Ground state solutions for Kirchhoff-type equations with a general nonlinearity in the critical growth

Theorem 1.1.

Assume that (f1)–(f4) hold, then for 2<q≤4 with M>0 sufficiently large or 4<q<6, problem (1.1) possesses a radial ground state solution.

Remark 1.2.

Conditions (f1)–(f4) were introduced in [25] to obtain the existence of ground state solutions for a class of Schrödinger–Poisson equations with critical growth. An interesting question now is whether the same existence results occur to the nonlocal problem (1.1) with critical growth. In this paper, we study problem (1.1) and give some positive answers. Theorem 1.1 extends the main result in [25] to the Kirchhoff-type equation.

Remark 1.3.

Set V=1 in (1.1). Compared to [16, Theorem 1.3], we need not consider the usual Ambrosetti–Rabinowitz (AR) condition (F4), which is restrictive. The lack of the (AR)-condition gives rise to two obstacles to the standard mountain pass arguments both in checking the geometrical assumptions in the functional and in proving the boundedness of its (PS) sequences. In the present paper, we use another method to obtain our results. Moreover, there exist some functions which satisfy our conditions (f1)–(f4), but do not satisfy the conditions of [16]. For example, the function f⁢(t)=μ⁢t5+M⁢t3, μ,M>0, satisfies our conditions (f1)–(f4), but does not satisfy conditions (F3) and (F4) in [16]. For this reason, Our main results can be viewed as a partial extension of [16].

Notations.

We use the following notations:

1. H1⁢(ℝ3) is the Sobolev space equipped with the norm

   ∥u∥H1⁢(ℝ3)=∫ℝ3(|∇⁡u|2+u2)⁢𝑑x.

2. Define

   ∥u∥2:=∫ℝ3(a⁢|∇⁡u|2+V⁢u2)⁢𝑑x for ⁢u∈H1⁢(ℝ3).

   Note that ∥⋅∥ is an equivalent norm on H1⁢(ℝ3).

3. For any 1≤s≤∞, we denote by

   ∥u∥Ls:=(∫ℝ3|u|s⁢𝑑x)1s

   the usual norm of the Lebesgue space Ls⁢(ℝ3).

4. For any x∈ℝ3, we set

   Br⁢(x)≜{y∈ℝ3:|x-y|<r}.

5. Let D1,2⁢(ℝ3):={u∈L6⁢(ℝ3):∇⁡u∈L2⁢(ℝ3)} be the Sobolev space equipped with the norm

   ∥u∥D1,2⁢(ℝ3)2:=∫ℝ3|∇⁡u|2⁢𝑑x.

6. S denotes the best Sobolev constant

   S:=infu∈D1,2⁢(ℝ3)∖{0}⁡∫ℝ3|∇⁡u|2⁢𝑑x(∫ℝ3u6⁢𝑑x)13.

7. C denotes various positive constants.

Theorem 2.1 (See [14]).

Let (X,∥⋅∥) be a Banach space and h⊂R+ an interval. Consider the following family of C1 functionals on X:

with B nonnegative and either A⁢(u)→+∞ or B⁢(u)→+∞ as ∥u∥→∞. We assume there are two points v1, v2 in X such that

where

Then for almost every λ∈h there is a sequence {un}⊂X such that

1. {un} is bounded,

2. Iλ⁢(un)→cλ,

3. Iλ′⁢(un)→0 in the dual X-1 of X.

Moreover, the map λ→cλ is continuous from the left.

Lemma 3.1.

If uλ, for λ∈[12,1], is a critical point of Iλ, then uλ satisfies the following Pohožaev-type identity:

Lemma 3.2.

Assume that conditions (f1), (f2)–(f4) are satisfied. Then the conclusions of Theorem 2.1 hold.

Proof.

By Theorem 2.1, we set

By (f3)–(f4), we obtain that f⁢(u) is odd and by the definition of A⁢(u), we can see that B⁢(u)≥0 for u∈E and A⁢(u)→+∞ as ∥u∥→∞. By (f1)–(f3), for any ε>0, there exists C⁢(ε)>0 such that

Then, by the Sobolev embedding theorem, there holds

which implies that there exists ρ>0 small enough and α>0 such that

By (f4) and (2.1), for ϕ∈E, ϕ≥0 and ϕ≢0, we have

Taking v2=t1⁢ϕ with t1>0 large enough, we have ∥v2∥>ρ and

On the other hand, Iλ⁢(0)=0. Set v1=0; then inequalities (3.2) and (3.3) imply that the conclusions of Theorem 2.1 hold. ∎

Lemma 3.3.

Assume that (f1)–(f4) hold. Then

for q∈(2,4] with M sufficiently large or q∈(4,6).

Proof.

For ε,r>0, define

where φ∈C0∞⁢(B2⁢r⁢(0)), 0≤φ≤1 and φ|Br⁢(0)≡1. Using the method of [9], we obtain

and

where K1, K2, K are positive constants. Moreover, the best Sobolev constant is S=K1⁢K2-1/3. By (3.4), we have

By Lemma 3.2 and the definition of cλ, we can deduce that cλ≤supt≥0⁡Iλ⁢(t⁢uε). Let

Note that h⁢(t) attains its maximum at

Then

for ε>0 small enough.

Obviously, we see that there exists 0<t1<1 such that for ε<1, we have

By (f4) , we have

It follows from (3.4), (3.5) and (3.7) that there exists 0<ε0<1 such that for ε<ε0, we have

Thus, there exists t2>0 such that

From (3.7) and the definition of h⁢(t), we have

For q∈(2,4], fix ε∈(0,ε0), it follows from (3.9) that

For q∈(4,6), by (3.5) and (3.9), we obtain

Since 6-q4<12, then there exists ε1∈(0,ε0) small enough such that for ε∈(0,ε1), we have

By (3.6), (3.8) and (3.10)–(3.12), the proof of Lemma 3.3 is complete. ∎

Lemma 3.4.

Set

If Jλ′⁢(u)=0, where λ∈[12,1], then Jλ⁢(u)≥0.

Proof.

Since 〈Jλ′⁢(u),u〉=0, by Lemma 3.1, we get the following the Pohožaev-type identity:

Then we get that

Combining (3.13) with the definition of Jλ⁢(u), we obtain that

Lemma 3.5.

For s,t>0 , the system

has a unique solution (t0,s0). Moreover, if

then t≥t0 and s≥s0, where

Lemma 3.6.

Assume that (f1)–(f3) hold. If {un}⊂E, for λ∈[12,1], is a sequence such that ∥un∥<C, Iλ⁢(un)→cλ, Iλ′⁢(un)→0, and, moreover, cλ<cλ*, then {un} has a strong convergent subsequence in E.

Proof.

Since ∥un∥<C in E, there exists a u∈E such that

Set

By Lemma 3.4 and the fact that cλ<cλ*, we have

Using an argument similar to [7, Radial Lemma A.II.], by the boundedness of {un}, we have lim|x|→∞⁡un⁢(x)=0. Since

we also get

By the compactness lemma of Strass [22], one has

Similarly,

Setting ωn=un-u, due to Brezis–Lieb (see [8]), we have

and

Then, from (3.15) and (3.16), we have

and

Then, by (3.16) and the fact that Iλ′⁢(un)→0, we have

We can assume that there exists li≥0 (i=1,2,3) such that

Then by (3.18) and (3.17), we have

By the definition of S, we see that