defined as in Lemma 2.3.
We consider, for N > 2 , θ ∈ ( 0, N ) , 2 ≤ p < q < N + θ N − 2 ,
− Δ v + a v = Y 1 b 1 ( v ) + Y 2 b 2 ( v ) , v ∈ H 1 ( ℝ N ) , (3.1)
where a > 0 , b i > 0 , i = 1 , 2 , Y 1 b 1 ( v ) : = b 1 2 ( I θ ∗ | v | p ) | v | p − 2 v , Y 2 b 2 ( v ) : = b 2 2 ( I θ ∗ | v | q ) | v | q − 2 v , and
− Δ v + V ε a ( x ) v = Y 1 ε b 1 ( v ) + Y 2 ε b 2 ( v ) , v ∈ H 1 ( ℝ N ) , (3.2)
where τ ≤ a ≤ τ ∞ , k ∞ ≤ b ≤ k , Y 1 ε b 1 ( v ) : = W 1 ε b 1 ( x ) [ I θ ∗ ( W 1 ε b 1 | v | p ) ] | v | p − 2 v , Y 2 ε b 2 ( v ) : = W 2 ε b 2 ( x ) [ I θ ∗ ( W 2 ε b 2 | v | q ) ] | v | q − 2 v with
V a ( x ) : = max { a , V ( x ) } , V ε a ( x ) : = V a ( ε x ) ,
W i b i ( x ) : = min { b i , W i ( x ) } , W i ε b i ( x ) : = W i b i ( ε x ) , i = 1,2.
The solutions v ∈ H 1 ( ℝ N ) of Equation (3.1) and Equation (3.2) can be obtained as critical points of the energy functionals
J a b ( v ) : = 1 2 ∫ ℝ N ( | ∇ v | 2 + a v 2 ) − 1 2 p ∫ ℝ N Y 1 ( v ) v − 1 2 q ∫ ℝ N Y 2 ( v ) v ,
J ε a b ( v ) : = 1 2 ∫ ℝ N ( | ∇ v | 2 + V ε a ( x ) v 2 ) − 1 2 p ∫ ℝ N Y 1 ε b 1 ( v ) v − 1 2 q ∫ ℝ N Y 2 ε b 2 ( v ) v ,
respectively. And the Nehari manifolds are denoted by N a b , N ε a b ; the least energies by E a b : = inf N a b J a b , E ε a b : = inf N ε a b J ε a b ; and the sets of least energy solutions by T a b , T ε a b , respectively. In particular, we define
J ∞ : = J τ ∞ k ∞ , N ∞ : = N τ ∞ k ∞ , E ∞ : = E τ ∞ k ∞ , V ε ∞ : = V ε τ ∞ ,
J ε ∞ : = J ε τ ∞ k ∞ , N ε ∞ : = N ε τ ∞ k ∞ , E ε ∞ : = E ε τ ∞ k ∞ , W i ε ∞ : = W i ε k ∞ , i = 1,2.
Lemma 3.1. There exist ρ > 0 and σ > 0 such that J a b ( v ) > σ for all ‖ v ‖ 1 = ρ . Moreover, l i m t → + ∞ J a b ( t v ) = − ∞ , if v ≠ 0 .
Lemma 3.2. Let Ψ a b : = { γ ∈ C ( [ 0 , 1 ] , H 1 ( ℝ N ) ) : γ ( 0 ) = 0 , J a b ( γ ( 1 ) ) < 0 } , then
E a b = inf v ∈ H 1 ( ℝ N ) \ { 0 } max t ≥ 0 J a b ( t v ) = inf γ ∈ Ψ a b max t ∈ [ 0,1 ] J a b ( γ ( t ) ) > 0.
Lemma 3.3. E a b is attained and T a b is compact in H 1 ( ℝ N ) .
Proof. We set the equivalent norm ‖ v ‖ 1 = ( ∫ ℝ N ( | ∇ v | 2 + a v 2 ) ) 1 2 for any v ∈
H 1 ( ℝ N ) . Obviously, N a b ≠ ∅ , we set v n ∈ N a b with v n ≥ 0 and
J a b ( v n ) → E a b as n → ∞ . On the basis of the Schwarz symmetrization and Theorem 3.1.5 in [
v n * ≠ 0 . We can know that ‖ v n * ‖ 1 2 ≤ ∫ ℝ N Y 1 b 1 ( v n * ) v n * + Y 2 b 2 ( v n * ) v n * . If ‖ v n * ‖ 1 2 = ∫ ℝ N Y 1 b 1 ( v n * ) v n * + Y 2 b 2 ( v n * ) v n * , then v n * ∈ N a b . If
‖ v n * ‖ 1 2 < ∫ ℝ N Y 1 b 1 ( v n * ) v n * + Y 2 b 2 ( v n * ) v n * , then there exists t n ∈ ( 0,1 ) such that t n v n * ∈
N a b and
E a b ≤ J a b ( t n v n * ) < p − 1 2 p ‖ v n ‖ 1 2 + q − p 2 p q ∫ ℝ N Y 2 b 2 ( v n ) v n = J a b ( v n ) → E a b as n → ∞ ,
which implies J a b ( t n v n * ) → E a b as n → ∞ . Define w n : = t n v n * , then w n ∈ N a b , w n ≥ 0 and
J a b ( w n ) → E a b as n → ∞ . (3.3)
By Lemma 2.2, one can check that { w n } is bounded in H 1 ( ℝ N ) . Along a subsequence, we may assume w n ⇀ w as n → ∞ . According to Lemma 1.4, w n → w in L r ( ℝ N ) for r ∈ ( 2,2 * ) as n → ∞ . Due to w n ∈ N a b and Lemma
2.2, ‖ w n ‖ 1 2 ≤ C ( ‖ w n ‖ 1 2 p + ‖ w n ‖ 1 2 q ) , which implies
∫ ℝ N Y 1 b 1 ( w n ) w n + Y 2 b 2 ( w n ) w n > C > 0 . By contradiction method, we get w ≠ 0. We can know ‖ w ‖ 1 2 ≤ lim inf n → ∞ ( ∫ ℝ N Y 1 b 1 ( w n ) w n + Y 2 b 2 ( w n ) w n ) = ∫ ℝ N Y 1 b 1 ( w ) w + Y 2 b 2 ( w ) w by the weakly lower semi-continuity of norm. By contradiction method, we can get w ∈ N a b and by (3.3),
E a b ≤ J a b ( w ) ≤ lim inf n → ∞ ( p − 1 2 p ‖ w n ‖ 1 2 + q − p 2 p q ∫ ℝ N Y 2 b 2 ( w n ) w n ) = lim inf n → ∞ J a b ( w n ) = E a b ,
which implies E a b = J a b ( w ) is attained. In the end, we have ( J a b ) ′ ( w ) = 0 ,
where w ∈ T a b is positive and radially symmetric. With similar arguments as above, T a b is compact in H 1 ( ℝ N ) . ,
In view of Theorem 3 in [
Lemma 3.4. If there exists a least energy solution v ∈ H 1 ( ℝ N ) for Equation (3.1), then v ∈ L 1 ( ℝ N ) ∩ C ∞ ( ℝ N ) , v is either positive or negative, and v is radially symmetric up to translations.
Lemma 3.5. Let a i > 0 and b i 1 , b i 2 > 0 for i = 1 , 2 .
(i) If min { a 2 − a 1 , b 1 1 − b 2 1 , b 1 2 − b 2 2 } ≥ 0 , then E a 1 b 1 ≤ E a 2 b 2 .
(ii) If min { a 2 − a 1 , b 1 1 − b 2 1 , b 1 2 − b 2 2 } ≥ 0 and max { a 2 − a 1 , b 1 1 − b 2 1 , b 2 1 − b 2 2 } > 0 , then E a 1 b 1 < E a 2 b 2 .
Lemma 3.6. If v is a groundstate solution of
− Δ v + τ ∞ v = Y 1 k 1 ∞ ( v ) + Y 2 k 2 ∞ ( v ) , v ∈ H 1 ( ℝ N ) , (3.4)
with the energy E ∞ , where Y 1 k 1 ∞ ( v ) : = k 1 ∞ 2 ( I θ ∗ | v | p ) | v | p − 2 v ,
Y 2 k 2 ∞ ( v ) : = k 2 ∞ 2 ( I θ ∗ | v | q ) | v | q − 2 v Letting u ( x ) : = λ v ( ( a τ ∞ ) 1 2 x ) , then Equation
(3.1) is equivalent to
− Δ u + a u = ( k 1 ∞ 2 b 1 2 ( a τ ∞ ) θ + 2 2 λ 2 − 2 p ) Y 1 b 1 ( u ) + ( k 2 ∞ 2 b 2 2 ( a τ ∞ ) θ + 2 2 λ 2 − 2 q ) Y 2 b 2 ( u ) , (3.5)
where u ∈ H 1 ( ℝ N ) , with the energy E λ = λ 2 ( a τ ∞ ) 1 − N 2 E ∞ .
Proof. Clearly, we can know v is a solution of Equation (3.4) if and only if u is a solution of Equation (3.5). Indeed,
− Δ u + a u = λ a τ ∞ ( − Δ v ( ( a τ ∞ ) 1 2 x ) + τ ∞ v ( ( a τ ∞ ) 1 2 x ) ) .
We can verify that v ∈ N ∞ if and only if u ∈ N λ a b , then
E λ = λ 2 ( a τ ∞ ) 1 − N 2 E ∞ . ,
Lemma 3.7 Assume that a ≤ τ ∞ , b ≥ k ∞ . Then m ( a , b ) E a b ≤ E ∞ .
Proof. Noticing that if λ > 0 satisfy
max { k 1 ∞ 2 b 1 2 ( a τ ∞ ) 2 + θ 2 λ 2 − 2 p , k 2 ∞ 2 b 2 2 ( a τ ∞ ) 2 + θ 2 λ 2 − 2 q } ≤ 1 , we can know E a b ≤ E λ .
According to the definition of m ( a , b ) , we can find two situations:
k 2 ∞ b 2 ≤ ( a τ ∞ ) 2 + θ 4 q − p p − 1 ( k 1 ∞ b 1 ) q − 1 p − 1 (3.6)
or
k 1 ∞ b 1 < ( a τ ∞ ) 2 + θ 4 p − q q − 1 ( k 2 ∞ b 2 ) p − 1 q − 1 . (3.7)
If (3.6) holds, let λ = [ k 1 ∞ b 1 ( a τ ∞ ) 2 + θ 4 ] 1 p − 1 , then E λ = ( a τ ∞ ) θ + 2 p 2 ( p − 1 ) − N 2 ( k 1 ∞ b 1 ) 2 p − 1 E ∞ , we obtain m ( a , b ) E a b ≤ E ∞ . If (3.7) holds, set λ = [ k 2 ∞ b 2 ( a τ ∞ ) 2 + θ 4 ] 1 q − 1 , then E λ = ( a τ ∞ ) θ + 2 q 2 ( q − 1 ) − N 2 ( k 2 ∞ b 2 ) 2 q − 1 E ∞ , we obtain m ( a , b ) E a b ≤ E ∞ . ,
Lemma 3.8. If τ < τ ∞ , k v ≥ k ∞ , then m ( τ , k v ) > 1 and E τ k v < E ∞ . If τ w ≤ τ ∞ , k > k ∞ , then m ( τ w , k ) ≥ 1 and E τ w k < E ∞ .
Proof. Set a = τ , b i = k i v , i = 1 , 2 in Equation (3.1), Equations (3.5)-(3.7), respectively. By the definition of m ( τ , k v ) , we get m ( τ , k v ) > 1 . By Lemma 3.7, we obtain E τ k v < E ∞ .
Similarly, we let a = τ w , b i = k i , i = 1 , 2 in Equation (3.1), Equations (3.5)-(3.7), respectively. Obviously, we have m ( τ w , k ) ≥ 1 . If (3.6) holds, we pick
λ = [ k 1 ∞ k 1 ( τ w τ ∞ ) 2 + θ 4 ] 1 p − 1 , then E τ w k ≤ E λ ≤ E ∞ by Lemmas 3.5, 3.6. If k 1 > k 1 ∞ ,
then E τ w k ≤ E λ < E ∞ by Lemma 3.6. If k 2 > k 2 ∞ , then E τ w k < E λ ≤ E ∞ by Lemma
3.5. Thus E τ w k < E ∞ . If (3.7) holds, we choose λ = [ k 2 ∞ k 2 ( τ w τ ∞ ) 2 + θ 4 ] 1 q − 1 , then
E τ w k ≤ E λ ≤ E ∞ . If k 1 > k 1 ∞ , then E τ w k < E λ ≤ E ∞ . If k 2 > k 2 ∞ , then E τ w k ≤ E λ < E ∞ . Thus, E τ w k < E ∞ . ,
Now we establish some results for Equation (3.2).
Lemma 3.9. There exist ρ > 0 , σ > 0 both independent of ε , a , b and just dependent on N , θ , p , τ , k , such that J ε a b ( v ) > σ for all ‖ v ‖ 1 = ρ . Moreover, l i m t → + ∞ J ε a b ( t v ) = − ∞ , if v ≠ 0 .
Lemma 3.10. Set Ψ ε a b : = { γ ∈ C ( [ 0 , 1 ] , H 1 ( ℝ N ) ) : γ ( 0 ) = 0 , J ε a b ( γ ( 1 ) ) < 0 } , then
E ε a b = inf v ∈ H 1 ( ℝ N ) \ { 0 } max t ≥ 0 J ε a b ( t v ) = inf γ ∈ Ψ ε a b max t ∈ [ 0,1 ] J ε a b ( γ ( t ) ) > 0.
Lemma 3.11. If J ε ∞ possesses a ( P S ) c sequence, then either c = 0 or c ≥ E ε ∞ . Besides, E ε ∞ ≥ E ∞ .
Proof. Let { v n } ⊂ H 1 ( ℝ N ) and J ε ∞ ( v n ) → c , ( J ε ∞ ) ′ ( v n ) → 0 in H − 1 ( ℝ N ) as n → ∞ . Assume c ≠ 0 , we will prove c ≥ E ε ∞ .
Since { v n } is bounded in H 1 ( ℝ N ) , we may assume v n ⇀ v in H 1 ( ℝ N ) as n → ∞ along a subsequence. Set z n : = v n − v . By the Brézis-Lieb lemma, we obtain
∫ ℝ N ( | ∇ v n | 2 + V ε ∞ ( x ) v n 2 ) = ∫ ℝ N ( | ∇ v | 2 + V ε ∞ ( x ) v 2 ) + ∫ ℝ N ( | ∇ z n | 2 + V ε ∞ ( x ) z n 2 ) + o ( 1 ) . (3.8)
By the proof of Lemma 3.5 in [
∫ ℝ N Y i ε ∞ ( v n ) v n = ∫ ℝ N Y i ε ∞ ( v ) v + ∫ ℝ N Y i ε ∞ ( z n ) z n + o ( 1 ) , i = 1,2, (3.9)
where Y 1 ε ∞ ( v ) : = W 1 ε ∞ ( x ) [ I θ ∗ ( W 1 ε ∞ | v | p ) ] | v | p − 2 v ,
Y 2 ε ∞ ( v ) : = W 2 ε ∞ ( x ) [ I θ ∗ ( W 2 ε ∞ | v | q ) ] | v | q − 2 v , and for any φ ∈ H 1 ( ℝ N ) ,
∫ ℝ N Y i ε ∞ ( v n ) φ = ∫ ℝ N Y i ε ∞ ( v ) φ + ∫ ℝ N Y i ε ∞ ( z n ) φ + o ( 1 ) ‖ φ ‖ 1 , i = 1,2. (3.10)
As the proof of Lemma 3.6 in [
J ε ∞ ( z n ) → c − J ε ∞ ( v ) , ( J ε ∞ ) ′ ( z n ) → 0 in H − 1 ( ℝ N ) as n → ∞ . (3.11)
Case 1 If there exists z n k ≡ 0 , that is v n k ≡ v , then J ε ∞ ( v ) = c ≠ 0 and v ∈ N ε ∞ . Thus c ≥ E ε ∞ .
Case 2 If z n ≠ 0 for all n ∈ ℕ , then there exists t n > 0 such that t n z n ∈ N ε ∞ . Hence
J ε ∞ ( t n z n ) ≥ E ε ∞ . (3.12)
It follows from 〈 ( J ε ∞ ) ′ ( t n z n ) , t n z n 〉 = 0 and 〈 ( J ε ∞ ) ′ ( z n ) , z n 〉 = o ( 1 ) that
( 1 − t n 2 p − 2 ) ∫ ℝ N Y 1 ε ∞ ( z n ) z n + ( 1 − t n 2 q − 2 ) ∫ ℝ N Y 2 ε ∞ ( z n ) z n = o ( 1 ) . (3.13)
Additionally, ‖ z n ‖ 1 2 ≤ C ∫ ℝ N Y 1 ε ∞ ( z n ) z n + Y 2 ε ∞ ( z n ) z n + o ( 1 ) . If
∫ ℝ N ( I θ ∗ | z n | p ) | z n | p → 0 and ∫ ℝ N ( I θ ∗ | z n | q ) | z n | q → 0 as n → ∞ , then
‖ z n ‖ 1 → 0 as n → ∞ . Thus v n → v in H 1 ( ℝ N ) as n → ∞ and
c = J ε ∞ ( v ) ≥ E ε ∞ . If ∫ ℝ N ( I θ ∗ | z n | p ) | z n | p ≥ δ > 0 or ∫ ℝ N ( I θ ∗ | z n | q ) | z n | q ≥ δ > 0 ,
then t n → 1 as n → ∞ by (3.13). Hence J ε ∞ ( t n z n ) → c − J ε ∞ ( v ) as n → ∞ by (3.11), which implies c ≥ J ε ∞ ( v ) + E ε ∞ ≥ E ε ∞ by (3.12).
Finally, it follows from V ε ∞ ( x ) ≥ τ ∞ and W i ε ∞ ( x ) ≤ k i ∞ , i = 1 , 2 for any x ∈ ℝ N that J ε ∞ ( v ) ≥ J ∞ ( v ) for all v ∈ H 1 ( ℝ N ) . Thus, E ε ∞ ≥ E ∞ . ,
Remark 3.12. Similarly, if J ε a b has a ( P S ) c sequence, then either c = 0 or c ≥ E ε ∞ .
Lemma 3.13. J ε a b satisfies the ( P S ) c condition for all c < E ε ∞ .
Proof. Let { v n } ⊂ H 1 ( ℝ N ) and J ε a b ( v n ) → c , ( J ε a b ) ′ ( v n ) → 0 in H − 1 ( ℝ N ) as n → ∞ .
Since { v n } is bounded in H 1 ( ℝ N ) , we assume v n ⇀ v in H 1 ( ℝ N ) as n → ∞ . Then ( J ε a b ) ′ ( v ) = 0 by Lemma 2.4. Set z n : = v n − v . Then z n ⇀ 0 in H 1 ( ℝ N ) and
z n → 0 in L loc t ( ℝ N ) as n → ∞ for t ∈ [ 2,2 * ) . (3.14)
Combine with the classical Brézis-Lieb lemma and Lemma 2.3, we have
J ε a b ( z n ) → c − J ε a b ( v ) , ( J ε a b ) ′ ( z n ) → 0 in H − 1 ( ℝ N ) as n → ∞ . (3.15)
Now we attest J ε ∞ ( z n ) → c − J ε a b ( v ) , ( J ε ∞ ) ′ ( z n ) → 0 in H − 1 ( ℝ N ) as n → ∞ . By definition, for any δ > 0 , there is R > 0 such that | V ε ∞ ( x ) − V ε a ( x ) | ≤ δ , | W i ε ∞ ( x ) − W i ε b i ( x ) | ≤ δ , i = 1,2 for all | x | > R . Hence, according to Lemma 2.2 and the Hölder inequality, we get
| J ε ∞ ( z n ) − J ε a b ( z n ) | ≤ ( p − 2 2 p | z n | 2 2 + ( q − p ) k 2 p q | z n | 2 N q / ( N + θ ) 2 q ) δ + C ( | z n | L 2 ( B R ) 2 + | z n | L 2 N q / ( N + θ ) ( B R ) q ) ,
which together with (3.14) and (3.15), imply that
J ε ∞ ( z n ) → c − J ε a b ( v ) as n → ∞ . (3.16)
For any φ ∈ H 1 ( ℝ N ) , by the Hölder inequality and Lemma 2.1, we have
| 〈 ( J ε ∞ ) ′ ( z n ) − ( J ε a b ) ′ ( z n ) , φ 〉 | ≤ C 1 δ ( | z n | 2 + | z n | 2 N p / ( N + θ ) 2 p − 1 + | z n | 2 N q / ( N + θ ) 2 q − 1 ) ‖ φ ‖ 1 + C 2 ( | z n | L 2 ( B R ) + | z n | L 2 N p / ( N + θ ) ( B R ) 2 p − 1 + | z n | L 2 N q / ( N + θ ) ( B R ) 2 q − 1 ) ‖ φ ‖ 1 ,
which combining with (3.14) and (3.15), implies that
( J ε ∞ ) ′ ( z n ) → 0 in H − 1 ( ℝ N ) as n → ∞ . (3.17)
It follows from (3.16) and (3.17) that { z n } is a ( P S ) c − J ε a b ( v ) sequence of
J ε ∞ . According to Lemma 3.11, either c = J ε a b ( v ) or c ≥ J ε a b ( v ) + E ε ∞ . But the latter contradicts with the assumption c < E ε ∞ . Thus c = J ε a b ( v ) and
J ε a b ( v n ) → J ε a b ( v ) as n → ∞ . (3.18)
We show below that v n → v in H 1 ( ℝ N ) as n → ∞ . According to (3.18), J ε a b ( z n ) → 0 as n → ∞ . Due to
J ε a b ( z n ) = p − 1 2 p ∫ ℝ N ( | ∇ z n | 2 + V ε a ( x ) z n 2 ) + q − p 2 p q ∫ ℝ N Y 2 ε b 2 ( z n ) z n + o ( 1 ) ,
we obtain ∫ ℝ N ( | ∇ z n | 2 + V ε a ( x ) z n 2 ) → 0 as n → ∞ , which means that ‖ z n ‖ 1 → 0 as n → ∞ . By using the Brézis-Lieb lemma, ‖ v n ‖ 1 → ‖ v ‖ 1 as n → ∞ . Hence, v n → v in H 1 ( ℝ N ) as n → ∞ . ,
Lemma 3.14. lim sup ε → 0 E ε a b ≤ E α β , where α = V a ( 0 ) , β i = W i b i ( 0 ) , i = 1 , 2 , β : = ( β 1 , β 2 ) . Meanwhile, if V ( 0 ) ≤ a , W i ( 0 ) ≥ b i , i = 1 , 2 , then lim ε → 0 E ε a b = E a b .
Proof. Set V ¯ ε ( x ) : = V ε a ( x ) − α and W ¯ i ε ( x ) : = β i − W i ε b i ( x ) , i = 1 , 2 . Thus
V ¯ ε ( x ) → 0, W ¯ i ε ( x ) → 0, i = 1,2 a .e . on ℝ N as ε → 0. (3.19)
Meanwhile,
J ε a b ( v ) = J α β ( v ) + 1 2 ∫ ℝ N V ¯ ε ( x ) v 2 + β 1 p ∫ ℝ N W ¯ 1 ε ( x ) ( I θ ∗ | v | p ) | v | p − 1 2 p ∫ ℝ N Y ¯ 1 ε ( v ) v + β 2 q ∫ ℝ N W ¯ 2 ε ( x ) ( I θ ∗ | v | q ) | v | q − 1 2 q ∫ ℝ N Y ¯ 2 ε ( v ) v , (3.20)
where Y ¯ 1 ε ( v ) : = W ¯ 1 ε ( x ) ( I θ ∗ W ¯ 1 ε | v | p ) | v | p − 2 v ,
Y ¯ 2 ε ( v ) : = W ¯ 2 ε ( x ) ( I θ ∗ W ¯ 2 ε | v | q ) | v | q − 2 v . By Lemma 3.3, there is e ∈ T α β . Set
r ε > 0 satisfy r ε e ∈ N ε a b , we get
max r ≥ 0 J ε a b ( r e ) = J ε a b ( r ε e ) ≥ E ε a b . (3.21)
Since J ε a b ( r e ) → − ∞ as r → + ∞ , there exists R 0 > 0 such that J ε a b ( r e ) < 0 , for all r > R 0 . Hence we get r ε ≤ R 0 . We posit r ε → r 0 as ε → 0 . It follows from (3.19), (3.20), (3.21) and the Lebesgue dominated convergence theorem that
E ε a b ≤ J α β ( r ε e ) + t ε 2 2 ∫ ℝ N V ¯ ε ( x ) e 2 + β 1 ⋅ t ε 2 p p ∫ ℝ N W ¯ 1 ε ( x ) ( I θ ∗ | e | p ) | e | p − t ε 2 p 2 p ∫ ℝ N Y ¯ 1 ε ( e ) e + β 2 ⋅ t ε 2 q q ∫ ℝ N W ¯ 2 ε ( x ) [ I θ ∗ | e | q ] | e | q − t ε 2 q 2 q ∫ ℝ N Y ¯ 2 ε ( e ) e → J α β ( r 0 e ) ≤ J α β ( e ) = E α β as ε → 0.
Thus lim sup ε → 0 E ε a b ≤ E α β .
Eventually, if V ( 0 ) ≤ a , W i ( 0 ) ≥ b i , then α = a , β i = b i , i = 1 , 2 . Hence V ¯ ε ( x ) ≥ 0 , W ¯ i ε ( x ) ≥ 0, i = 1,2 for all x ∈ ℝ N . we get J ε a b ( v ) ≥ J α β ( v ) for all v ∈ H 1 ( ℝ N ) by (3.20). Thus, E ε a b ≥ E α β . Due to
E α β ≤ lim inf ε → 0 E ε a b ≤ lim sup ε → 0 E ε a b ≤ E α β , we obtain lim ε → 0 E ε a b = E α β = E a b .
Lemma 3.15. If τ ≤ a < τ ∞ , k ≥ b ≥ k ∞ or τ ≤ a ≤ τ ∞ , k ≥ b > k ∞ , then there exists ε a b > 0 such that for all ε ≤ ε a b , E ε a b is attained at v ε a b > 0 .
Proof. Noting Lemma 3.8, we have E α β < E ∞ , where α = V a ( 0 ) and β i = W i b i ( 0 ) , i = 1 , 2 . By Lemmas 3.14 and 3.11, there exists ε a b > 0 such that E ε a b < E ∞ ≤ E ε ∞ for all ε ≤ ε a b . By Lemma 3.13, J ε a b satisfies the ( P S ) E ε a b condition for all ε ≤ ε a b , which together with Lemmas 3.9 and 3.10 imply that E ε a b is attained at v ε a b ∈ H 1 ( ℝ N ) . Since J ε a b ( v ) = J ε a b ( | v | ) for any v ∈ H 1 ( ℝ N ) , we may assume that v a b ≥ 0 . By bootstrap method and elliptic regularity theory, v ε a b ∈ C 2 ( ℝ N ) . By strong maximum principle, v ε a b > 0 . ,
Setting v ( x ) : = w ( ε x ) , the Equation (1.1) is equivalent to
− Δ v + V ( ε x ) v = Y 1 ( v ) + Y 2 ( v ) , v ∈ H 1 ( ℝ N ) , (4.1)
where Y 1 ( v ) : = W 1 ( ε x ) [ I θ ∗ ( W 1 ( ε x ) | v | p ) ] | v | p − 2 v ,
Y 2 ( v ) : = W 2 ( ε x ) [ I θ ∗ ( W 2 ( ε x ) | v | q ) ] | v | q − 2 v . If v ε ( x ) is a solution of Equation (4.1), then w ε ( x ) = v ε ( x ε ) is a solution of Equation (1.1).
Noting V ( ε x ) = V ε τ ( x ) , W i ( ε x ) = W i ε k i ( x ) , i = 1 , 2 , we find that Equation (4.1) is particular form of Equation (3.2). We set
J ε : = J ε τ k , N ε : = N ε τ k , E ε : = E ε τ k , T ε : = T ε τ k , V ε : = V ε τ , W i ε : = W i ε k i , i = 1,2.
Without loss of generality, we assume x i v = 0 . Then V ( 0 ) = τ , W i ( 0 ) = k i v , i = 1 , 2 .
Lemma 4.1. There exists an m-dimensional subspace D r m of H 1 ( ℝ N ) such that sup v ∈ D r m J ε ( v ) < E ∞ , for all r ≥ r m , ε ≤ ε m , where r m and ε m are existing constants depending on m.
Proof. Choose a = τ , b i = k i v , i = 1 , 2 , in Equation (3.1). By Lemma 3.3, there exists v ∈ T τ k v and v ( x ) = v ( | x | ) > 0 . Let r > 0 , χ r ∈ C 0 ∞ ( ℝ + ) satisfy χ r ( t ) = 1 for t ≤ r and χ r ( t ) = 0 for t ≥ r + 1 with | χ ′ r ( t ) | ≤ 2 . Set v r ( x ) : = χ r ( | x | ) v ( x ) for x ∈ ℝ N . It follows from
‖ v r − v ‖ 1 2 ≤ C ¯ ( ∫ | x | > r | ∇ v | 2 + v 2 ) → 0 as r → ∞ , that v r → v in H 1 ( ℝ N ) ,
v r → v in L 2 N s N + θ ( ℝ N ) and ∫ ℝ N ( I θ ∗ v r s ) v r s → ∫ ℝ N ( I θ ∗ v s ) v s for s = p , q as r → ∞ . There exists d r > 0 such that d r v r ∈ N τ k v and d r → 1 as r → ∞ . Hence
max d ≥ 0 J τ k v ( d v r ) = ( d r 2 2 − d r 2 p 2 p ) ∫ ℝ N Y 1 k 1 v ( v r ) v r + ( d r 2 2 − d r 2 q 2 q ) ∫ ℝ N Y 2 k 2 v ( v r ) v r → p − 1 2 p ∫ ℝ N Y 1 k 1 v ( v ) v + q − 1 2 q ∫ ℝ N Y 2 k 2 v ( v ) v ( r → ∞ ) = max d ≥ 0 J τ k v ( d v ) = J τ k v ( v ) = E τ k v = E τ k v . (4.2)
Additionally,
V ε ( x ) → V ( 0 ) = τ , W i ε ( x ) → W i ( 0 ) = k i v , i = 1,2 as ε → 0 (4.3)
uniformly on any bounded set of x. There exists d ^ r > 0 such that d ^ r v r ∈ N ε and d ^ r → 1 as r → ∞ . Therefore, (4.2) and (4.3) mean that
max d ≥ 0 J ε ( d v r ) = ( d ^ r 2 2 − d ^ r 2 p 2 p ) ∫ ℝ N Y 1 ε ( v r ) v r + ( d ^ r 2 2 − d ^ r 2 q 2 q ) ∫ ℝ N Y 2 ε ( v r ) v r → ( d ^ r 2 2 − d ^ r 2 p 2 p ) ∫ ℝ N Y 1 k 1 v ( v r ) v r + ( d ^ r 2 2 − d ^ r 2 q 2 q ) ∫ ℝ N Y 2 k 2 v ( v r ) v r ( ε → 0 ) → max d ≥ 0 J τ k v ( d v r ) → E τ k v ( r → ∞ ) . (4.4)
According to lemma 3.8, we get m ( τ , k v ) > 1 . We let m v = m ( τ , k v ) . For the maximal integer m ∈ ℤ + with m < m v , we have m ≥ 1 . Define η r j ( x ) : = v r ( x 1 − 2 j ( x + 1 ) , x 2 , ⋯ , x N ) for j = 0,1, ⋯ , m − 1 and set D r m : = s p a n { η r j ( x ) : j = 0,1, ⋯ , m − 1 } . We can get ( η r i , η r j ) 1 = 0 if i ≠ j . Hence dim D r m = m . Similarly as (4.4), for all j = 1 , 2 , ⋯ , m − 1 , we get
max d ≥ 0 J ε ( d ψ r j ) = ( d ^ r 2 2 − d ^ r 2 p 2 p ) ∫ ℝ N Y 1 ε ( ψ r j ) ψ r j + ( d ^ r 2 2 − d ^ r 2 q 2 q ) ∫ ℝ N Y 2 ε ( ψ r j ) ψ r j → ( d ^ r 2 2 − d ^ r 2 p 2 p ) ∫ ℝ N Y 1 k 1 v ( v r ) v r + ( d ^ r 2 2 − d ^ r 2 q 2 q ) ∫ ℝ N Y 2 k 2 v ( v r ) v r ( ε → 0 ) → max d ≥ 0 J τ k v ( d v r ) → E τ k v ( r → ∞ ) .
Thus, for all δ > 0 , there exist r δ > 0 , ε δ > 0 such that
max d ≥ 0 J ε ( d ψ r j ) ≤ E τ k v + δ , for all r ≥ r δ and ε ≤ ε δ , j = 0,1, ⋯ , m − 1 . For any
v ∈ D r m , we posit v = ∑ j = 0 m − 1 d j ψ r j , where d j ∈ ℝ for j = 0,1, ⋯ , m − 1 . Thus, we have J ε ( v ) ≤ ∑ j = 0 m − 1 J ε ( d j ψ r j ) ≤ ∑ j = 0 m − 1 max d ≥ 0 J ε ( d ψ r j ) ≤ m ( E τ k v + δ ) for all r ≥ r δ and
ε ≤ ε δ , which implies that sup v ∈ D r m J ε ( v ) ≤ m ( E τ k v + δ ) . Due to Lemma 3.7, we set 0 < δ < E ∞ m − E τ k v , then there is r m > 0 , ε m > 0 such that sup v ∈ D r m J ε ( v ) < E ∞ , for
all r ≥ r m , ε ≤ ε m . ,
Lemma 4.2. Equation (4.1) has at least m pairs of semiclassical solutions.
Proof. Let us consider the symmetric group ℤ 2 = { i d , − i d } and set Σ : = { T ⊂ D : T is closed and T = − T } . For any T ∈ Σ , the Krasnoselskii genus of T is denoted by
gen ( T ) : = inf { n : thereexists g ∈ C ( T , ℝ n \ { 0 } ) and g is odd } .
Set H : = { h ∈ C ( D , D ) : h is an odd home omorphism } and for any T ∈ Σ , define Benci pseudo-index of T by
i ( T ) : = min h ∈ H gen ( h ( T ) ∩ ∂ B ρ ) ,
where ρ > 0 is a constant defined in Lemma 3.9. Let ς j : = inf i ( T ) ≥ j sup v ∈ T J ε ( v ) ,
j = 1,2, ⋯ , m . We can easily to verify that ς 1 ≤ ς 2 ≤ ⋯ ≤ ς m .
When j = 1 , for any T ∈ Σ and i ( T ) ≥ 1 , we have gen ( T ∩ ∂ B ρ ) ≥ 1 , which means T ∩ ∂ B ρ ≠ ∅ . By Lemma 3.9 that
sup v ∈ T J ε ( v ) > σ and ς 1 ≥ σ .
When j = m , taking into account that the Krasnoselskii genus satisfies the dimension property [
h ∈ H , which implies i ( D r m ) = m . Hence ς m ≤ sup v ∈ D r m J ε ( v ) . Due to Lemmas 4.4,
3.11, we have that for any r ≥ r m , ε ≤ ε m ,
σ ≤ ς 1 ≤ ς 2 ≤ ⋯ ≤ ς m ≤ sup v ∈ D r m J ε ( v ) < E ∞ ≤ E ε ∞ . (4.5)
Next we are going to prove ς j ( j = 1,2, ⋯ , m ) are critical values of J ε by
using Theorem 1.4 in [
( J ε ) c : = { v ∈ H 1 ( ℝ N ) : J ε ( v ) ≤ c } , K c : = { v ∈ H 1 ( ℝ N ) : J ε ( v ) = c , ( J ε ) ′ ( v ) = 0 } .
Since J ε is an even fuctional, ( J ε ) c ∈ Σ , K c ∈ Σ , for all c ∈ [ ς 0 , ς ∞ ] . According to (4.5) and Lemma 3.13, J ε satisfies the ( P S ) c condition for any c ∈ [ ς 0 , ς ∞ ] , which means that K c is compact in H 1 ( ℝ N ) , for any c ∈ [ ς 0 , ς ∞ ] . For any c ∈ [ ς 0 , ς ∞ ] , d > 0 and
( K c ) d : = { v ∈ H 1 ( ℝ N ) : dist ( v , K c ) < d } , choose δ = d 4 , then by the contradiction method we can get that there exists ε ˜ > 0 such that ‖ ( J ε ) ′ ( v ) ‖ ≥ 8 ε ˜ δ , for all
v ∈ J ε − 1 ( [ c − 2 ε ˜ , c + 2 ε ˜ ] ) \ ( K c ) d / 2 ¯ .
On the basis of Lemma 2.3 in [
μ ˜ ∈ C ( [ 0,1 ] × H 1 ( ℝ N ) , H 1 ( ℝ N ) ) such that μ ˜ ( 1, ( J ε ) c + ε ˜ ∩ S ) ⊂ ( J ε ) c − ε ˜ and
μ ˜ ( t , ⋅ ) is an odd homeomorphism on H 1 ( ℝ N ) for any t ∈ [ 0,1 ] . Set μ ( ⋅ ) : = μ ˜ ( 1, ⋅ ) , then μ is an odd homeomorphism on H 1 ( ℝ N ) and
μ ( ( J ε ) c + ε ˜ \ ( K c ) d ) ⊂ ( J ε ) c − ε ˜ . (4.6)
For any T ∈ Σ and T ⊂ ( J ε ) ς 0 = ( J ε ) σ , then J ε ( v ) ≤ σ for any v ∈ T . By Lemma 3.9, we have T ∩ ∂ B ρ = ∅ . As a result, gen ( T ∩ ∂ B ρ ) = 0 and
i ( T ) = min h ∈ H gen ( h ( T ) ∩ ∂ B ρ ) = 0. (4.7)
Then, we get
D r m ⊂ ( J ε ) ς ∞ and i ( D r m ) = m ≥ 1. (4.8)
Combining (4.6), (4.7) and (4.8), we have that ς 1 , ς 2 , ⋯ , ς m are critical values of J ε , and gen ( K c ) ≥ r + 1 if c : = ς j = ς j + 1 = ⋯ = ς j + r with j ≥ 1 and j + r ≤ m . Since J ε is even, we infer that J ε has at least m pairs of critical points which are also solutions of Equation (4.1). ,
Lemma 4.3. Equation (4.1) has at least one positive and one negative least energy solution for m ≥ 1 .
Proof. Choose a = τ , b i = k i , i = 1 , 2 in Equation (3.1), then α = V τ ( 0 ) = V ( 0 ) = τ , β i = W i k i ( 0 ) = W i ( 0 ) = k i , i = 1 , 2 . Due to Lemmas 3.7, 3.11, 3.14, 3.13, J ε has a ( P S ) E ε sequence and satisfies ( P S ) E ε condition. According to Lemma 3.15, there exists ε 0 > 0 such that E ε is attained at v ε > 0 for all ε ≤ ε 0 . Hence, v ε and − v ε are positive and negative least energy solutions of Equation (4.1), respectively. ,
This completes the proof.
We can assume without loss of generality that x w = 0 . Then V ( 0 ) = τ ω , W i ( 0 ) = k i , i = 1 , 2 . Setting a = τ ω , b i = k i , i = 1 , 2 in Equation (3.1), there is v ∈ T τ ω k . Due to Lemma 3.8, m ( τ ω , k ) ≥ 1 . We set
m w = ( m ( τ ω , k ) if m ( τ ω , k ) > 1, 3 2 if m ( τ ω , k ) = 1.
For the maximal integer m < m w , we get m ≥ 1 . Because of Lemma 3.7, m E τ ω k < E ∞ . The remaining proof of this theorem is similar to the proof of Theorem 1.1 and other details are omitted.
In general, we assume x i v = 0 . Then V ( 0 ) = τ , W i ( 0 ) = k i v , i = 1 , 2 . We can verify that the condition of ( A 3 )(i) implies that (1.5) holds. It follows from Theorem 1.1 that Equation (1.1) has a positive groundstate solution w ε ( x ) and Equation (4.1) has a positive least energy solution v ε ( x ) = w ε ( ε x ) . Next, we will prove the case ( A 3 )(i), the other case can be handled similarly.
Lemma 4.4. v ε → v as ε → 0 in the sence of sequence after translations.
Proof. Set ε j → 0 as j → ∞ , v j : = v ε j ∈ T ε j with v j > 0 . Thus, we have
E ε j = J ε j ( v j ) = p − 1 2 p ∫ ℝ N ( | ∇ v j | 2 + V ε j ( x ) v j 2 ) + q − p 2 p q ∫ ℝ N Y 2 ε j ( v j ) v j ≥ C ‖ v j ‖ 1 2 ,
due to Lemma 3.14, we know that { v j } is bounded in H 1 ( ℝ N ) . Let
l i m j → ∞ sup y ∈ ℝ N ∫ B 1 ( y ) v j 2 = 0 , by Lemmas 1.5, 2.1, we obtain v j → 0 in L 2 N r / ( N + θ ) ( ℝ N ) ,
∫ ℝ N ( I θ ∗ v j r ) v j r → 0 as j → ∞ for r = p , q , which together with v j ∈ N ε j imply that ‖ v j ‖ 1 → 0 as j → ∞ . It is a contradiction with ‖ v j ‖ 1 ≥ C > 0 . Thus, there is δ > 0 and y ′ j ∈ ℝ N such that
∫ B 1 ( y ′ j ) v j 2 ≥ δ . (4.9)
Define v ^ j ( x ) : = v j ( x + y ′ j ) , V ^ ε j ( x ) : = V ε j ( x + y ′ j ) , W ^ i ε j ( x ) : = W i ε j ( x + y ′ j ) , i = 1,2 . Thus, v ^ j is the solution of
− Δ v ^ j + V ^ ε j ( x ) v ^ j = Y ^ 1 ε j ( v ^ j ) + Y ^ 2 ε j ( v ^ j ) , v ^ j > 0 (4.10)
with least energy
E ^ ε j = J ^ ε j ( v ^ j ) : = p − 1 2 p ∫ ℝ N Y ^ 1 ε j ( v ^ j ) v ^ j + q − 1 2 q ∫ ℝ N Y ^ 2 ε j ( v ^ j ) v ^ j . (4.11)
where Y ^ 1 ε j ( v ^ j ) : = W ^ 1 ε j ( x ) [ I θ ∗ ( W ^ 1 ε j v ^ j p ) ] v ^ j p − 1 , Y ^ 2 ε j ( v ^ j ) : = W ^ 2 ε j ( x ) [ I θ ∗ ( W ^ 2 ε j v ^ j q ) ] v ^ j q − 1 . Additonally, [ I θ ∗ ( W 1 ε j v j p ) ] ( x + y ′ j ) = [ I θ ∗ ( W ^ 1 ε j v ^ j p ) ] ( x ) , [ I θ ∗ ( W 2 ε j v j q ) ] ( x + y ′ j ) = [ I θ ∗ ( W ^ 2 ε j v ^ j q ) ] ( x ) for any x ∈ ℝ N , which imply that
E ^ ε j = J ^ ε j ( v ^ j ) = J ε j ( v j ) = E ε j . (4.12)
Due to the boundedness of { v ^ j } , we can suppose without loss of generality that
v ^ j ⇀ v in H 1 ( ℝ N ) as j → ∞ , (4.13)
v ^ j → v in L l o c r ( ℝ N ) as j → ∞ for r ∈ [ 2,2 * ) , (4.14)
which combine with (4.9) imply that v ≠ 0 .
According to V and W i , i = 1 , 2 are bounded, we posit
V ε j ( y ′ j ) → V 0 and W i ε j ( y ′ j ) → W i 0 , i = 1,2 as j → ∞ . (4.15)
Because of ∇ V : | ∇ V ( x ) | ≤ M for all x ∈ ℝ N , we have that for any r > 0 , | V ^ ε j ( x ) − V ε j ( y ′ j ) | ≤ ε j M r , for all x ∈ B r ( 0 ) . Hence V ^ ε j → V 0 , W ^ i ε j → W i 0 , i = 1 , 2 as j → ∞ uniformly on any bounded set of x. Using the proof of Lemma 3.14, we have
lim sup j → ∞ E ^ ε j ≤ E V 0 W 0 . (4.16)
Uniting (4.10), (4.13), (4.15), we get that for any φ ∈ C 0 ∞ ( ℝ N ) ,
0 = lim j → ∞ ∫ ℝ N [ ∇ v ^ j ∇ φ + V ^ ε j ( x ) v ^ j φ − Y ^ 1 ε j ( v ^ j ) φ − Y ^ 2 ε j ( v ^ j ) φ ] = ∫ ℝ N [ ∇ v ∇ φ + V 0 v φ − Y 10 ( v ) φ − Y 20 ( v ) φ ] ,
with Y 10 ( v ) : = W 10 [ I θ ∗ ( W 10 v p ) ] v p − 2 v , Y 20 ( u ) : = W 20 [ I θ ∗ ( W 20 v q ) ] v q − 2 v , which means v solves
− Δ v + V 0 v = Y 10 ( v ) + Y 20 ( v ) , v > 0 (4.17)
with energy
J V 0 W 0 ( v ) : = 1 2 ∫ ℝ N ( | ∇ v | 2 + V 0 v 2 ) − 1 2 p ∫ ℝ N Y 10 ( v ) v − 1 2 q ∫ ℝ N Y 20 ( v ) v ≥ E V 0 W 0 . (4.18)
Due to Fatou’s Lemma, we obtain
∫ ℝ N Y i 0 ( v ) v ≤ lim inf j → ∞ ∫ ℝ N Y ^ i ε j ( v ^ j ) v ^ j , i = 1,2. (4.19)
Combining (4.11), (4.16), (4.18) and (4.19),
E V 0 W 0 ≤ J V 0 W 0 ( v ) ≤ lim inf j → ∞ J ^ ε j ( v ^ j ) ≤ lim sup j → ∞ E ^ ε j ≤ E V 0 W 0 . Hence,
lim j → ∞ E ^ ε j = E V 0 W 0 = J V 0 W 0 ( v ) . (4.20)
Choose ξ ∈ C 0 ∞ ( ℝ + ) satisfy supp ξ ( t ) ⊂ B 2 and ξ ≡ 1 on B 1 with
| ξ ′ ( t ) | ≤ 2 . Define μ ˜ j ( x ) : = ξ ( x j ) v ( x ) and z j ( x , y ) : = v ^ j ( x ) − μ ˜ j ( x ) for
x ∈ ℝ N . Thus as j → ∞ , μ ˜ j → v in H 1 ( ℝ N ) , μ ˜ j → v in L r ( ℝ N ) for
r ∈ [ 2, N + θ N − 2 ] , μ ˜ j → v a.e. on ℝ N and z j ⇀ 0 in H 1 ( ℝ N ) , z j → 0 in L l o c r ( ℝ N ) for r ∈ [ 2, N + θ N − 2 ) , z j → 0 a.e. on ℝ N .
Next, our main goal is to obtain J ^ ε j ( z j ) → 0 and 〈 ( J ^ ε j ) ′ ( z j ) , z j 〉 → 0 as
j → ∞ , where
J ^ ε j ( z j ) : = 1 2 ∫ ℝ N | ∇ z j | 2 + V ^ ε j ( x ) | z j | 2 − 1 2 p ∫ ℝ N Y ^ 1 ε j ( z j ) z j − 1 2 q ∫ ℝ N Y ^ 2 ε j ( z j ) z j .
Indeed, similar to the proof of Theorem 1.3 in [
‖ z j ‖ 1 2 = ‖ v ^ j ‖ 1 2 − ‖ μ ˜ j ‖ 1 2 + o ( 1 ) , (4.21)
∫ ℝ N V ^ ε j ( x ) | z j | 2 = ∫ ℝ N V ^ ε j ( x ) | v ^ j | 2 − ∫ ℝ N V ^ ε j ( x ) | μ ˜ j | 2 + o ( 1 ) , (4.22)
∫ ℝ N Y ^ 1 ε j ( z j ) z j = ∫ ℝ N Y ^ i ε j ( v ^ j ) v ^ j − ∫ ℝ N Y ^ i ε j ( μ ˜ j ) μ ˜ j + o ( 1 ) , i = 1,2. (4.23)
According to the Lebesgue dominated convergence theorem, we get that
∫ ℝ N V ^ ε j ( x ) μ ˜ j 2 = ∫ ℝ N V 0 v 2 + o ( 1 ) , (4.24)
∫ ℝ N Y ^ i ε j ( μ ˜ j ) μ ˜ j = ∫ ℝ N Y i 0 ( v ) v + o ( 1 ) , i = 1,2. (4.25)
Additionally,
| ∇ μ ˜ j | 2 2 = | ∇ v | 2 2 + o ( 1 ) . (4.26)
By (4.21), (4.22), (4.23), (4.24), (4.25), (4.26), (4.20), (4.10) and (4.17), we have
J ^ ε j ( z j ) = E ^ ε j − J V 0 W 0 ( v ) + o ( 1 ) = o ( 1 ) , 〈 ( J ^ ε j ) ′ ( z j ) , z j 〉 = 〈 ( J ^ ε j ) ′ ( v j ) , v j 〉 − 〈 ( J V 0 W 0 ) ′ ( v ) , v 〉 + o ( 1 ) = o ( 1 ) . (4.27)
Due to (4.27), we get that o ( 1 ) = J ^ ε j ( z j ) − 1 2 p 〈 ( J ^ ε j ) ′ ( z j ) , z j 〉 ≥ C ‖ z j ‖ 1 2 ,
which means z j → 0 in H 1 ( ℝ N ) as j → ∞ . Hence ‖ v ^ j − v ‖ 1 ≤ ‖ z j ‖ 1 + ‖ μ ˜ j − v ‖ 1 as j → ∞ . ,
Lemma 4.5. v ^ j ( x ) → 0 as | x | → ∞ uniformly in j ∈ ℕ .
Proof. We have that there are δ > 0 , x n ∈ ℝ N , | x n | → ∞ as n → ∞ such that | v ^ j n ( x n ) | ≥ δ by contradiction method. Meanwhile, there exists C 0 > 0
which independent of j such that | v ^ j n ( x n ) | ≤ C 0 ( ∫ B 1 ( x n ) v ^ j n 2 ) 1 2 . Thus by applying
the Minkowski inequality, we have
δ ≤ | v ^ j n ( x n ) | ≤ C 0 ( ∫ ℝ N | v ^ j n − v | 2 ) 1 2 + C 0 ( ∫ B 1 ( x n ) | v | 2 ) 1 2 → 0 as n → ∞ ,
which is impossible. ,
Lemma 4.6. { ξ j y ′ j } j is bounded on ℝ N .
Proof. Assume by contradiction that there is | ε j y ′ j | → ∞ as j → ∞ along a subsequence. Therefore V 0 ≥ τ ∞ > τ and W i 0 ≤ k i ∞ ≤ k i v , i = 1 , 2 , which together with Lemma 3.5, imply that E V 0 W 0 > E τ k v . However, due to (4.12), (4.20) and
Lemma 3.14, we have E V 0 W 0 = lim j → ∞ E ε j ≤ lim sup j → ∞ E ε j ≤ E τ k v , which is a contradiction.
Hence, without loss of generality we may posit
ε j y ′ j → x 0 as j → ∞ . (4.28)
By (4.15), we obtain
V 0 = V ( x 0 ) , W i 0 = W i ( x 0 ) , i = 1 , 2. (4.29)
Noticing (4.17), we claim v is a least energy solution of Equation (1.7). ,
Lemma 4.7. { ε y ε } ε is bounded, where y ε ∈ ℝ N is a maximum point of v ε .
Proof. Suppose there exists ε j → 0 with | ε j y j | → ∞ as j → ∞ where y j : = y ε j is a maximum point of v j : = v ε j . By Lemmas 4.4, 4.5, 4.6, we can obtain that there is y ′ j ∈ ℝ N such that v ^ j = v j ( ⋅ + y ′ j ) → v ≠ 0 in H 1 ( ℝ N ) as j → ∞ and v ^ j ( x ) → 0 as | x | → ∞ uniformly in j ∈ ℕ , { ε j y ′ j } j is bounded on ℝ N . Hence | ε j y j − ε j y ′ j | ≥ | ε j y j | − | ε j y ′ j | → ∞ as j → ∞ , which means that
| y j − y ′ j | → ∞ as j → ∞ . Therefore max ℝ N v j = v j ( y j ) = v ^ j ( y j − y ′ j ) → 0 as
j → ∞ . Due to v ^ j > 0 , we get v ^ j → 0 as j → ∞ uinformly in x ∈ ℝ N , which contradicts with v ≠ 0 .
Lemma 4.8. lim ε → 0 dist ( ε y ε , S v ) = 0 .
Proof. According to Lemma 4.7, we get there is ε j → 0 with ε j y j → y 0 as j → ∞ , where y j : = y ε j is the maximum point of v j : = v ε j . We just require to attest y 0 ∈ S v . By Lemmas 4.4, 4.6, there exists y ′ j ∈ ℝ N satisfying v ^ j ( x ) = v j ( x + y ′ j ) and (4.28). Due to Lemma 4.5, we can suppose v ^ j ( x ′ j ) = max ℝ N v ^ j and { x ′ j } j is bounded on ℝ N . Hence y j = x ′ j + y ′ j and ε j y j − ε j y ′ j = ε j x ′ j → 0 as j → ∞ . And combining with (4.28), (4.29), mean that
y 0 = x 0 , V ( y 0 ) = V 0 , W i ( y 0 ) = W i 0 , i = 1 , 2. (4.30)
Assume by contradiction that y 0 ∉ S v , then we have V ( y 0 ) = τ , W 1 ( y 0 ) < k 1 v , W 2 ( y 0 ) = k 2 v or V ( y 0 ) = τ , W 1 ( y 0 ) = k 1 v , W 2 ( y 0 ) < k 2 v or V ( y 0 ) > τ , W i ( y 0 ) ≤ k i v , i = 1 , 2 . Due to Lemma 3.5, E V ( y 0 ) W ( y 0 ) > E τ k v . Combining (4.12), (4.20), (4.30), and Lemma 3.14, we have
lim j → ∞ E ε j = lim j → ∞ E ^ ε j = E V 0 W 0 = E V ( y 0 ) W ( y 0 ) > E τ k v ≥ lim sup j → ∞ E ε j , which is a contradiction.
Particularly, if V ∩ ( W 1 ∩ W 2 ) ≠ ∅ , then x 0 ∈ S v = V ∩ ( W 1 ∩ W 2 ) , we can
get lim ε → 0 dist ( ε y ε , V ∩ ( W 1 ∩ W 2 ) ) = 0 and V ( x 0 ) = τ , W i ( x 0 ) = k i , i = 1 , 2 ,
which combine with Equation (1.7) mean that v is a least energy solution of Equation (1.8). ,
Lemma 4.9. For p , q ∈ ( 2, N + θ N − 2 ) , there is C > 0 and R ^ > 0 such that for all small ε > 0 , v ε ( x ) ≤ C | x | 1 − N 2 exp ( − τ 2 | x | ) for all | x | ≥ R ^ .
Proof. We check its correctness for any sequence. By Lemma 4.5, we obtain
lim | x | → ∞ W ^ 1 ε j ( x ) ( I θ ∗ ( W ^ 1 ε j v ^ j p ) ) ( x ) ( v ^ j ( x ) ) p − 2 + W ^ 2 ε j ( x ) ( I θ ∗ ( W ^ 2 ε j v ^ j q ) ) ( x ) ( v ^ j ( x ) ) q − 2 = 0
uniformly in j ∈ ℕ , which means that there exists R ^ > 0 such that for any | x | ≥ R ^ and j ∈ ℕ ,
W ^ 1 ε j ( x ) ( I θ ∗ ( W ^ 1 ε j v ^ j p ) ) ( x ) ( v ^ j ( x ) ) p − 2 + W ^ 2 ε j ( x ) ( I θ ∗ ( W ^ 2 ε j v ^ j q ) ) ( x ) ( v ^ j ( x ) ) q − 2 ≤ 3 4 τ . (4.31)
Thus, by (4.10) and (4.31), we have − Δ v ^ j + τ 4 v ^ j ≤ 0 for any | x | ≥ R ^ and
j ∈ ℕ .
Similar to the proof of Theorem 1.3 in [
and j ∈ ℕ , v ^ j ( x ) ≤ C | x | 1 − N 2 exp ( − τ 2 | x | ) . ,
Set x ε = ε y ε . Then w ε ( x ε ) = v ε ( y ε ) . Due to Lemma 4.7, x ε is a maximum point of w ε and { x ε } ε is bounded on ℝ N . According to Lemma 4.8, lim ε → 0 dist ( x ε , S v ) = 0 . On the basis of Lemmas 4.4, 4.5, v ^ ε ( x ) = v ε ( x + y ′ ε ) = w ε ( ε x + x ε − ε x ′ ε ) , where x ′ ε = y ε − y ′ ε is a maximum point of v ^ ε with ε x ′ ε → 0 as ε → 0 . Finally, we obtain that,
w ε ( x ) ≤ C ε N − 1 2 | x − x ε | 1 − N 2 exp ( − τ 4 ε | x − x ε | ) , for all | x | ≥ R , by Lemma 4.9,
where R : = R ^ + sup ε | x ε | .
The proof of Theorem 1.3 is completed.
By making reasonable assumptions about potentials, we use pseudo-index theory to prove the multiplicity of semiclassical solutions to Equation (1.1). The existence of groundstate solutions are proved using Nehari method. In addition, we also demonstrate the concentration and convergence of the positive groundstate solution.
All of the data and material is owned by the authors.
We declare that there are no competing interests that might be perceived to influence the results reported in this paper.
Zhao, X.Y. (2023) Multiplicity and Concentration of Solutions for Choquard Equation with Competing Potentials via Pseudo-Index Theory. Open Access Library Journal, 10: e11026. https://doi.org/10.4236/oalib.1111026