Ground states to a Kirchho ff equation with fractional Laplacian

: The aim of this paper is to deal with the Kirchho ff type equation involving fractional Laplacian operator

The fractional Laplacian operator (−∆) s has a wide range of applications arising in some physical phenomena such as fractional quantum mechanics, flames propagation, etc. (see [10,13]). In recent years, problems involving fractional Laplacian operators and Kirchhoff-type nonlocal terms have been discussed by lots of researchers for their broad applications. Some remarkable results have been yielded, see [1, 2, 6-8, 12, 14, 15] and the references therein. In particular, when α = 1, β = 0 and R 3 is replaced by R N (N ≥ 2), equation (1.1) turns into the classical fractional Laplacian problem (1.2) In [4], employing the constrained variational methods, Dipierro et al. studied the existence and symmetry of nontrivial solutions for (1.2) as s ∈ (0, 1) and p ∈ (2, 2N N−2s ). In this paper, we intend to consider the fractional Kirchhoff equation (1.1) with p ∈ (2, 2 * s ) using variational arguments, and we encounter several difficulties to overcome. First, note that solutions of Eq (1.1) correspond to critical points of the following functional: Since the nonlocal term R 3 |(−∆) s 2 ψ| 2 dx 2 included in the energy functional E(ψ) is homogeneous of degree 4, and the nonlinearity |ψ| p−2 ψ does not satisfy the global Ambrosetti-Rabinowitz type condition for p ∈ (2, 2 * s ), it would bring about more difficulties to establish the boundedness of (PS)-sequence for E(ψ) when p ≤ 4. Second, in general, from ψ n ⇀ ψ in H s (R 3 ), we do not know whether there holds which is vital when we consider the convergence of the (PS)-sequence.
We now give the main result.
By Lemma 2.1, for all ψ ∈ H s (R 3 ), we know that ∥ψ∥ p L p (R 3 ) ≤ c∥ψ∥ p for some positive constant c > 0, and noting that p > 2, we deduce that Next, we expound that there is a Palais-Smale-Pohozaev sequence ((PSP)-sequence, for short) at the minimax level c mp defined by (2.1).
To sum up, we have obtained a sequence {ψ n } ⊂ H s (R 3 ) that satisfies The proof is completed.

Proof of Theorem 1.1. The result of Lemma 2.3 reveals that there is a (PSP)-sequence {ψ
Moreover, by Lemma 2.4 the (PSP)-sequence {ψ n } must be bounded in H s (R 3 ). Then, passing to a subsequence if necessary, we may suppose that ψ n ⇀ ψ 0 weekly in H s (R 3 ), and ψ n (x) → ψ 0 (x) a.e. in x ∈ R 3 . (2.11) Next, we divide our arguments into several steps.
Step 3: We estimate the decay properties of ψ 0 (x). Following [3], by the standard regularity arguments we can deduce that ψ 0 (x) ∈ H 2s (R 3 ) ∩ C r (R 3 ) for all r ∈ (0, 2s) and lim |x|→∞ ψ 0 (x) = 0. Note that p > 2. Then, we can pick ρ > 0 such that for all |x| ≥ ρ, where L > 0 such that ∥ψ 0 ∥ 2 H s (R 3 ) ≤ L, and we conclude that Therefore, and for some suitable R 1 > 0. Let R = max{ρ, R 1 }, and set Then, by the maximum principle we infer that U(x) ≥ 0 for all |x| ≥ R. In addition, by the definition of U(x), obviously, U(x) ≥ 0 for |x| ≤ R. Thus, we get U(x) ≥ 0 for all x ∈ R 3 , furthermore, we have The proof of Theorem 1.1 is finished.

Conclusions
In this paper, we are interested in the existence and decay property of ground state solutions for a Kirchhoff equation involving fractional Laplacian operator. Since the nonlocal term R 3 |(−∆) s 2 ψ| 2 dx 2 included in the energy functional E(ψ) is homogeneous of degree 4, when p ≤ 4, it brings about two obstacles to the standard mountain-pass arguments both in checking the geometrical assumptions in the corresponding energy functional and in proving the boundedness of the Palais-Smale sequence for E(ψ). By constructing a Palais-Smale-Pohozaev sequence at the minimax value c mp , the existence of ground state solutions to this equation for all p ∈ (2, 2 * s ) is established by variational arguments. Furthermore, the decay property of the ground state solution is also investigated. Our result extends and improves the recent results in the literature. We believe that the proposed approach in the present paper can also be applied to studying other related variational problems.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.