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In this paper, the existence of a pair of ordered solutions for the following class of equations in (1) was studied. A bounded (PS) (Palais-Smale) sequence was constructed and the related variational principle was used to prove the existence of the positive solution. The existence of the ordered solutions is finally found.

In recent years, studies about the nontrivial solutions of Schrödinger equations are very popular, involving differential equations, linear algebra and many subjects. The solution of these problems cannot only develop new methods, such as minimizations [^{ }reveal new laws, but also have important academic value and wide application prospects [

In this paper, we consider the existence of ordered solutions for the following quasilinear Schrödinger equations:

− Δ u + V ( x ) u − Δ ( u 2 ) u = g ( x , u ) in ℝ N (2)

we make the following assumptions:

(V_{1}) V ∈ C 1 ( ℝ N , ℝ ) ;

(V_{2}) 0 < 2 C 0 < V 0 ≤ V ( x ) ≤ V 1 < ∞ , for all x ∈ ℝ N ;

(V_{3}) V ( x ) is symmetrical radially, which is V ( x ) = V ( | x | ) ;

(V_{4}) There exists γ ∈ [ 1 , 2 ) , such that ( ∇ V ( x ) ⋅ x ) + ∈ L 2 * / 2 * − γ ( ℝ N ) , where

( ∇ V ( x ) ⋅ x ) + = max { ∇ V ( x ) ⋅ x , 0 } ;

(g_{1}) g ∈ C ( ℝ , ℝ ) , for any t ≤ 0 , g ( x , t ) = 0 , g ( x , t ) = o ( t ) as t → 0 + ;

(g_{2}) There exist C 0 > 0 and q ∈ ( 4 , 2 ⋅ 2 * ) , such that | g ( x , t ) | ≤ C 0 ( t + t q − 1 ) as t ∈ ℝ + ;

(g_{3}) lim t → + ∞ g ( x , t ) t = + ∞ ;

(g_{4}) G ( x , t ) = ∫ 0 t g ( x , s ) d s ≥ 0 for all t ∈ ℝ + ;

(g_{5}) There exists K > 0 , such that | ∫ ℝ N x ⋅ ∇ x G d x | < K , where

∇ x G = ( ∂ G ( x , t ) ∂ x 1 , ∂ G ( x , t ) ∂ x 2 , ⋯ , ∂ G ( x , t ) ∂ x N ) ;

(g_{6}) There exists C * > 0 , such that ∫ ℝ N | g t ( x , t ) | d x < C * ;

(g_{7}) There exists μ ≥ 4 , such that 0 < μ G ( x , s ) < g ( x , s ) s for any s > 0 .

We are now state the main results of the paper:

Theorem 1.1 Assume conditions (V_{1})-(V_{4}), (g_{1})-(g_{7}) are satisfied, there is at least one positive solution to Equation (2).

Theorem 1.2 Assume conditions (V_{1})-(V_{4}), (g_{1})-(g_{7}) are satisfied, there is at least one pair of ordered positive solutions to Equation (2).

We observe that formally problem (2) is the Euler-Lagrange equation associated of the natural energy functional given by

J ( u ) = 1 2 ∫ ℝ N ( | ∇ u | 2 + V ( x ) u 2 ) d x + ∫ ℝ N u 2 | ∇ u | 2 d x − ∫ ℝ N G ( x , u ) d x . (3)

It is well known that J is not well defined in general in H 1 ( ℝ N ) . To overcome this difficulty, we make the change of variables developed in [

f ′ ( t ) = 1 1 + 2 f 2 ( t ) , f ( 0 ) = 0 , on t ∈ [ 0 , + ∞ ) , (4)

and

f ( t ) = − f ( − t ) , on t ∈ ( − ∞ , 0 ] .

Thus we can write J ( u ) as

I ( v ) : = J ( f ( u ) ) = 1 2 ∫ ℝ N ( | ∇ v | 2 + V ( x ) f 2 ( v ) ) d x − ∫ ℝ N G ( x , f ( v ) ) d x , (5)

which is well defined in the space

H r 1 ( ℝ N ) : = { u ∈ H 1 ( ℝ N ) | u is symmetrical radially } .

We can see that the nontrivial critical points of I ( v ) are precise weak solutions for

− Δ v = − V ( x ) f ( v ) f ′ ( v ) + g ( x , f ( v ) ) f ′ ( v ) , x ∈ ℝ N . (6)

Lemma 3.1 (see in [

1) | f ′ ( t ) | ≤ 1 ;

2) | f ( t ) | ≤ | t | ;

3) | f ( t ) | ≤ 2 1 4 | t | 1 2 ;

4) 1 2 f ( t ) ≤ t f ′ ( t ) ≤ f ( t ) , for all t ∈ ℝ + ;

5) 1 2 f 2 ( t ) ≤ f ( t ) f ′ ( t ) t ≤ f 2 ( t ) ;

6) lim t → 0 f ( t ) t = 1 ;

7) lim t → ∞ f ( t ) t = 2 1 4 ;

8) There exists a positive constant C, such that

| f ( t ) | ≥ { C | t | , if | t | ≤ 1 , C | t | 1 2 , if | t | ≥ 1 ;

9) For each λ > 1 , we have f 2 ( λ t ) ≤ λ 2 f 2 ( t ) , for all t ∈ ℝ .

Proof: The proofs of (1)-(3) and (6) only require the knowledge of calculus. The reader can refer to the literature [

Let l ( t ) = 2 t − 1 + 2 f 2 ( t ) f ( t ) , there is l ( 0 ) = 0 obviously, and

l ′ ( t ) = 2 − 2 ( f ′ ( t ) ) 2 f 2 ( t ) − 1 + 2 f 2 ( t ) f ′ ( t ) = 1 − 2 f 2 ( t ) ( f ′ ( t ) ) 2 = 1 1 + 2 f 2 ( t ) > 0.

Thus l ( t ) ≥ 0 for all t ∈ ℝ + , so we have 2 t f ′ ( t ) ≥ f ( t ) ( t ∈ ℝ + ). t f ′ ( t ) ≤ f ( t ) ( t ∈ ℝ + ) can be proved similarly. (5) can be derived from (4) easily.

From the conclusion (4), we can get f ( t ) ≥ f ( 1 ) t for any t > 1 . Thus we have lim t → + ∞ f ( t ) = + ∞ , and

lim t → + ∞ f 2 ( t ) t = lim t → + ∞ 2 f ′ ( t ) f ( t ) = 2 lim t → + ∞ f ( t ) 1 + 2 f 2 ( t ) = 2 .

Therefore

lim t → + ∞ f ( t ) t = 2 1 4 .

(8) can be derived from (6) (7).

Finally we prove (9). For any t > 0 , we have the following inequality by (5)

( f 2 ( t ) ) ′ t f 2 ( t ) = 2 f ( t ) f ′ ( t ) t f 2 ( t ) ≤ 2.

Then we have

ln f 2 ( λ t ) f 2 ( t ) = ∫ t λ t ( f 2 ( s ) ) ′ f 2 ( s ) d s ≥ ∫ t λ t 2 s d s = 2 ln λ t t = ln λ 2 .

Thus

f 2 ( λ t ) ≤ λ 2 f 2 ( t ) .

For all t ∈ ℝ , we have f 2 ( λ t ) ≤ λ 2 f 2 ( t ) , because f 2 ( x ) is even function.

Remark 3.1. To convenience, we note support as supp, and superior as sup.

Proposition 3.1. (Rellich-Kondrachov theorem) Let Ω ⊆ ℝ N be an open, bounded Lipschitz domain, and let 1 ≤ p < n . Set

p ∗ : = n p n − p .

Then the Sobolev space W 1 , p ( Ω , ℝ ) is continuously embedded in the L q ( Ω , ℝ ) space and is compactly embedded in L q ( Ω , ℝ ) for every 1 ≤ q < p ∗ . In symbols, W 1 , p ( Ω , ℝ ) embedding in L p ∗ ( Ω , ℝ ) , and W 1 , p ( Ω ) ⊂ L q ( Ω ) for 1 ≤ q < p ∗ .

Proposition 3.2. (Hölder inequality) Let ( S , Σ , μ ) be a measure space and let p , q ∈ [ 1 , + ∞ ) with 1 / p + 1 / q = 1 . Then, for all measurable real or complex-valued functions f and g on S,

‖ f g ‖ 1 ≤ ‖ f ‖ p ‖ g ‖ q .

Proposition 3.3. (Sobolev inequality) Assume that u is a continuously differentiable real-valued function on ℝ n with compact support. Then for 1 ≤ p < n there is a constant C depending only on n and p such that

‖ u ‖ L p ∗ ( ℝ n ) ≤ C ‖ D u ‖ L p (ℝn)

with

1 / p ∗ = 1 / p − 1 / n .

Lemma 3.2 X is Banach space, and ‖ ⋅ ‖ X is a norm of this space. ς ⊂ ℝ + is a range. The family of functionals { Φ λ ( v ) } λ ∈ ς of class C 1 in X satisfy:

1) For all λ ∈ ς , there is Φ λ ( v ) = A ( v ) − λ B ( v ) . There is A ( v ) → + ∞ or B ( v ) → + ∞ as ‖ v ‖ X → ∞ ;

2) For each λ ∈ ς and for all v ∈ X , there is B ( v ) ≥ 0 ;

3) There exist two points v 1 , v 2 ∈ X , such that

c λ = inf γ ∈ Γ max t ∈ [ 0 , 1 ] Φ λ ( γ ( t ) ) > max { Φ λ ( v 1 ) , Φ λ ( v 2 ) } , foreach λ ∈ ς (7)

where Γ = { γ ∈ C ( [ 0 , 1 ] , X ) | γ ( 0 ) = v 1 , γ ( 1 ) = v 2 } .

Thus there exists a sequence { v n ( λ ) } ⊂ X , for a.e. λ ∈ ς , we have

1) { v n ( λ ) } is bounded;

2) { Φ λ ( v n ( λ ) ) } → c λ ;

3) { Φ ′ λ ( v n ( λ ) ) } → 0 .

In order to use Lemma 3.2, in the following discussion, we take

X = H r 1 ( ℝ N ) : = { u ∈ H 1 ( ℝ N ) | u is symmetrical radially } ,

and consider the following family of functional

I λ ( v ) = 1 2 ∫ ℝ N ( | ∇ v | 2 + V ( x ) f 2 ( v ) ) d x − λ ∫ ℝ N G ( x , f ( v ) ) d x , v ∈ X ,

where λ ∈ [ 1 2 , 1 ] .

Define

A ( v ) = 1 2 ∫ ℝ N ( | ∇ v | 2 + V ( x ) f 2 ( v ) ) d x ,

B ( v ) = ∫ ℝ N G ( x , f ( v ) ) d x ,

so that

I λ ( v ) = A ( v ) − λ B ( v ) ,

The following lemma shows that I λ ( v ) satisfies the conditions of Lemma 3.2.

Lemma 3.3 Assume conditions (V_{1}) and (g_{1})-(g_{4}) are satisfied, we have

1) B ( v ) ≥ 0 for all v ∈ X ;

2) A ( v ) → ∞ as ‖ v ‖ → ∞ ;

3) There exists v 0 ∈ X independent on l, such that I λ ( v 0 ) < 0 for each λ ∈ [ 1 2 , 1 ] ;

4) c λ = inf γ ∈ Γ max t ∈ [ 0 , 1 ] I λ ( γ ( t ) ) > max { I λ ( 0 ) , I λ ( v 0 ) } , for each λ ∈ [ 1 2 , 1 ] , where Γ = { γ ∈ C ( [ 0 , 1 ] , X ) | γ ( 0 ) = 0 , γ ( 1 ) = v 0 } .

Proof: (1) can be directly obtained from (g_{4}). Let’s prove (2) by Lemma 3.1 and embedding theorem, we infer that

‖ v ‖ 2 = ∫ ℝ N | ∇ v | 2 d x + ∫ { x | | v ( x ) | ≤ 1 } v 2 d x + ∫ { x | | v ( x ) | > 1 } v 2 d x ≤ ∫ ℝ N | ∇ v | 2 d x + C ∫ { x | | v ( x ) | ≤ 1 } f 2 ( v ) d x + ∫ { x | | v ( x ) | > 1 } | v | 2 * d x ≤ ∫ ℝ N | ∇ v | 2 d x + C 1 ∫ ℝ N V ( x ) f 2 ( v ) d x + C 2 ( ∫ ℝ N | ∇ v | 2 d x ) 2 * 2 ≤ C 3 ( A ( v ) + A ( v ) 2 * 2 ) .

Therefore, A is convex.

To prove (3), firstly, we let

J λ ( u ) = 1 2 ∫ ℝ N ( | ∇ u | 2 + V ( x ) u 2 ) d x + ∫ ℝ N u 2 | ∇ u | 2 d x − λ ∫ ℝ N G ( x , u ) d x

and U = { x ∈ ℝ N | u ( x ) ≠ 0 } ( m e a s U > 0 ). Then fixing a non-negative radial symmetry function u ∈ C 0 ∞ ( ℝ N ) \ { 0 } , for all t > 0 , we have

J 1 2 ( t u ( x t ) ) = t N 2 ∫ ℝ N | ∇ u | 2 d x + t N + 2 2 ∫ ℝ N V ( t x ) u 2 d x + t N + 2 ∫ ℝ N u 2 | ∇ u | 2 d x − t N 2 ∫ ℝ N G ( x , t u ) d x ≤ t N + 2 2 ( 1 t 2 ∫ ℝ N | ∇ u | 2 d x + ∫ ℝ N V 1 u 2 + 2 ∫ ℝ N u 2 | ∇ u | 2 d x − ∫ ℝ N G ( x , t u ) t 2 d x ) .

By (g_{3}): lim t → + ∞ g ( x , t ) t = + ∞ , we have

lim t → + ∞ G ( x , t u ) t 2 = lim t → + ∞ ∫ 0 t u g ( x , s ) d s t 2 = lim t → + ∞ u g ( x , t u ) 2 t → + ∞ , a . e . x ∈ U .

J 1 2 ( t u ( x t ) ) < 0 when t is large enough.

Thus there exists v 0 = f − 1 ( u 0 ) ∈ X ( v 0 is independent on l), such that I λ ( v 0 ) = J λ ( u 0 ) ≤ J 1 2 ( u 0 ) < 0 , for each λ ∈ [ 1 2 , 1 ] .

Finally, we prove (4). Define G ^ ( x , t ) = − V 0 2 f 2 ( t ) + G ( x , f ( t ) ) . By (g_{1}), (g_{2}) and Lemma 3.1, we have lim t → 0 G ^ ( x , t ) t 2 = − V 0 2 , lim t → 0 G ^ ( x , t ) | t | 2 * = 0 .

Hence, there exists C > 0 , such that G ^ ( x , t ) ≤ − V 0 4 t 2 + C | t | 2 * , for all t ∈ ℝ .

Then

I λ ( v ) ≥ 1 2 ∫ ℝ N | ∇ v | 2 d x + 1 2 ∫ ℝ N V 0 f 2 ( v ) d x − ∫ ℝ N G ( x , f ( v ) ) ≥ 1 2 ∫ ℝ N | ∇ v | 2 d x + 1 2 ∫ ℝ N V 0 f 2 ( v ) d x + 1 4 ∫ ℝ N V 0 v 2 d x − C ∫ ℝ N | v | 2 * d x − 1 2 ∫ ℝ N V 0 f 2 ( v ) d x ≥ 1 2 ∫ ℝ N | ∇ v | 2 d x + V 0 4 ∫ ℝ N v 2 d x − C ∫ ℝ N | v | 2 * d x ≥ min { 1 2 , V 0 4 } ‖ v ‖ 2 − C | v | L 2 * 2 * .

It follows that I λ ( v ) > 0 with 0 ≤ ‖ v ‖ < ρ . We also have I λ ( 0 ) = 0 , I λ ( v 0 ) < 0 , thus c λ = inf γ ∈ Γ max t ∈ [ 0 , 1 ] I λ ( γ ( t ) ) > 0 > max { I λ ( 0 ) , I λ ( v 0 ) } .

By Lemma 3.2 and Lemma 3.3, we can construct the (PS) sequence of I λ ( u ) . Specifically, there exists ς 1 ⊂ [ 1 2 , 1 ] ( m e a s ς 1 = 0 ), for each λ ∈ [ 1 2 , 1 ] \ ς 1 , then we have a sequence { v n } ⊂ X , satisfy

1) { v n } ⊂ X is bounded;

2) I λ ( v n ) → c λ ;

3) I ′ λ ( v n ) → 0 .

Lemma 3.4 If { v n } is a (PS) sequence of I λ , then there exists a subsequence, still denoted by { v n } , which convergence to the positive critical point v λ of I λ .

Proof: Since { v n } ⊂ X is bounded, by Rellich-Kondrachov theorem, there exists v λ ∈ X , such that

i) v n → v λ in X;

ii) v n → v λ in L q 2 ( ℝ N ) ;

iii) v n → v λ a.e. x ∈ ℝ N .

By (i) and (ii), we obtain I ′ λ ( v λ ) = 0 .

Next we prove v n → v λ in X. Firstly, let

H ( x , t ) = 1 2 V ( x ) t 2 − 1 2 V ( x ) f 2 ( t ) + λ G ( x , f ( t ) ) .

Hence I λ ( v ) is transformed into

I λ ( v ) = 1 2 ∫ ℝ N ( | ∇ v | 2 + V ( x ) v 2 ) d x − ∫ ℝ N H ( x , v ) d x .

Let h ( x , t ) = d H ( x , t ) d t , so that

h ( x , t ) = V ( x ) t − V ( x ) f ( t ) f ′ ( t ) + λ g ( x , f ( t ) ) f ′ ( t ) .

By (g_{1}), (g_{2}) and Lemma 3.1, there exists C 1 , C 2 > 0 , for every x ∈ ℝ N and for all t ∈ ℝ , such that

| h ( x , t ) | ≤ V ( x ) | t | + V ( x ) | f ( t ) | f ′ ( t ) + λ | g ( x , f ( t ) ) | f ′ ( t ) ≤ 2 V 1 | t | + λ C 0 | f ( t ) | f ′ ( t ) + λ C 0 | f ( t ) | q − 1 f ′ ( t ) ≤ C 1 | t | + C 2 | t | q − 1 2 . (8)

By (8) and v n → v λ in L q 2 ( ℝ N ) , we get

lim n → ∞ ∫ ℝ N ( h ( x , v n ) − h ( x , v λ ) ) ( v n − v λ ) d x = 0.

Thus

o ( 1 ) = 〈 I ′ λ ( v n ) − I ′ λ ( v λ ) , v n − v λ 〉 = ∫ ℝ N ( | ∇ ( v n − v λ ) | 2 + V ( x ) ( v n − v λ ) 2 ) d x − ∫ ℝ N ( h ( x , v n ) − h ( x , v λ ) ) ( v n − v λ ) ≥ min { 1 , V 0 } ‖ v n − v λ ‖ 2 + o ( 1 ) ,

that is v n → v λ in X. Therefore, v λ > 0 is the critical point of functional I λ , and I λ ( v λ ) = c λ . This completes the proof.

Lemma 3.5 Suppose that the conditions of Theorem 1.1 are satisfied. Then there exists { λ n } ⊂ [ 1 2 , 1 ] and corresponding critical point sequence { v n * } ⊂ X \ { 0 } , such that lim n → ∞ λ n = 1 and v n * > 0 , I λ n ( v n * ) = c λ n ≤ c 1 2 , I ′ λ n ( v n * ) = 0 , n = 1 , 2 , ⋯ .

Proof: Let λ 1 ∈ [ 1 2 , 1 ] \ ς 1 , by Lemma 3.1, there exists (PS) sequence { v 1 , m } ⊂ X , such that I λ 1 ( v 1 , m ) → c λ 1 , I ′ λ 1 ( v 1 , m ) → 0 as m → ∞ . By Lemma 3.4, we have v 1 , m → v 1 * , and I λ 1 ( v 1 * ) = c λ 1 , I ′ λ 1 ( v 1 * ) = 0 in X as m → ∞ .

Similarly, let λ 2 ∈ [ λ 1 + 1 2 , 1 ] \ ς 1 , we have I λ 2 ( v 2 , m ) → c λ 2 , I ′ λ 2 ( v 2 , m ) → 0 as m → ∞ , and v 2 , m → v 2 * , I λ 2 ( v 2 * ) = c λ 2 , I ′ λ 2 ( v 2 * ) = 0 in X.

Let λ n ∈ [ λ n − 1 + 1 2 , 1 ] \ ς 1 , we have I λ n ( v n , m ) → c λ n , I ′ λ n ( v n , m ) → 0 as m → ∞ , and v n , m → v n * , I λ n ( v n * ) = c λ n , I ′ λ n ( v n * ) = 0 in X.

Thus we get lim n → ∞ λ n = 1 , and since I λ ( v ) is monotonically decreasing with l, so that I λ n ( v n * ) = c λ n ≤ c 1 / 2 . This completes the proof.

Lemma 3.6 If u ∈ X is a critical point of I λ , then

N − 2 2 ∫ ℝ N | ∇ u | 2 d x + N 2 ∫ ℝ N V ( x ) f 2 ( u ) d x + 1 2 ∫ ℝ N ∇ V ( x ) ⋅ x f 2 ( u ) d x − λ N ∫ ℝ N G ( x , f ( u ) ) d x − ∫ ℝ N x ⋅ ∇ G x d x = 0.

Proof: Multiply the two sides of the equation

− Δ v + V ( x ) f ( v ) f ′ ( v ) = λ g ( x , f ( v ) ) f ′ (v)

by x ⋅ ∇ v , we have

0 = ( Δ v − V ( x ) f ( v ) f ′ ( v ) + λ g ( x , f ( v ) ) f ′ ( v ) ) ( x ⋅ ∇ v ) = d i v ( ∇ v ( x ⋅ ∇ v ) − | ∇ v | 2 − x ⋅ ∇ ( | ∇ v | 2 2 ) ) + λ x ⋅ ∇ G ( x , f ( v ) ) + 1 2 x f 2 ( v ) ∇ V ( x ) + N 2 V ( x ) f 2 ( v ) = d i v ( ∇ v ( x ⋅ ∇ v ) − x ⋅ | ∇ v | 2 2 + λ x G ( x , f ( v ) ) ) + N − 2 2 | ∇ v | 2 − λ N G ( x , f ( v ) ) + 1 2 x f 2 ( v ) ∇ V ( x ) + N 2 V ( x ) f 2 ( v ) .

Finally, we integrate the equation on ℝ N , and then the improved Pohozaev type identity can be obtained.

Lemma 3.7 The critical point sequence obtained in Lemma 3.2.7 is bounded in Lemma 3.5.

Proof: For convenience, we let { v n } denote { v n * } of Lemma 3.5. By I λ n ( v n ) = c λ n ≤ c 1 / 2 in Lemma 3.5, Lemma 3.6, Hölder inequality, Sobolev inequality, (g_{5}) and (V_{4}), then there exists γ ∈ [ 1 , 2 ) , such that

∫ ℝ N | ∇ v n | 2 d x ≤ 1 2 ∫ ℝ N x ⋅ ∇ V ( x ) f 2 ( v n ) d x + N c 1 / 2 − ∫ ℝ N x ⋅ ∇ x G d x ≤ 1 2 | ( ∇ V ( x ) ⋅ x ) + | 2 * / ( 2 * − γ ) ( ∫ ℝ N f 2 ⋅ 2 * γ ( v n ) d x ) γ 2 * + N c 1 / 2 + K ≤ C 1 ( ∫ ℝ N | v n | 2 * d x ) γ 2 * + N c 1 / 2 + K ≤ C 2 ( ∫ ℝ N | ∇ v n | 2 d x ) γ 2 * + N c 1 / 2 + K . (9)

Therefore, ∫ ℝ N | ∇ v n | 2 d x is bounded.

Next we prove ∫ ℝ N V ( x ) f 2 ( v n ) d x is bounded. By (g_{1}), (g_{2}) and Lemma 3.1, we have lim t → 0 | g ( x , f ( t ) ) f ′ ( t ) t | f 2 ( t ) = 0 and lim t → ∞ | g ( x , f ( t ) ) f ′ ( t ) t | | t | 2 * = 0 .

Thus, for any ε > 0 , there exists C ( ε ) > 0 , such that

| g ( x , f ( t ) ) f ′ ( t ) t | ≤ ε f 2 ( t ) + c ( ε ) | t | 2 * forall t ∈ ℝ . (10)

By using 〈 I ′ λ n ( v n ) , v n 〉 = 0 and Lemma 3.1, we get

∫ ℝ N | ∇ v n | 2 d x + 1 2 ∫ ℝ N V ( x ) f 2 ( v n ) d x ≤ ∫ ℝ N | ∇ v n | 2 d x + ∫ ℝ N V ( x ) f ( v n ) f ′ ( v n ) v n d x = λ n ∫ ℝ N g ( x , f ( v n ) ) f ′ ( v n ) v n d x ≤ ε ∫ ℝ N | f ( v n ) | 2 d x + C ( ε ) ∫ ℝ N | v n | 2 * d x ≤ ε V 0 ∫ ℝ N V ( x ) f 2 ( v n ) d x + C ′ ( ε ) ( ∫ ℝ N | ∇ v n | 2 d x ) 2 * 2 .

Choosing enough small ε ( 0 < ε < V 0 2 ) , we obtain that ∫ ℝ N V ( x ) f 2 ( v n ) d x is bounded.

Proof of Theorem 1.1. By Lemma 3.5 and Lemma 3.7, there exist { λ n } ⊂ [ 1 2 , 1 ] and a bounded sequence { v n } ⊂ X \ { 0 } , such that lim n → ∞ λ n = 1 , I λ n ( v n ) = c λ n , I ′ λ n ( v n ) = 0 .

Then by the fact that the map λ → c λ is left continuous, we have

lim n → ∞ I ( v n ) = lim n → ∞ ( I λ n ( v n ) + ( λ n − 1 ) ∫ ℝ N G ( x , f ( v n ) ) d x ) = lim n → ∞ c λ n = c 1 .

Similarly, we obtain I ′ ( v n ) → 0 in space X * yields that { v n } is a bounded (PS) sequence of functional I and lim n → ∞ I ( v n ) = c 1 . By Lemma 3.4, a positive critical point v can be obtained.

To prove Theorem 1.2, we need to prove the following lemmas.

Lemma 4.1 The trivial solution of Equation (2) is a local minimizer for I ( v ) in H r 1 ( ℝ N ) , and there exists a constant C > 0 (dependent on V 0 and embedding constant), such that the every non-negative solution v of Equation (2) satisfies the inequality

‖ ∇ v ‖ ≥ C . (11)

Proof: By (V_{2}), (g_{2}), and Lemma 3.1 (3), we have

I ( v ) = 1 2 ∫ ℝ N ( | ∇ v | 2 + V ( x ) f 2 ( v ) ) d x − ∫ ℝ N G ( x , f ( v ) ) d x ≥ 1 2 ∫ ℝ N ( | ∇ v | 2 + V 0 f 2 ( v ) ) d x − C 0 ∫ ℝ N f 2 ( v ) d x − C 0 ∫ ℝ N f q ( v ) d x ≥ 1 2 ∫ ℝ N | ∇ v | 2 d x + V 0 − 2 C 0 2 ∫ ℝ N f 2 ( v ) d x − C 0 ∫ ℝ N f q ( v ) d x ≥ C 1 ‖ v ‖ 2 − C 2 ‖ v ‖ q 2 .

Thus I ( v ) > 0 as ‖ v ‖ is enough small and q > 4 . In conclusion, the trivial solution v = 0 is a local minimizer for I ( v ) in H r 1 ( ℝ N ) .

v ∈ H r 1 ( ℝ N ) is the non-negative of Equation (2), so that 〈 I ′ ( v ) , v 〉 = 0 . By (V_{1}), Lemma 3.1 (2) (4) (5) and the embedding between L p ( ℝ N ) and H r 1 ( ℝ N ) , we have

0 = ∫ ℝ N | ∇ v | 2 + ∫ ℝ N V ( x ) f ( v ) f ′ ( v ) v − ∫ ℝ N g ( x , f ( v ) ) f ′ ( v ) v ≥ ∫ ℝ N | ∇ v | 2 + V 1 2 ∫ ℝ N f 2 ( v ) − 1 2 ∫ ℝ N g ( x , f ( v ) ) f ( v ) ≥ ∫ ℝ N | ∇ v | 2 + V 1 2 ∫ ℝ N f 2 ( v ) − C 0 2 ∫ ℝ N ( f 2 ( v ) + f q ( v ) ) ≥ | ∇ v | 2 2 − C 0 2 ∫ ℝ N v q ≥ | ∇ v | 2 2 − C 0 2 | ∇ v | 2 q .

This implies that inequality (11) is satisfied. This completes the proof.

Lemma 4.2 Suppose that the conditions of theorem 1.2 are satisfied, Equation (2) admits a positive solution v, and v is a local minimizer for I ( v ) in H r 1 ( ℝ N ) .

Proof: According to the reference [

M = { u ∈ H r 1 : u _ ≤ u ≤ u ¯ } .

Let v be a solution of Equation (2) on M, then

I ( v ) = inf ξ ∈ M I ( ξ ) . (12)

Next, we prove that v is a local minimizer for I ( v ) in H r 1 ( ℝ N ) Suppose by contradiction that v is not the local minimizer for I ( v ) in H r 1 ( ℝ N ) . Then there exists a sequence { u n } in H r 1 ( ℝ N ) , such that ‖ u n − v ‖ → 0 as n → ∞ and I ( u n ) < I ( v ) . Put

v n = max { u _ , min { u ¯ , u n } } = { u _ , u n < u _ , u n , u _ < u n < u ¯ , u ¯ , u n > u ¯ ,

ω n = ( u n − u ¯ ) + = { 0 , u n < u ¯ , u n − u ¯ , u n ≥ u ¯ ,

z n = ( u _ − u n ) + = { 0 , u n > u _ , u _ − u n , u n ≤ u _ .

Therefore, u n = v n − z n + ω n , v n ∈ M , and ω n and z n have disjoint support.

The following defines some sets and functions:

R n = { x ∈ ℝ N : u _ ≤ u n ≤ u ¯ } ,

S n = s u p p ( ω n ) ,

T n = s u p p ( z n ) ,

L ( x , u ) = − 1 2 V ( x ) f 2 ( u ) + G ( x , f ( u ) ) .

And then I ( u n ) is transformed into

I ( u n ) = 1 2 ∫ S n | ∇ u n | 2 d x − ∫ S n L ( x , u n ) d x + 1 2 ∫ T n | ∇ u n | 2 d x − ∫ T n L ( x , u n ) d x + 1 2 ∫ R n | ∇ u n | 2 d x − ∫ R n L ( x , u n ) d x . (13)

Obviously, v n = u ¯ on S n , so that

∫ S n ( 1 2 | ∇ u n | 2 − L ( x , u n ) ) d x = ∫ S n ( 1 2 | ∇ ( v n + ω n ) | 2 − L ( x , v n + ω n ) ) d x = ∫ S n ( 1 2 | ∇ ( u ¯ + ω n ) | 2 − L ( x , u ¯ + ω n ) ) d x .

Similarly, by v n = u _ on T n , we have

∫ T n ( 1 2 | ∇ u n | 2 − L ( x , u n ) ) d x = ∫ T n ( 1 2 | ∇ ( v n + z n ) | 2 − L ( x , v n + z n ) ) d x = ∫ T n ( 1 2 | ∇ ( u _ + z n ) | 2 − L ( x , u _ + z n ) ) d x .

Since v n = u n on R n , we get

∫ R n ( 1 2 | ∇ u n | 2 − L ( x , u n ) ) d x = ∫ R n ( 1 2 | ∇ v n | 2 − L ( x , v n ) ) d x = I ( v n ) − ∫ S n ( 1 2 | ∇ u ¯ | 2 − L ( x , u ¯ ) ) d x − ∫ T n ( 1 2 | ∇ u _ | 2 − L ( x , u _ ) ) d x .

Consequently, we have

I ( u n ) = I ( v n ) + ∫ S n | ∇ ( u ¯ + ω n ) | 2 − | ∇ u ¯ | 2 2 d x − ∫ S n ( L ( x , u ¯ + ω n ) − L ( x , u ¯ ) ) d x + ∫ T n | ∇ ( u _ − z n ) | 2 − | ∇ u _ | 2 2 d x − ∫ T n ( L ( x , u _ − z n ) − L ( x , u _ ) ) d x . (14)

Since u _ is a sub-solution, we obtain

− Δ u _ ≤ f ′ ( u _ ) ( g ( x , f ( u _ ) ) − V ( x ) f ( u _ ) ) = L u ( x , u _ ) ,

Yields that − Δ u _ ≤ L u ( x , u _ ) .

Similarly, u ¯ is a sup-solution, so that − Δ u ¯ ≥ L u ( x , u ¯ ) .

Hence,

∫ ℝ N ∇ u _ ∇ ( − z n ) d x ≥ ∫ ℝ N L u ( x , u _ ) ( − z n ) d x ,

∫ ℝ N ∇ u ¯ ∇ ω n d x ≥ ∫ ℝ N L u ( x , u ¯ ) ω n d x .

In addition, we noticed that

1 2 | ∇ ( u ¯ + ω n ) | 2 − | ∇ u ¯ | 2 = ∇ u ¯ ∇ ω n + 1 2 | ∇ ω n | 2 ,

1 2 | ∇ ( u _ − z n ) | 2 − | ∇ u _ | 2 = ∇ u _ ∇ ( − z n ) + 1 2 | ∇ z n | 2 ,

So that by (14), we have

I ( u n ) ≥ I ( v n ) + 1 2 ∫ ℝ N | ∇ ω n | 2 d x + 1 2 ∫ ℝ N | ∇ z n | 2 d x − ∫ S n ( L ( x , u ¯ + ω n ) − L ( x , u ¯ ) − L u ( x , u ¯ ) ω n ) d x − ∫ T n ( L ( x , u _ − z n ) − L ( x , u _ ) − L u ( x , u _ ) ( − z n ) ) d x .

To complete the Lemma, we still need to prove the following claim: as n → ∞ ,

∫ S n ( L ( x , u ¯ + ω n ) − L ( x , u ¯ ) − L u ( x , u ¯ ) ω n ) d x ≤ o ( 1 ) ∫ ℝ N | ∇ ω n | 2 d x , (15)

∫ T n ( L ( x , u _ − z n ) − L ( x , u _ ) − L u ( x , u _ ) ( − z n ) ) d x ≤ o ( 1 ) ∫ ℝ N | ∇ z n | 2 d x . (16)

Since the proofs of inequalities (15) and (16) are similar, we only prove (15) Firstly, note

L n ( x ) = L ( x , u ¯ + ω n ) − L ( x , u ¯ ) − L u ( x , u ¯ ) ω n ,

Split

L n ( x ) = L 0 n ( x ) − L 1 n ( x ) ,

where

L 0 n ( x ) = − 1 2 V ( x ) ( f 2 ( u ¯ + ω n ) − f 2 ( u ¯ ) ) + V ( x ) f ( u ¯ ) f ′ ( u ¯ ) ω n ,

L 1 n ( x ) = G ( x , f ( u ¯ + ω n ) ) − G ( x , f ( u ¯ ) ) − g ( x , f ( u ¯ ) ) f ′ ( u ¯ ) ω n .

By the define of f, for any t ∈ ℝ , we have

( f ( t ) f ′ ( t ) ) ′ ≤ 1 , (17)

f ″ ( t ) ≤ 2 . (18)

By differential mean value theorem, Lemma 3.1(1) and (17), we have

L 0 n ( x ) = − 1 2 V ( x ) ( f 2 ( u ¯ + ω n ) − f 2 ( u ¯ ) ) + V ( x ) f ( u ¯ ) f ′ ( u ¯ ) ω n = − V ( x ) f ( u ¯ + θ 1 ω n ) f ′ ( u ¯ + θ 1 ω n ) ω n + V ( x ) f ( u ¯ ) f ′ ( u ¯ ) ω n = V ( x ) ω n 2 θ 1 ( f ( u ¯ + θ 1 θ 2 ω n ) f ′ ( u ¯ + θ 1 θ 2 ω n ) ) ′ ≤ V ( x ) ω n 2 , ( 0 ≤ θ 1 ≤ 1 , 0 ≤ θ 2 ≤ 1 ) .

Then, by Hölder inequality and Sobolev inequality, we obtain

∫ S n L 0 n ( x ) ≤ ∫ S n | L 0 n ( x ) | ≤ ∫ S n V ( x ) | ω n | 2 ≤ β ∫ S n ω n 2 ≤ β | S n | 2 N ( ∫ ℝ N ( ω n 2 ) N N − 2 ) N − 2 N = β | S n | 2 N | ω n | 2 * 2 ≤ β | S n | 2 N | ω n | 2 2 .

Moreover, by the define of S n , we have lim n → ∞ | S n | = 0 . In fact, for any ε > 0 , there exists δ ( ε ) > 0 , such that | { u ¯ ≤ v + δ } | < ε , since v < u ¯ in ℝ N . Thus

S n ⊂ { u ¯ ≤ v + δ } ∪ { v + δ < u ¯ < u n } .

Again since | u n − v | 2 → 0 as n → ∞ , there exists n 0 , such that for n ≥ n 0

ε δ 2 ≥ ∫ ℝ N ( u n − v ) 2 ≥ ∫ { u n > v + δ } ( u n − v ) 2 ≥ ∫ { u n > v + δ } δ 2 = δ 2 | { u n > v + δ } |

Therefore

| S n | ≤ | { u n ≤ v + δ } | + | v + δ < u ¯ < u n | ≤ 2 ε ,

and then as n → ∞ , we have

∫ ℝ N L 0 n ( x ) ≤ o ( 1 ) ‖ ω n ‖ 2 .

Set

g 2 ( x , t ) = G s ( x , f ( t ) ) = g ( x , f ( t ) ) f ′ ( t ) ,

It follows from differential mean value theorem that

L 1 n ( x ) = G ( x , f ( u ¯ + ω n ) ) − G ( x , f ( u ¯ ) ) − g ( x , f ( u ¯ ) ) f ′ ( u ¯ ) ω n = g ( x , f ( u ¯ + θ 1 ( x ) ω n ) ) f ′ ( u ¯ + θ 2 ( x ) ω n ) ω n − g ( x , f ( u ¯ ) ) f ′ ( u ¯ ) ω n .

To be continue, set

P ( x , u ) = g ( x , f ( u ) ) f ′ ( u ) ,

p ( x , u ) = P s ( x , u ) = g s ( x , f ( u ) ) ( f ′ ( u ) ) 2 + g ( x , f ( u ) ) f ″ (u)

Again by differential mean value theorem, we have

L 1 n = ( P ( x , u ¯ + θ 1 ( x ) ω n ) − P ( x , u ¯ ) ) ω n = p ( x , u ¯ + θ 1 ( x ) θ 2 ( x ) ω n ) θ 1 ( x ) ω n 2 .

Set u 0 = u ¯ + θ 1 ( x ) θ 2 ( x ) ω n , so that by Lemma 3.1 (3), (g_{2}), (g_{6}), (17), (18), Hölder inequality and Sobolev inequality, we get

∫ S n L 1 n ( x ) d x ≤ ∫ S n p ( x , u ¯ + θ 1 ( x ) θ 2 ( x ) ω n ) θ 1 ω n 2 d x ≤ ∫ S n ( c ( f ′ ( u 0 ) ) 2 + g ( x , f ( u 0 ) ) f ″ ( u 0 ) ) ω n 2 d x ≤ ∫ S n ( g ′ u ( x , f ( u 0 ) ) ( f ′ ( u 0 ) ) 2 d x + C 0 f ( u 0 ) f ″ ( u 0 ) d x + C 0 f q − 1 ( u 0 ) f ″ ( u 0 ) ) ω n 2 d x ≤ C 1 ∫ S n ω n 2 d x + C 2 ( ∫ ℝ N | u ¯ | q − 1 2 2 * 2 * − 2 d x ) 2 * − 2 2 * ( ∫ { u n ≥ u ¯ } ω n 2 2 * 2 d x ) 2 2 * + C 3 ∫ S n ω n q + 3 2 d x ≤ o ( 1 ) ( ∫ ℝ N | u ¯ | q − 1 2 2 * 2 * − 2 d x ) 2 * − 2 2 * ( ∫ ℝ N ω n 2 * d x ) 2 2 * ≤ o ( 1 ) | ∇ ω n | 2 2 ,

which implies that (15) is satisfied.

By (15) and (16), as n → ∞ we have

1 2 ∫ ℝ N | ∇ ω n | 2 d x + 1 2 ∫ ℝ N | ∇ z n | 2 d x ≤ I ( u n ) − I ( v n ) + ∫ S n ( L ( x , u ¯ + ω n ) − L ( x , u ¯ ) − L u ( x , u ¯ ) ω n ) d x + ∫ T n ( L ( x , u _ − z n ) − L ( x , u _ ) − L u ( x , u _ ) z n ) d x

< ∫ S n ( L ( x , u ¯ + ω n ) − L ( x , u ¯ ) − L u ( x , u ¯ ) ω n ) d x + ∫ T n ( L ( x , u _ − z n ) − L ( x , u _ ) − L u ( x , u _ ) z n ) d x ≤ o ( 1 ) | ∇ ω n | 2 2 + o ( 1 ) | ∇ z n | 2 2 .

Since ω n and z n have disjoint support, as n → ∞

∫ ℝ N | ∇ ω n | 2 d x ≤ o ( 1 ) | ∇ ω n | 2 2 ,

∫ ℝ N | ∇ z n | 2 d x ≤ o ( 1 ) | ∇ z n | 2 2 .

Then z n ( x ) = ω n ( x ) = 0 a.e. x ∈ ℝ N , which implies u n = v n a.e. x ∈ ℝ N . By (12), we have

I ( v ) ≤ I ( v n ) = I ( u n ) .

Contradiction. Thus, the proof is complete.

Define a set

Π = { u ∈ H r 1 ( ℝ N ) : 0 ≤ u ≤ v , a . e . x ∈ ℝ N } ,

where v is a positive solution in Lemma 4.1. The critical point in II is also the critical point in H r 1 ( ℝ N ) of I [

Lemma 4.3 Suppose that the conditions of theorem 1.2 are satisfied, then I satisfies (PS) condition on II.

Proof: Firstly, we need to prove the boundedness of any (PS) sequence { v n } on P. Assume { v n } ⊂ Π is a (PS) sequence, then

I ( v n ) = 1 2 ∫ ℝ N | ∇ v n | 2 d x + 1 2 ∫ ℝ N V ( x ) f 2 ( v n ) d x − ∫ ℝ N G ( x , f ( v n ) ) d x = c + o ( 1 ) , (19)

〈 I ′ ( v n ) , ϕ 〉 = ∫ ℝ N ∇ v n ∇ ϕ d x + ∫ ℝ N V ( x ) f ( v n ) f ′ ( v n ) ϕ d x − ∫ ℝ N g ( x , f ( v n ) ) f ′ ( v n ) ϕ d x = o ( 1 ) ‖ ϕ ‖ . (20)

By (g_{7}) and (19), there exists μ ≥ 4 , such that

c + o ( 1 ) = I ( v n ) > 1 2 ∫ ℝ N | ∇ v n | 2 d x + 1 2 ∫ ℝ N V ( x ) f 2 ( v n ) d x − 1 μ ∫ ℝ N g ( x , f ( v n ) ) f ( v n ) d x . (21)

Specially, choose ϕ = f ( v n ) f ′ ( v n ) = 1 + 2 f 2 ( v n ) f ( v n ) . By Lemma 3.1 (2) (3), we have

| ϕ | = 1 + 2 f 2 ( v n ) | f ( v n ) | ≤ 2 1 + f 2 ( v n ) | f ( v n ) | ≤ 2 ( 1 + | f ( v n ) | ) | f ( v n ) | ≤ 2 ( | f ( v n ) | + | f ( v n ) | 2 ) ≤ 2 | v n | + 2 | v n | ≤ 4 | v n | .

Again since

| ∇ ϕ | = ( 1 + 4 f 2 ( v n ) 1 + 2 f 2 ( v n ) ) | ∇ v n | ≤ 2 | ∇ v n | ,

implies that ‖ φ ‖ ≤ C ‖ v n ‖ . (20) is transformed to

∫ ℝ N 1 + 4 f 2 ( v n ) 1 + 2 f 2 ( v n ) | ∇ v n | 2 + ∫ ℝ N V ( x ) f 2 ( v n ) − ∫ ℝ N g ( x , f ( v n ) ) f ( v n ) = o ( 1 ) ‖ v n ‖ ,(22)

(22) implies

1 μ ∫ ℝ N g ( x , f ( v n ) ) f ( v n ) d x = 1 μ ∫ ℝ N 1 + 4 f 2 ( v n ) 1 + 2 f 2 ( v n ) | ∇ v n | 2 + 1 μ ∫ ℝ N V ( x ) f 2 ( v n ) − 1 μ ∫ ℝ N g ( x , f ( v n ) ) f ( v n ) + o ( 1 ) ‖ v n ‖ . (23)

Substituting (23) into (21), we obtain

c + o ( 1 ) = I ( v n ) > ( 1 2 − 1 μ 1 + 4 f 2 ( v n ) 1 + 2 f 2 ( v n ) ) ∫ ℝ N | ∇ v n | 2 d x + ( 1 2 − 1 μ ) ∫ ℝ N V ( x ) f 2 ( v n ) d x − o ( 1 ) ‖ v n ‖ .

Thus

‖ v n ‖ 2 ≤ c + o ( 1 ) ( 1 + ‖ v n ‖ ) , μ ≥ 4

This implies that ‖ v n ‖ 2 is bounded.

Next, let’s prove (PS) sequence { v n } satisfies (PS) condition. Since { v n } ⊂ H r 1 ( ℝ N ) is bounded in H r 1 ( ℝ N ) , by Rellich-Kondrachov theorem, there exists v ∈ H r 1 ( ℝ N ) , such that

i) v n → Weakconvergence v in H r 1 ( ℝ N ) ;

ii) v n → v in L q 2 ( ℝ N ) ;

iii) v n → v a.e. in ℝ N .

By (i) and (ii), we obtain I ′ λ ( v λ ) = 0 .

The following we prove v n → v in H r 1 ( ℝ N ) . Set

H ( x , t ) = 1 2 V ( x ) t 2 − 1 2 V ( x ) f 2 ( t ) + G ( x , f ( t ) ) .

Then I ( v ) is transformed to

I λ ( v ) = 1 2 ∫ ℝ N ( | ∇ v | 2 + V ( x ) v 2 ) d x − ∫ ℝ N H ( x , v ) d x .

Set h ( x , t ) = d H ( x , t ) d t , then

h ( x , t ) = V ( x ) t − V ( x ) f ( t ) f ′ ( t ) + g ( x , f ( t ) ) f ′ ( t ) .

By (g_{1}), (g_{2}) and Lemma 3.1, there exists C 1 , C 2 > 0 , for any x ∈ ℝ N and t ∈ ℝ , such that

| h ( x , t ) | ≤ C 1 | t | + C 2 | t | q − 1 2 . (24)

By (24) and v n → v in L q 2 ( ℝ N ) , we have

lim n → ∞ ∫ ℝ N ( h ( x , v n ) − h ( x , v ) ) ( v n − v ) d x = 0.

Thus

o ( 1 ) = 〈 I ′ ( v n ) − I ′ ( v λ ) , v n − v 〉 = ∫ ℝ N ( | ∇ ( v n − v ) | 2 + V ( x ) ( v n − v ) 2 ) d x − ∫ ℝ N ( h ( x , v n ) − h ( x , v ) ) ( v n − v ) d x ≥ min { 1 , V 0 } ‖ v n − v ‖ 2 + o ( 1 ) ,

Which implies that v n → v in H r 1 ( ℝ N ) . The proof is complete.

Lemma 4.4. (see ( [

Proof of Theorem 1.2. Applying Lemma 4.4, we arrive to the following dichotomy

1) I ( v ) = I ( 0 ) and v and 0 may be connected in any neighborhood of the set of local minima of I to II, or

2) I admits a critical point u ˜ in II which is not a local minimum.

But Lemma 4.1 ensures that the trivial solution is an isolated solution of problem (2). Hence a second independent solution of problem (2) should exist since the solution found in Lemma 4.2 is a local minimum of I. In conclusion, problem (2) admits one pair of ordered positive solutions to equation. The proof is complete.

Wu, J. and Jia, G. (2018) Existence of Ordered Solutions to Quasilinear Schrödinger Equations with General Nonlinear Term. Journal of Applied Mathematics and Physics, 6, 770-786. https://doi.org/10.4236/jamp.2018.64068