Multiplicity and asymptotic behavior of solutions for Kirchhoff type equations involving the Hardy–Sobolev exponent and singular nonlinearity

In this paper, we study a class of critical elliptic problems of Kirchhoff type: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \biggl[a+b \biggl( \int_{\mathbb{R}^{3}}\vert \nabla u\vert ^{2}-\mu \frac{u^{2}}{\vert x\vert ^{2}}\,dx \biggr)^{\frac{2-\alpha }{2}} \biggr]\biggl(-\Delta u- \mu \frac{u}{\vert x\vert ^{2}}\biggr) = \frac{\vert u\vert ^{2^{*}(\alpha )-2}u }{\vert x\vert ^{\alpha }}+\lambda \frac{f(x)\vert u\vert ^{q-2}u }{\vert x\vert ^{\beta }}, $$\end{document}[a+b(∫R3|∇u|2−μu2|x|2dx)2−α2](−Δu−μu|x|2)=|u|2∗(α)−2u|x|α+λf(x)|u|q−2u|x|β, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a,b>0$\end{document}a,b>0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu \in [0,1/4)$\end{document}μ∈[0,1/4), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha , \beta \in [0,2)$\end{document}α,β∈[0,2), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q\in (1,2)$\end{document}q∈(1,2) are constants and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{*}(\alpha )=6-2\alpha $\end{document}2∗(α)=6−2α is the Hardy–Sobolev exponent in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{3}$\end{document}R3. For a suitable function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x)$\end{document}f(x), we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b>0$\end{document}b>0 as a parameter to obtain the convergence property of solutions for the given problem as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b\searrow 0^{+}$\end{document}b↘0+ by the mountain pass theorem and Ekeland’s variational principle.


Introduction and main results
In the present paper, we consider the following Schrödinger equation: where a, b > 0, μ ∈ [0, 1/4), α, β ∈ [0, 2), and q ∈ (1, 2) are constants and 2 * (α) = 6 -2α is the critical Hardy-Sobolev exponent. We call (1.1) a Schrödinger equation of Kirchhoff type because of the appearance of the term b( R 3 |∇u| 2μu 2 |x| -2 dx) (2-α)/2 which makes the study of (1.1) interesting. Indeed, if we choose μ = α = 0 and let |u| 4 u + f (x)|u| q-2 u|x| -β = k(x, u) -V (x)u, then (1.1) transforms to the following classical Kirchhoff type equation: which is degenerate if b = 0 and non-degenerate otherwise. Equation (1.2) arises in a meaningful physical context. In fact, if we set V (x) = 0 and replace R 3 by a bounded domain ⊂ R 3 , then we get the following Dirichlet problem: which is related to the stationary analogue of the equation proposed by Kirchhoff in [16] as an extension of the classical D' Alembert's wave equation for free vibrations of elastic strings. This model takes the changes in length of the string produced by transverse vibrations into account. After J. L. Lions in his pioneer work [21] presented an abstract functional analysis framework to (1.2), this problem has been widely studied in extensive literature such as [8,11,12,19,20,24,25].
In their celebrated paper, Ambrosetti et al. [2] studied the following semilinear elliptic equation with concave-convex nonlinearities: ⎧ ⎨ ⎩ u = |u| p-2 u + ξ |u| q-2 u, in , where is a bounded domain in R N , ξ > 0 and 1 < q < 2 < p ≤ 2 * = 2N/(N -2) with N ≥ 3. By the variational method, they obtained the existence and multiplicity of positive solutions to the above problem. Subsequently, an increasing number of researchers have paid attention to semilinear elliptic equations with critical exponent and concave-convex nonlinearities; for example, see [1,5,13,14,27,29] and the references therein. Using the Nehari manifold and fibering maps, Chen et al. [6] extended the above analysis to the subcritical semilinear elliptic problem of Kirchhoff type: where M is the so-called Kirchhoff function depending on 1 < q < 2 < p < 2 * , is a bounded domain with a smooth boundary in R N and the weight functions h, g ∈ C( ) satisfy some specified conditions f ± = max{±f , 0} = 0 and g ± = max{±g, 0} = 0, they proved the existence of multiple solutions of it. In the critical case, Lei et al. [19] considered the following Kirchhoff problem in three dimensions: where > 0 is a sufficiently small constant, and they employed the mountain pass theorem to show that the problem admits at least two different positive solutions. Some other related and important results can be found in [18,23] and the references therein. Before stating our main results, we introduce some function spaces. Throughout the paper, L p (R 3 ) (1 ≤ p ≤ +∞) is the usual Lebesgue space with the standard norm |u| p , and we consider the Hilbert space D 1,2 (R 3 ) equipped with its usual inner product and norm By the well-known Hardy inequality [17] we derive that the induced inner product and norm are equivalent to the usual inner product and norm on D 1,2 (R 3 ) for any μ ∈ [0, 1/4). As a special case of [15,Lemma 2.3], for any μ ∈ [0, 1/4) and s ∈ [0, 2), we can define We also know that S μ,s can be attained by a positive function U ∈ D 1,2 (R 3 ) satisfying Motivated by all the works mentioned above, we are interested in the multiplicity and asymptotic behavior of solutions of (1.1) whose natural variational functional is Note that we can adopt the idea used in [28] to prove that J(u) is well-defined on D 1,2 (R 3 ) and of class C 1 . Furthermore, any solution of (1.1) is a critical point of J(u). Hence we obtain the solutions of it by finding the critical points of the functional J(u). To this aim, we assume the following condition: and there exists R 0 > 0 such that supp f ∈ B R 0 (0).
Since supp f ⊂ B R 0 (0), using Hölder's inequality and (1.3), we have (1.5) For the convenience of narration, we set max qλ 1 / 2(4α), qλ 2 /2 , and Inspired by the works in [8,24,25], we prefer to study the asymptotic behavior of multiple solutions to (1.1) because the solutions depend on the parameter b. By analyzing the convergence property, we establish the following result in this paper. 2), and q ∈ (1, 2), then (1.1) has at least two positive solutions u 1 b and u 2 b for any λ ∈ (0, λ M ). Moreover, let λ ∈ (0, λ M ) and a > 0 be fixed constants, then there exist subsequences still denoted by themselves (1.6) Remark 1.5 A natural question is why we do not study the convergence of solutions obtained in Theorem 1.2. In fact, if we do this step by step, we can only prove that equation (1.6) has at least one nontrivial solution. The main reason for this phenomenon is that we cannot prove there exists d 1 < 0 independent of b such that m + < d 1 (see Lemma 2.5 for details). To explain this in a little more detail, we assume there exists a sequence The outline of this paper is as follows. In Sect. 2, we present some preliminary results. In Sect. 3, we obtain the existence of two local minimax solutions of (1.1). In Sect. 4, we prove the convergence property on the parameter b > 0.
Notations Throughout this paper we shall denote by C and C i (i = 1, 2, . . .) various positive constants whose exact value may change from lines to lines but are not essential to the analysis of problem. We use "→" and " " to denote the strong and weak convergence in the related function space, respectively. For any ρ > 0 and any x ∈ R 3 , B ρ (x) denotes the ball of radius ρ centered at x, that is, B ρ (x) := {y ∈ R 3 : |y -x| < ρ}.
Let (X, · ) be a Banach space with its dual space (X * , · * ), and be its functional on X. The Palais-Smale sequence at level d ∈ R ((PS) d sequence in short) corresponding to satisfies that (x n ) → d and (x n ) → 0 as n → ∞, where {x n } ⊂ X.

Nehari manifold and fibering map
In this section, we study the so-called Nehari manifold because the variational functional J(u) is not bounded from below on D 1,2 (R 3 ). Let us define and then any nontrivial solution of (1.1) belongs to N . Obviously, u ∈ N if and only if The following lemma tells us the behavior of J(u) on N .

Lemma 2.1 The functional J(u) is coercive and bounded from below on N .
Proof For any u ∈ N , since α ∈ (0, 2) and q ∈ (1, 2), we get which yields that J(u) is coercive and bounded from below on N .
The Nehari manifold N is closely linked to the functions ϕ u (t) = J(tu) for any t > 0. As we all know, the above maps were introduced by Drábek and Pohozaev [9] and discussed in Brown and Zhang [4] (or Chen et al. [6]). For any u ∈ D 1,2 (R 3 ), we have It is easy to see that for any u ∈ D 1,2 (R 3 )\{0} and t > 0 we obtain which gives that ϕ u (t) = 0 if and only if tu ∈ N . In particular, ϕ u (1) = 0 if and only if u ∈ N . Arguing as Brown and Zhang [4], we split N into three parts: Therefore, for any u ∈ N , we have It is similar to the argument in Brown and Zhang [4, Theorem 2.3] that we can derive the following result.
is a local minimizer for J(u) on N and u / ∈ N 0 , then J (u) = 0 in (D 1,2 (R 3 )) * . Inspired by the above lemma, we will study when N 0 = ∅ is established.
To find solutions of (1.1), it is necessary to consider whether N ± are nonempty.

Lemma 2.4
Assume (F) and for any 0 < λ < 1 , then for any u ∈ D 1,2 (R 3 )\{0} there exist t 0 > 0 and unique t + and twith 0 < t + < t 0 < tsuch that t ± u ∈ N ± and Proof Compared with the results in [6], the proof is standard after some simple modifications and we omit it.
From Lemma 2.3, we know that N = N + ∪ Nfor any 0 < λ < 1 max{λ 1 , λ 2 }. Moreover, by Lemma 2.4 we have N ± = ∅ and by Lemma 2.1 we may define Then we have the following result.

Lemma 2.5 Under the assumptions of Theorem
which implies that Thus we obtain that m + < 0.
(ii) To end the proof, we split it into the following two cases.

Proof of Theorem 1.2
In this section, we prove Theorem 1.2. Using Ekeland's variational principle [10] and the argument in [6, Lemma 5.2], we have the following result. The following lemma provides the interval where the (PS) condition holds for J(u).

Lemma 3.2 If λ ∈ (0, * ), any (PS) c sequence of J(u) contains a strongly convergent subsequence whenever c
and C 0 is a positive constant given by Lemma 3.3 below.
Proof Let {u n } ⊂ D 1,2 (R 3 ) be a (PS) c sequence of J(u), and we conclude that {u n } is bounded in D 1,2 (R 3 ). In fact which yields that {u n } is bounded in D 1,2 (R N ) since 1 < q < 2. Up to a subsequence if necessary, there exists u ∈ D 1,2 (R 3 ) such that u n u in D 1,2 (R 3 ), u n → u in L r loc (R 3 ) for r ∈ [1, 2 * (α)) and u n → u a.e. in R 3 . Next we prove that u n → u in D 1,2 (R 3 ).
By the concentration compactness principle [22], there exist a countable set , a set of different points {x j } ⊂ R 3 \{0}, nonnegative real numbers μ x j , ν x j for j ∈ , and nonnegative real numbers μ 0 , γ 0 , and ν 0 such that where δ x is the Dirac mass at x ∈ R 3 . Without loss of generality, we only consider the possibility of concentration at the singular point 0 ∈ R 3 . To do it, for any > 0, we let x j / ∈ B (0) for all j ∈ and choose ϕ to be a smooth cut-off function such that 0 ≤ ϕ ≤ 1, Since {u n } is bounded, using (3.2) we have In view of Sobolev inequality (1.3), that is, which gives that a contradiction! Hence we have which together with (1.5) implies Hence there holds which yields that u n → u in D 1,2 (R 3 ). The proof is complete.
To apply in Lemma 3.2, we have the following result.
Since U ∈ D 1,2 (R 3 )\{0}, by Lemma 2.4 there exists unique t ± U such that t ± U U ∈ N ± . Consequently, we have m -≤ J(t -U U) ≤ max t≥0 J(tU), which completes the proof. Now, we establish the existence of a local minimum for J(u) on N . By Lemma 2.1, we know that {u n } is bounded in D 1,2 (R 3 ). Going to a subsequence if necessary, there exists u λ ∈ D 1,2 (R 3 ) such that u n u λ in D 1,2 (R 3 ). It follows from the definitions of m and m ± that m ≤ m ± . Hence u n → u λ in D 1,2 (R 3 ) by Lemmas 3.2-3.3, then J(u λ ) = m and J (u λ ) = 0. Since m ≤ m + < 0, we can derive u λ is a nontrivial solution of (1.1) by Lemma 2.2. By the fact that J(u) is translation invariant, we know that J(|u λ |) = J(u λ ) = m and J (|u λ |) = J (u λ ) = 0. By using Harnack's inequality [26], it follows that u λ (x) > 0 in R 3 and then u λ is a positive solution of (1.1). We now claim that u λ ∈ N + . Indeed, we argue it indirectly and assume u λ ∈ Nby Lemma 2.3. It follows from Lemma 2.4 that there exist unique t + λ and tλ such that t ± λ u λ ∈ N ± with 0 < t + λ < tλ ≡ 1. By the same idea used in [6,Lemma 4.2], we know that ϕ u λ (t) = J(tu λ ) is strictly increasing on (t + λ , tλ ) and hence a contradiction! So, we can obtain u λ ∈ N + , which implies that m + ≤ J(u λ ) = m ≤ m + . Consequently, the proof of (i) is complete.
By Lemma 4.1 and the mountain pass theorem in [28], a (PS) sequence of the functional J(u) at the level can be constructed, where the set of paths is defined as In other words, there exists a sequence {u n } ⊂ D 1,2 (R 3 ) such that Remark 4.2 By (4.1), we can conclude that c b < c μ,α -C 0 λ 2/(2-q) for any λ ∈ (0, λ M ). In fact, in view of the proof of Lemma 3.3, we obtain for any λ ∈ (0, λ M ). As the proof of Lemma 4.1(ii), there exists sufficiently large t U > 0 such that J b (t U U) < 0. Hence let γ 0 (t) = tt U U ∈ b , then c b ≤ sup t≥0 J(tU), which yields c b < c μ,α -C 0 λ 2/(2-q) for any λ ∈ (0, λ M ).
To obtain a solution with negative energy, we introduce the following lemma.
Now, we establish the existence of multiple solutions of (1.1).
Summing the above three steps, we obtain that u 1 and u 2 are two nontrivial solutions of (1.6). The proof is complete.

Conclusion
This paper is concerned with the qualitative analysis of solutions of a nonlocal problem with Sobolev-Hardy exponent of Kirchhoff type. Meanwhile, it seems that the study of Kirchhoff type equation involving Hardy term and singular nonlinearity via the Nehari manifold and fibering maps is new.