Existence and multiplicity of solutions for Kirchhof-type problems with Sobolev–Hardy critical exponent

In this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.


Introduction and main results
In this paper, we investigate the following Kirchhof-type problem with Soblev-Hardy critical exponent: related to the stationary analogue of the equation which was first proposed by Kirchhoff [1] as an extension of the classical D' Alembert wave equation for free vibrations of elastic strings. Equation (1.2) has aroused widespread concern after the work of Lions [2], which proposes a function analysis framework. After that, many interesting results have been obtained such as [3][4][5][6][7][8][9]. For instance, Xu and Chen [10] studied Kirchhoff-type equations with a general nonlinearity in the critical growth. Under certain conditions, the existence of ground state solutions were proved by using variational methods. In particular, they do not use the classical Ambrosetti-Rabinowitz condition. Fiscella et al. [11] dealt with the existence of nontrivial solutions for critical Hardy-Schrdinger-Kirchhoff systems driven by the fractional p-Laplacian operator. The existence was derived as an application of the mountain pass theorem and the Ekeland variational principle. The authors extend the existence results recently obtained for fractional systems to entire solutions with critical nonlinear terms and generalized the systems driven by the p-Laplacian operator to the fractional Hardy-Schrödinger-Kirchhoff case. Xiang and Vicentiu [12] investigated the existence of solutions for critical Schrödinger-Kirchhof-type systems driven by nonlocal integro-differential operators. By applying the mountain pass theorem and Ekeland's variational principle, the existence and asymptotic behavior of solutions for the problem under some suitable assumptions were obtained. A distinguished feature of their paper is that the systems are degenerate, that is, the Kirchhoff function could vanish at zero. This is the first time of exploiting the existence of solutions for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities in R N . In the case k(x, u) = f (x, u) + u 5 , Xie et al. [6] studied the nondegenerate and degenerate cases and proved the existence and multiplicity of solutions by using the Brezis-Lieb lemma and mountain pass theorem. Naimen [8] further discussed this problem in the case of k(x, u) = μg(x, u) + u 5 under different conditions of g(x, u) and μ ∈ R. In the meantime, the results were expanded in [6] by establishing the existence and nonexistence of positive solutions by using the second concentration compactness lemma and mountain pass theorem.
Problem (1.1) in the case of a = 1 and b = 0 can be reduced to the classic semilinear elliptic problem with critical exponents, for which the existence and multiplicity of solutions was proved by Ding and Tang [9].
Inspired by the results of the above paper, the purpose of this paper is to consider the existence and multiplicity of solutions to problem (1.1). The main difficulty in this paper is that it contains the Sobolev-Hardy critical exponent term, which leads to the energy functional not satisfying the Palais-Smale condition.
In order to state our main results, let F(x, u) = u 0 f (x, t) dt. We introduce the following assumptions: (  [6]. We overcome the compactness problem with concentration compactness principle. Lei [7] studied another special case of problem (1.1) with f (x, u) = λf (x)|u| q-2 u|x| -β for a suitable function f (x) and 1 < q < 2. By using the Nehari manifold and fibering maps, they obtained two positive solutions. We observe that the term λf (x)|u| q-2 u|x| -β has to be a homogeneous function; however, it does not satisfy the assumptions we give in this paper.
The rest of this paper is organized as follows. In Sect. 2, we give some preliminary results. In Sect. 3, we establish the proofs of our main results.

Preliminaries
In this part, we give some information to support this paper. Otherwise stated, C, C 0 , C 1 , . . . represent positive constants, and "→" and " " represent the strong convergence and weak convergence in the corresponding space, respectively. Let H 1 0 ( ) be the usual Hilbert space endowed with the usual inner product and norm (u, v) H 1 0 ( ) = ∇u∇v dx and u H 1 0 ( ) = |∇u| 2 dx 1 2 .
By the well-known Hardy inequality [13] respectively, which are equivalent to the usual inner product and norm on H 1 0 ( ) for any μ ∈ [0, 1/4).
We also define the best Sobolev-Hardy constant S inf (2.1) From Lemma 2.2 in [14] we find that S is independent of , and when = R 3 , it is obtained by the functions for all > 0 andμ = 1/4. In addition, the function y (x) is the solution to the equation Since 0 ∈ , let R 0 be a positive constant such that B 2R 0 (0) ⊂ . We take a cut-off function Then we have the following results [14]: Now we define the functional I on H 1 0 ( ) by Obviously, the functional I belongs to the class C 1 (H 1 0 ( ), R). Furthermore,

Proofs of our main results
In this section, we consider the existence and multiplicity of solutions to problem (1.1).
We first verify that the functional I(u) satisfies the local (PS) condition. Proof Suppose that {u n } is a (PS) c sequence. Then, for c ∈ (0, 0 ), First, we prove that {u n } is a bounded sequence. From (3.1) we have where θ = min{ρ, 2 * (s)}. Thus we conclude that {u n } is a bounded sequence in H 1 0 ( ). By the continuity of embedding we have |u n | 2 * (s) 2 * (s) ≤ C 1 < ∞ (denoting the usual L p ( ) norm by | · | p ). Up to subsequences if necessary, there exists u ∈ H 1 0 ( ) such that u n u weakly in H 1 0 ( ), u n → u in L q ( ) for q ∈ 1, 2 * (s) , u n → u a.e. in .
Then we prove that u n → u in H 1 0 ( ). By (f 1 ), for any > 0, there exists a( ) such that Let δ 1 = 2a( ) . When E ⊂ and mes E < δ 1 , we have Hence { f (x, u n )u n dx, n ∈ N} is equiabsolutely continuous. It is easy to get the following from the Vitali convergence theorem: Similarly, we can prove that Further, by the concentration compactness principle [15] there exist a countable set , a set of different points {x j } ⊂ \ {0}, nonnegative real numbers μ x j , ν x j for j ∈ , and nonnegative real numbers μ 0 , γ 0 , ν 0 such that where δ x is the Dirac mass at x ∈ . For any > 0, we let x j / ∈ B (0) for all j ∈ and choose φ to be s smooth cut-off function such that 0 ≤ φ ≤ 1, φ ≡ 0 for x ∈ B c (0), φ ≡ 1 for x ∈ B /2 (0), and |∇φ| ≤ 4/ . Then The proofs of (3.4) and (3.5) are similar to that of Theorem 2.3 in [8] and are omitted here. Since {u n } is bounded, by (3.1) we have Combining this with (2.1), we have that S 2 * (s)/2 ν 0 ≤ (μ 0μγ 0 ) 2 * (s)/2 , and we deduce that Therefore we get a contradiction. Thus we obtain Combining this with (3.2), we find which shows that u n → u in H 1 0 ( ). The proof is completed. Proof We consider the functions By (f 1 ), obviously, Therefore we obtain and t 4-2s + |t | 4-2s |v | 2 * (s) dx = t 4-2s 1 + |v | 2 * (s) dx ≤ 3 2 t 4-2s . (3.8) Thanks to (2.3), when is small enough, we conclude from d( ) |v | 2 dx → 0 as → 0 that From (3.7)-(3.9) we get Combining this with (2.2), we obtain Consequently, the function g 0 (t) attains its maximum at t 0 and continuously increases in the interval [0, t 0 ]. From this, together with (2.2) and the inequality F(x, t) ≥ C 2 |t| ρ , which is directly obtained from (f 2 ), we derive that In addition, from (2.3) it follows that Thanks to (f 2 ), we have Choosing small enough, we conclude sup t≥0 I(tv ) = g(t ) < .
This completes the proof of Lemma 3.2.
Next, we prove that the functional I(u) satisfies the mountain pass geometry.
Proof (i) By (f 1 ), for any > 0, there exists C 3 such that f (x, t) ≤ t + C 3 t 5-2s for all t ∈ R + and x ∈ . By the definition of F(x, u) we get F(x, t) ≤ 1 2 t 2 + C 4 t 2 * (s)