Positive solutions for a Kirchhoff type problem with fast increasing weight and critical nonlinearity

In this paper, we study the following Kirchhoff type problem − ( a + b ∫ R3 K(x)|∇u|2dx ) div(K(x)∇u) = λK(x)|x|β|u|q−2u + K(x)|u|4u, x ∈ R3, where K(x) = exp(|x|α/4) with α ≥ 2, β = (α − 2)(6 − q)/4 and the parameters a, b, λ > 0. When 6− 4 α < q < 6, we obtain a positive ground state solution for any λ > 0. When 2 < q < 4, we obtain a positive solution for λ > 0 small enough. In the proof we use variational methods.

It is well known that Kirchhoff type problems are presented by Kirchhoff in [9] as an extension of the classical d'Alembert wave equation for free vibrations of elastic strings.When K(x) ≡ 1, the general Kirchhoff type problem involving critical exponent has been studied by many researchers.Under different assumptions on f (x, u), some interesting studies for (1.2) can be found in [12-14, 23, 25].There are also several existence results Corresponding author.Email: qianxiaotao1984@163.com for (1.2) on a bounded domain Ω ⊂ R 3 .For this case, we refer the interested readers to [4,10,11,16,22].
In [5], Furtado et al. concerned the following equation where 2 * = 2N/(N − 2), N ≥ 3, γ = (α − 2) (2 * −q) (2 * −2) and 2 < q < 2 * .In that article, the authors obtained the existence of a positive solution for (1.3) by using Mountain Pass Theorem.In particular, when N = 3, they proved that there is a positive solution for large value of λ if 2 < q ≤ 6 − 4 α , and no restriction on λ if 6 − 4 α < q < 6.Subsequently, Furtado et al. [6] studied the number of solutions for the following problem where f (u) is superlinear and subcritical.More precisely, for any given k ∈ N, the authors shown that there exists λ * = λ * (k) > 0 such that (1.4) has at least k pairs of solutions for λ ∈ 0, λ * (k) .But they can not give any information about the sign of these solutions.Recently, we investigated the following Kirchhoff type of problem with concave-convex nonlinearities and critical exponent (see [21]) where 1 < q < 2, and > 0 is small enough.Under some conditions on f (x), we gave the existence of two positive solutions and obtained uniform lower estimates for extremal values for the problem.For more results of related problem, please see [2,7,[17][18][19][20] and the references therein.
From these results above, we do not see any existence of positive solutions for problem (1.1) in the case of 2 < q < 6, the term K(x)|∇u| 2 dx and critical nonlinearity, hence it is natural to ask what the case would be.Our aim of this paper is to show how variational methods can be employed to establish some existence of positive solutions for the Kirchhoff type problem (1.1).
In order to state our main results, let H denote the Hilbert space obtained as the completion of C ∞ 0 (R 3 ) with respect to the norm Define the weighted Lebesgue spaces for each q ∈ [2, 6] . By [5, Proposition 2.1], we have that the embedding H → L q K (R 3 ) is continuous for 2 ≤ q ≤ 6, and compact for 2 ≤ q < 6.This enables us to define for each q ∈ [2, 6] (1.5) In particular, when q = 6, we put S = S 6 for simplicity.It is worth mentioning that this constant is equal to the best constant of the embedding D 1,2 (R 3 ) → L 6 (R 3 ), see [2].By the above embedding, it is easy to see the following functional associated to (1.1) is well defined on H and I ∈ C 1 (H, R).It is commonly known that there exists a one to one correspondence between the critical points of I and the weak solutions of (1.1).Here, we say u ∈ H is a weak solution of (1.1), if for any φ ∈ H, there holds Additionally, we say a nontrivial solution u ∈ H to (1.1) is a ground state solution, if I(u) ≤ I(v) for any nontrivial solution v ∈ H to (1.1).
Our main results for (1.1) are the following theorems.
Kirchhoff type problems are often treated as nonlocal in view of the presence of the term K(x)|∇u| 2 dx which implies that equation (1.1) is no longer a pointwise identity.And so, the methods employed in [5] cannot be used here.For Theorem 1.1, motivated by [23] (see also [15]), we shall use Nehari Manifold method to prove the existence of a positive ground state solution for problem (1.1).For Theorem 1.2, we cannot proceed as in proof of Theorem 1.1 since 2 < q < 4. We also remark that the method used in [5] by letting λ sufficiently large do not apply here, due to the appearance of the term K(x)|∇u| 2 dx.On the contrary, we overcome this difficulty by letting λ small enough, which is inspired by [12].
This paper is organized as follows.In the next section, we give some notations and preliminaries.Then we prove Theorem 1.1 in Section 3, and Theorem 1.2 in Section 4.

Notations and preliminaries
Throughout this paper, we write u instead of R 3 u(x)dx.B r (x) denotes a ball centered at x with radius r > 0. Let → denote strong convergence.Let denote weak convergence.O(ε t ) denotes |O(ε t )|/ε t ≤ C as ε → 0, and o(ε t ) denotes |o(ε t )|/ε t → 0 as ε → 0. All limitations hold as n → ∞ unless otherwise stated.C and C i denote various positive constants whose values may vary from line to line.Lemma 2.1.Let a, b > 0 and 2 < q < 6, then the functional I satisfies the mountain-pass geometry: (i) There exist ρ, θ > 0 such that I(u) ≥ θ > 0 for any u = ρ.
(ii) There exists e ∈ H with e > ρ such that I(e) < 0.
3 Positive ground state solution for 6 − 4 α < q < 6 In this section, we will employ Nehari method to prove the existence of a positive ground state solution of the considered problem for 6 − 4 α < q < 6.And, suppose that the assumptions of Theorem 1.1 hold throughout this section.
Define the Nehari manifold where Let be the infimum of I on the Nehari manifold.
Proof.For u ∈ Λ, it follows from Lemma 3.1 and 6 − 4 α < q < 6 that Thus, the coercivity and lower boundedness of I hold.The proof of Lemma 3.2 is completed.
Lemma 3.3.Given u ∈ Λ, there exist ρ u > 0 and a continuous function g ρ u : Proof.Fix u ∈ Λ and define F : R + × H → R as below Since u ∈ Λ, we have F(1, 0) = 0.Moreover, using Lemma 3.1, we also have for 6 Using the implicit function theorem for F at the point (1, 0), we can conclude that there exists ρ u > 0 satisfying for w ∈ H, w < ρ u , the equation F(t, w) = 0 has a unique continuous solution t = g ρ u (w) > 0 with g ρ u (0) = 1.Since F(g ρ u (w), w) = 0 for w ∈ H, w < ρ u , we get Furthermore, we have for all φ ∈ H, r > 0 and consequently Thus, This completes the proof of Lemma 3.3.
Proof.The proof is similar to [24, Lemma 4.1], and is omitted here.
, and set According to [2], we have that and Then, we obtain the following estimate In addition, we also have whenever 6α 2+α < q < 6.This and (3.4) imply that for 6α 2+α < q < 6 and ε small enough, we have To this goal, let Note that v ε 6 = 1.Thus, by Lemma 3.4, we know that g(t) has a unique maximum point t ε := t(v ε ) > 0. We claim that t ε ≥ C 0 > 0 for some positive constant C 0 and any ε > 0.
Otherwise, there is some sequence ε n → 0 satisfying t ε n → 0 and g(t Then, by Lemma 2.1 and the continuity of I, we conclude that which is a contradiction.Hence, the claim holds.By (3.2), we also have that Obviously, we have 6 − 4 α > 6α 2+α provided α ≥ 2. Furthermore, by using (3.5) and (3.6), we obtain for 6 − 4 α < q < 6 and ε small enough sup This completes the proof.Firstly, we show that u n is bounded.By (3.7), we have that for q > 4 which implies that u n is bounded.Up to a subsequence (still denoted by {u n }), we may assume that Since u n ∈ Λ, it then follows from Lemma 3.1 that ι 2 > 0.
Secondly, we prove that u * ≡ 0. If not, we have u * ≡ 0 and so K(x)|x| β |u n | q = o(1).On the other hand, by (3.7) and the boundedness of {u n }, we have and thus from (1.5), Letting n → ∞ in (3.8), we have Consequently, Finally, we claim that u n 2 → u * 2 .Indeed, if to the contrary, it follows from Fatou Lemma that Then, passing to the limit as n → ∞, we get Taking v = u * in the above equation, we obtain This together with (3.9) imply that I (u * ), u * < 0. By Lemma 3.4, then it is easy to see that there exists t 0 ∈ (0, 1) such that I (t 0 u * ), t 0 u * = 0. Therefore, This and the weak convergence of {u n } in H implies that u n → u * in H, and Lemma 3.6 is proved.
Lemma 3.7.For any λ > 0, there exists a sequence {u n } ⊂ Λ such that: Proof.In view of Lemma 3.2, we can apply Ekeland variational principle to construct a minimizing sequence {u n } ⊂ Λ satisfying the following properties: where ρ u n and g u n are defined according to Lemma 3.3.Let v ρ = ρu with u = 1.Fix n and let It then follows from the definition of Fréchet derivative that Thus, Therefore, From u = 1, Lemma 3.3 and the boundedness of {u n }, it follows that

Note that
Furthermore, for fixed n, since (u n ), u n = 0 and (u n − v ρ ) → u n as ρ → 0, by letting ρ → 0 in (3.10) we can deduce that which shows that I (u n ) → 0. This finishes the proof of Lemma 3.7.
With the previous preparations, we are now ready to prove Theorem 1.1.
Proof of Theorem 1.1.By Lemma 3.7, we see that there exists a minimizing sequence {u n } ⊂ Λ satisfying u n ≥ 0, I(u n ) → c * and I (u n ) → 0 for any λ > 0. It then follows from Lemma 3.6 that u n → u * , I(u * ) = c * and u * ≥ 0 is a weak solution of (1.1).By standard elliptic regularity argument and the strong maximum principle we have that u * > 0. This and the definition of c * imply that u * is a positive ground state solution of (1.1) and the proof is complete.
Thus, we have 3q b q 4−q (6 − q) 6q S −q/2 q which is a contradiction with our assumption c < c 1 − D 0 λ 4 4−q .Thus, the claim follows, that is, ũn → ũ * in H.This finishes the proof of Lemma 4.2.Now, we are in a position to prove Theorem 1.2.
Proof of Theorem 1.2.Using Lemma 2.1, we can apply Mountain Pass Theorem to obtain a sequence { ũn } ⊂ Λ satisfying I( ũn ) → c * and I ( ũn ) → 0. It then follows from Lemmas 4.1 and 4.2 that there is λ * > 0 such that ũn → ũ * for all λ ∈ (0, λ * ), and ũ * is a weak solution of (1.1).Furthermore, if we replace I by the following functional It is easy to see that all the above calculations can be repeated word for word.Then, we can infer from Ĩ ( ũ * ), ũ− * = 0 that ũ− * = 0.In turn, we obtain ũ * ≥ 0. By standard elliptic regularity argument and the strong maximum principle, we also have that ũ * > 0, that is, ũ * is a positive solution of (1.1).This completes the proof.

. 5 )
ByLemma 3.4  and the definition of c * , it is easy to see that Lemma 3.5 follows if we can show that sup t>0 I(tv ε ) < abS 3 4 + b 3 S 6 24 + (b 2 S 4 + 4aS)