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The Prevalence and Correlates of Gambling Participation among Community-Dwelling Chinese Older Adults in the U.S.

  • Received: 17 December 2014 Accepted: 17 May 2015 Published: 20 May 2015
  • This study aimed to examine the prevalence and correlates of gambling participation and problems among community-dwelling Chinese older adults in the U.S. Based on a community-based participatory research approach, the study enrolled 3,159 Chinese older adults aged 60 years and above in the greater Chicago area. Among the participants, 58.9% were women and the average age was 72.8 years. Overall, 467 older adults had engaged in gambling in the past twelve months and 65 older adults had experienced any risk of problem gambling. Visiting a casino was the most commonly reported type of gambling, whereas betting on Mahjong had the highest frequency. Being male, lower educational levels, higher income levels, having more children, living in the U.S. for a longer period of time, living in the community for a longer period of time, better health status, lower quality of life, and improved health over the past year were significantly correlated with any gambling in the past year. Younger age, being male, and living with more people were significantly correlated with experiencing any risk of problem gambling in the past year. Future studies should be conducted to better examine the health effects of gambling and problem gambling among Chinese older adults.

    Citation: Ruijia Chen, XinQi Dong. The Prevalence and Correlates of Gambling Participation among Community-Dwelling Chinese Older Adults in the U.S.[J]. AIMS Medical Science, 2015, 2(2): 90-103. doi: 10.3934/medsci.2015.2.90

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  • This study aimed to examine the prevalence and correlates of gambling participation and problems among community-dwelling Chinese older adults in the U.S. Based on a community-based participatory research approach, the study enrolled 3,159 Chinese older adults aged 60 years and above in the greater Chicago area. Among the participants, 58.9% were women and the average age was 72.8 years. Overall, 467 older adults had engaged in gambling in the past twelve months and 65 older adults had experienced any risk of problem gambling. Visiting a casino was the most commonly reported type of gambling, whereas betting on Mahjong had the highest frequency. Being male, lower educational levels, higher income levels, having more children, living in the U.S. for a longer period of time, living in the community for a longer period of time, better health status, lower quality of life, and improved health over the past year were significantly correlated with any gambling in the past year. Younger age, being male, and living with more people were significantly correlated with experiencing any risk of problem gambling in the past year. Future studies should be conducted to better examine the health effects of gambling and problem gambling among Chinese older adults.


    We study the following Neumann problem of Kirchhoff type equation with critical growth

    $ {(a+bΩ|u|2dx)Δu+u=Q(x)|u|4u+λP(x)|u|q2u,in  Ω,uv=0,on  Ω, $ (1.1)

    where $ \Omega $ $ \subset $ $ \mathbb{R}^3 $ is a bounded domain with a smooth boundary, $ a, b > 0 $, $ 1 < q < 2 $, $ \lambda > 0 $ is a real parameter. We assume that $ Q(x) $ and $ P(x) $ satisfy the following conditions:

    $ (Q_1) $ $ Q(x)\in C(\bar{\Omega}) $ is a sign-changing;

    $ (Q_2) $ there exists $ x_M\in \Omega $ such that $ Q_M = Q(x_M) > 0 $ and

    $ |Q(x)-Q_M| = o(|x-x_M|)\; \mathrm{as}\; x\rightarrow x_M; $

    $ (Q_3) $ there exists $ 0\in \partial\Omega $ such that $ Q_m = Q(0) > 0 $ and

    $ |Q(x)-Q_m| = o(|x|)\; \mathrm{as}\; x\rightarrow 0; $

    $ (P_1) $ $ P(x) $ is positive continuous on $ \bar{\Omega} $ and $ P(x_0) = \max_{x\in\bar{\Omega}}P(x) $;

    $ (P_2) $ there exist $ \sigma > 0 $, $ R > 0 $ and $ 3-q < \beta < \frac{6-q}{2} $ such that $ P(x)\geq \sigma |x-y|^{-\beta} $ for $ |x-y|\leq R $, where $ y $ is $ x_M\in\Omega $ or $ 0\in\partial\Omega $.

    In recent years, the following Dirichlet problem of Kirchhoff type equation has been studied extensively by many researchers

    $ {(a+bΩ|u|2dx)Δu=f(x,u),in  Ω,u=0,on  Ω, $ (1.2)

    which is related to the stationary analogue of the equation

    $ utt(a+bΩ|u|2dx)Δu=f(x,u) $ (1.3)

    proposed by Kirchhoff in [13] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. In (1.2) and (1.3), $ u $ denotes the displacement, $ b $ is the initial tension and $ f(x, u) $ stands for the external force, while $ a $ is related to the intrinsic properties of the string (such as Young's modulus). We have to point out that such nonlocal problems appear in other fields like biological systems, such as population density, where $ u $ describes a process which depends on the average of itself (see Alves et al. [2]). After the pioneer work of Lions [18], where a functional analysis approach was proposed. The Kirchhoff type Eq (1.2) with critical growth began to call attention of researchers, we can see [1,9,14,17,23,24,28,30] and so on.

    Recently, the following Kirchhoff type equation has been well studied by various authors

    $ {(a+bR3|u|2dx)Δu+V(x)u=f(x,u),inR3,u>0,uH1(R3). $ (1.4)

    There has been much research regarding the concentration behavior of the positive solutions of (1.4), we can see [10,11,12,25,33]. Many papers studied the existence of ground state solutions of (1.4), for example [5,8,15,16,21,22,24]. In addition, the authors established the existence of sign-changing solutions of (1.4) in [20,31]. In papers [27,32] proved the existence and multiplicity of nontrivial solutions of (1.4) by using mountain pass theorem.

    In particular, Chabrowski in [6] studied the solvability of the Neumann problem

    $ {Δu=Q(x)|u|22u+λf(x,u),in  Ω,uv=0,on  Ω, $

    where $ \Omega $ $ \subset $ $ \mathbb{R}^N $ is a smooth bounded domain, $ 2^* = \frac{2N}{N-2}(N\geq3) $ is the critical Sobolev exponent, $ \lambda > 0 $ is a parameter. Assume that $ Q(x)\in C(\overline{\Omega}) $ is a sign-changing function and $ \int_{\Omega}Q(x)dx < 0 $, under the condition of $ f(x, u) $. Using the space decomposition $ H^1(\Omega) = span 1\oplus V $, where $ V = \{v\in H^1(\Omega): \int_{\Omega}vdx = 0\} $, the author obtained the existence of two distinct solutions by the variational method.

    In [14], Lei et al. considered the following Kirchhoff type equation with critical exponent

    $ {(a+bΩ|u|2dx)Δu=u5+λuq1|x|β,in  Ω,u=0,on  Ω, $

    where $ \Omega $ $ \subset $ $ \mathbb{R}^3 $ is a smooth bounded domain, $ a, b > 0 $, $ 1 < q < 2 $, $ \lambda > 0 $ is a parameter. They obtained the existence of a positive ground state solution for $ 0\leq\beta < 2 $ and two positive solutions for $ 3-q\leq\beta < 2 $ by the Nehari manifold method.

    In [34], Zhang obtained the existence and multiplicity of nontrivial solutions of the following equation

    $ {(a+bΩ|u|2dx)Δu+u=λ|u|q2u+f(x,u)+Q(x)u5,in  Ω,uv=0,on  Ω, $ (1.5)

    where $ \Omega $ is an open bounded domain in $ \mathbb{R}^3 $, $ a, b > 0 $, $ 1 < q < 2 $, $ \lambda\geq0 $ is a parameter, $ f(x, u) $ and $ Q(x) $ are positive continuous functions satisfying some additional assumptions. Moreover, $ f(x, u)\thicksim|u|^{p-2}u $ with $ 4 < p < 6 $.

    Comparing with the above mentioned papers, our results are different and extend the above results to some extent. Specially, motivated by [34], we suppose $ Q(x) $ changes sign on $ \Omega $ and $ f(x, u)\equiv0 $ for (1.5). Since (1.1) is critical growth, which leads to the cause of the lack of compactness of the embedding $ H^1(\Omega)\hookrightarrow L^{6}(\Omega) $, we overcome this difficulty by using P.Lions concentration compactness principle [19]. Moreover, note that $ Q(x) $ changes sign on $ \Omega $, how to estimate the level of the mountain pass is another difficulty.

    We define the energy functional corresponding to problem (1.1) by

    $ Iλ(u)=12u2+b4(Ω|u|2dx)216ΩQ(x)|u|6dxλqΩP(x)|u|qdx. $

    A weak solution of problem (1.1) is a function $ u\in H^1(\Omega) $ and for all $ \varphi\in H^1(\Omega) $ such that

    $ Ω(auφ+uφ)dx+bΩ|u|2dxΩuφdx=ΩQ(x)|u|4uφdx+λΩP(x)|u|q2uφdx. $

    Our main results are the following:

    Theorem 1.1. Assume that $ 1 < q < 2 $ and $ Q(x) $ changes sign on $ \Omega $. Then there exists $ \Lambda_{0} > 0 $ such that for every $ \lambda\in(0, \Lambda_{0}) $, problem $(1.1)$ has at least one nontrivial solution.

    Theorem 1.2. Assume that $ 1 < q < 2 $, $ 3-q < \beta < \frac{6-q}{2} $ and $ Q(x) $ changes sign on $ \Omega $, there exists $ \Lambda_{*} > 0 $ such that for all $ \lambda\in(0, \Lambda_{*}) $. Then problem $(1.1)$ has at least two nontrivial solutions.

    Throughout this paper, we make use of the following notations:

    ● The space $ H^1(\Omega) $ is equipped with the norm $ \|u\|_{H^1(\Omega)}^2 = \int_{\Omega}(|\nabla u|^2+u^2)dx $, the norm in $ L^p(\Omega) $ is denoted by $ \|\cdot\|_p $.

    ● Define $ \|u\|^2 = \int_{\Omega}(a|\nabla u|^2+u^2)dx $ for $ u\in H^1(\Omega) $. Note that $ \|\cdot\| $ is an equivalent norm on $ H^1(\Omega) $ with the standard norm.

    ● Let $ D^{1, 2}(\mathbb{R}^3) $ is the completion of $ C_0^{\infty}(\mathbb{R}^3) $ with respect to the norm $ \|u\|_{D^{1, 2}(\mathbb{R}^3)}^2 = \int_{\mathbb{R}^3}|\nabla u|^2dx $.

    ● $ 0 < Q_M = \max_{x\in \bar{\Omega}}Q(x), $ $ 0 < Q_m = \max_{x\in \partial\Omega} Q(x) $.

    ● $ \Omega^+ = \{x\in \Omega:Q(x) > 0\} $ and $ \Omega^- = \{x\in \Omega:Q(x) < 0\} $.

    ● $ C, C_1, C_2, \dots $ denote various positive constants, which may vary from line to line.

    ● We denote by $ S_\rho $ (respectively, $ B_\rho $) the sphere (respectively, the closed ball) of center zero and radius $ \rho $, i.e. $ S_\rho = \{u\in H^{1}(\Omega): \|u\| = \rho\}, $ $ B_\rho = \{u\in H^{1}(\Omega): \|u\|\leq \rho\} $.

    ● Let $ S $ be the best constant for Sobolev embedding $ H^1(\Omega)\hookrightarrow L^{6}(\Omega) $, namely

    $ S=infuH1(Ω){0}Ω(a|u|2+u2)dx(Ω|u|6dx)1/3. $

    ● Let $ S_0 $ be the best constant for Sobolev embedding $ D^{1, 2}(\mathbb{R}^3)\hookrightarrow L^{6}(\mathbb{R}^3) $, namely

    $ S0=infuD1,2(R3){0}R3|u|2dx(R3|u|6dx)1/3. $

    In this section, we firstly show that the functional $ I_\lambda(u) $ has a mountain pass geometry.

    Lemma 2.1. There exist constants $ r, \rho, \Lambda_0 > 0 $ such that the functional $ I_{\lambda} $ satisfies the following conditions for each $ \lambda\in (0, \Lambda_0) $:

    $ (\mathrm{i}) $ $ I_{\lambda}|_{u\in S_\rho}\geq r > 0 $; $ \inf_{_{u\in B_\rho}}I_{\lambda}(u) < 0 $.

    $ (\mathrm{ii}) $ There exists $ e\in H^1(\Omega) $ with $ \|e\| > \rho $ such that $ I_{\lambda}(e) < 0 $.

    Proof. $ (\mathrm{i}) $ From $ (P_1) $, by the H$ \ddot{\mathrm{o}} $lder inequality and the Sobolev inequality, for all $ u\in H^1(\Omega) $ one has

    $ ΩP(x)|u|qdxP(x0)Ω|u|qdxP(x0)|Ω|6q6Sq2uq, $ (2.1)

    and there exists a constant $ C > 0 $, we get

    $ |ΩQ(x)|u|6dx|CΩ|u|6dxCS3u6. $ (2.2)

    Hence, combining (2.1) and (2.2), we have the following estimate

    $ Iλ(u)=12u2+b4(Ω|u|2dx)216ΩQ(x)|u|6dxλqΩP(x)|u|qdx12u2C6Ω|u|6dxλqP(x0)|Ω|6q6Sq2uquq(12u2qC6S3u6qλqP(x0)|Ω|6q6Sq2). $

    Set $ h(t) = \frac{1}{2}{t}^{2-q}-\frac{C}{6}{S^{-3}t}^{6-q} $ for $ t > 0 $, then there exists a constant $ \rho = \left(\frac{3(2-q)S^3}{C(6-q)}\right)^{\frac{1}{4}} > 0 $ such that $ \max_{t > 0}h(t) = h(\rho) > 0 $. Letting $ \Lambda_0 = \frac{qS^{\frac{q}{2}}}{P(x_0)|\Omega|^{\frac{6-q}{6}}}h(\rho) $, there exists a constant $ r > 0 $ such that $ I_{\lambda}|_{u\in S_{\rho}}\geq r $ for every $ \lambda\in (0, \Lambda_0) $. Moreover, for all $ u\in H^1(\Omega)\backslash\{0\} $, we have

    $ limt0+Iλ(tu)tq=λqΩP(x)|u|qdx<0. $

    So we obtain $ I_{\lambda}(tu) < 0 $ for every $ u\neq 0 $ and $ t $ small enough. Therefore, for $ \|u\| $ small enough, one has

    $ minfuBρIλ(u)<0. $

    $ (\mathrm{ii}) $ Let $ v\in H^1(\Omega) $ be such that supp $ v\subset\Omega^+ $, $ v\not\equiv 0 $ and $ t > 0 $, we have

    $ Iλ(tv)=t22v2+bt44(Ω|v|2dx)2t66ΩQ(x)|v|6dxλtqqΩP(x)|v|qdx $

    as $ t\rightarrow\infty $, which implies that $ I_{\lambda}(tv) < 0 $ for $ t > 0 $ large enough. Therefore, we can find $ e\in H^1(\Omega) $ with $ \|e\| > \rho $ such that $ I_{\lambda}(e) < 0 $. The proof is complete.

    Denote

    $ {Θ1=abS304QM+b3S6024Q2M+aS0b2S40+4aS0QM6QM+b2S40b2S40+4aS0QM24Q2M,Θ2=abS3016Qm+b3S60384Q2m+aS0b2S40+16aS0Qm24Qm+b2S40b2S40+16aS0Qm384Q2m. $

    Then we have the following compactness result.

    Lemma 2.2. Suppose that $ 1 < q < 2 $. Then the functional $ I_{\lambda} $ satisfies the $ (PS)_{c_{\lambda}} $ condition for every $ c_{\lambda} < c_* = $ min $ \{\Theta_1-D\lambda^{\frac{2}{2-q}}, \Theta_2-D\lambda^{\frac{2}{2-q}}\} $, where $ D = \frac{2-q}{3q}(\frac{6-q}{4}P(x_0)S^{-\frac{q}{2}}|\Omega|^{\frac{6-q}{6}})^{\frac{2}{2-q}} $.

    Proof. Let $ \{u_n\}\subset H^1(\Omega) $ be a $ (PS)_{c_\lambda} $ sequence for

    $ Iλ(un)cλandIλ(un)0asn. $ (2.3)

    It follows from (2.1), (2.3) and the H$ \ddot{\mathrm{o}} $lder inequality that

    $ cλ+1+o(un)Iλ(un)16Iλ(un),un13un2+b12(Ω|un|2dx)2λ(1q16)P(x0)Sq2|Ω|6q6unq13un2λ(6q)6qP(x0)Sq2|Ω|6q6unq. $

    Therefore $ \{u_n\} $ is bounded in $ H^1(\Omega) $ for all $ 1 < q < 2 $. Thus, we may assume up to a subsequence, still denoted by $ \{u_n\} $, there exists $ u\in H^{1}(\Omega) $ such that

    $ {unu,weaklyinH1(Ω),unu,stronglyinLp(Ω)(1p<6),un(x)u(x),a.e.inΩ, $ (2.4)

    as $ n\rightarrow\infty $. Next, we prove that $ u_n\rightarrow u $ strongly in $ H^1(\Omega) $. By using the concentration compactness principle (see [19]), there exist some at most countable index set $ J $, $ \delta_{x_j} $ is the Dirac mass at $ x_j\subset \bar{\Omega} $ and positive numbers $ \{\nu_j\} $, $ \{\mu_j\} $, $ j\in J $, such that

    $ |un|6dxdν=|u|6dx+jJνjδxj,|un|2dxdμ|u|2dx+jJμjδxj. $

    Moreover, numbers $ \nu_{j} $ and $ \mu_{j} $ satisfy the following inequalities

    $ S0ν13jμjifxjΩ,S0223ν13jμjifxjΩ. $ (2.5)

    For $ \varepsilon > 0 $, let $ \phi_{\varepsilon, j}(x) $ be a smooth cut-off function centered at $ x_j $ such that $ 0\leq\phi_{\varepsilon, j}\leq 1, $ $ |\nabla \phi_{\varepsilon, j}|\leq\frac{2}{\varepsilon} $, and

    $ ϕε,j(x)={1, in B(xj,ε2)ˉΩ,0, in ΩB(xj,ε). $

    There exists a constant $ C > 0 $ such that

    $ limε0limnΩP(x)|un|qϕε,jdxP(x0)limε0limnB(xj,ε)|un|qdx=0. $

    Since $ |\nabla \phi_{\varepsilon, j}|\leq\frac{2}{\varepsilon} $, by using the H$ \ddot{\mathrm{o}} $lder inequality and $ L^2(\Omega) $-convergence of $ \{u_n\} $, we have

    $ limε0limn(a+bΩ|un|2dx)Ωun,ϕε,jundxClimε0limn(Ω|un|2dx)12(Ω|un|2|ϕε,j|2dx)12Climε0(B(xj,ε)|u|6dx)16(B(xj,ε)|ϕε,j|3dx)13Climε0(B(xj,ε)|u|6dx)16(B(xj,ε)(2ε)3dx)13C1limε0(B(xj,ε)|u|6dx)16=0, $

    where $ C_1 > 0 $, and we also derive that

    $ limε0limnΩ|un|2ϕε,jdxlimε0Ω|u|2ϕε,jdx+μj=μj, $
    $ limε0limnΩQ(x)|un|6ϕε,jdx=limε0ΩQ(x)|u|6ϕε,jdx+Q(xj)νj=Q(xj)νj, $
    $ limε0limnΩu2nϕε,jdx=limε0Ωu2ϕε,jdxlimε0B(xj,ε)u2dx=0. $

    Noting that $ u_n\phi_{\varepsilon, j} $ is bounded in $ H^1(\Omega) $ uniformly for $ n $, taking the test function $ \varphi = u_n\phi_{\varepsilon, j} $ in (2.3), from the above information, one has

    $ 0=limε0limnIλ(un),unϕε,j=limε0limn{(a+bΩ|un|2dx)Ωun,(unϕε,j)dx+Ωu2nϕε,jdxΩQ(x)|un|6ϕε,jdxλΩP(x)|un|qϕε,jdx}=limε0limn{(a+bΩ|un|2dx)Ω(|un|2ϕε,j+un,ϕε,jun)dxΩQ(x)|un|6ϕε,jdx}limε0{(a+bΩ|u|2dx+bμj)(Ω|u|2ϕε,jdx+μj)ΩQ(x)|u|6ϕε,jdxQ(xj)νj}(a+bμj)μjQ(xj)νj, $

    so that

    $ Q(x_{j})\nu_{j}\geq(a+b\mu_{j})\mu_{j}, $

    which shows that $ \{u_n\} $ can only concentrate at points $ x_{j} $ where $ Q(x_{j}) > 0 $. If $ \nu_{j} > 0 $, by (2.5) we get

    $ ν13jbS20+b2S40+4aS0QM2QMifxjΩ,ν13jbS20+b2S40+16aS0Qm273QmifxjΩ. $ (2.6)

    From (2.5) and (2.6), we have

    $ μjbS30+b2S60+4aS30QM2QMifxjΩ,μjbS30+b2S60+16aS30Qm8QmifxjΩ. $ (2.7)

    To proceed further we show that (2.7) is impossible. To obtain a contradiction assume that there exists $ j_0\in J $ such that $ \mu_{j_0}\geq\frac{bS_0^3+\sqrt{b^2S_0^6+4aS_0^3Q_M}}{2Q_M} $ and $ x_{j_0}\in \Omega $. By (2.1), (2.3) and (2.4), one has

    $ cλ=limn{Iλ(un)16Iλ(un),un}=limn{a3Ω|un|2dx+b12(Ω|un|2dx)2+13Ωu2ndxλ6q6qΩP(x)|un|qdx}a3(Ω|u|2dx+jJμj)+b12(Ω|u|2dx+jJμj)2+13Ωu2dxλ6q6qP(x0)Sq2|Ω|6q6uqa3μj0+b12μ2j0+13u2λ6q6qP(x0)Sq2|Ω|6q6uq. $

    Set

    $ g(t)=13t2λ6q6qP(x0)Sq2|Ω|6q6tq,t>0, $

    then

    $ g(t)=23tλ6q6P(x0)Sq2|Ω|6q6tq1=0, $

    we can deduce that $ \min_{t\geq0}g(t) $ attains at $ t_0 > 0 $ and

    $ t0=(λ6q4P(x0)Sq2|Ω|6q6)12q. $

    Consequently, we obtain

    $ cλabS304QM+b3S6024Q2M+aS0b2S40+4aS0QM6QM+b2S40b2S40+4aS0QM24Q2MDλ22q=Θ1Dλ22q, $

    where $ D = \frac{2-q}{3q}\left(\frac{6-q}{4}P(x_0)S^{-\frac{q}{2}}|\Omega|^{\frac{6-q}{6}}\right)^{\frac{2}{2-q}} $. If $ \mu_{j_0}\geq\frac{bS_0^3+\sqrt{b^2S_0^6+16aS_0^3Q_m}}{8Q_m} $ and $ x_{j_0}\in \partial\Omega $, then, by the similar calculation, we also get

    $ cλabS3016Qm+b3S60384Q2m+aS0b2S40+16aS0Qm24Qm+b2S40b2S40+16aS0Qm384Q2mDλ22q=Θ2Dλ22q. $

    Let $ c_* = \min\{\Theta_1-D\lambda^{\frac{2}{2-q}}, \Theta_2-D\lambda^{\frac{2}{2-q}}\} $, from the above information, we deduce that $ c_\lambda\geq c_* $. It contradicts our assumption, so it indicates that $ \nu_j = \mu_j = 0 $ for every $ j\in J $, which implies that

    $ Ω|un|6dxΩ|u|6dx $ (2.8)

    as $ n\rightarrow\infty $. Now, we may assume that $ \int_{\Omega}|\nabla u_n|^2dx\rightarrow A^2 $ and $ \int_{\Omega}|\nabla u|^2dx\leq A^2 $, by (2.3), (2.4) and (2.8), one has

    $ 0=limnIλ(un),unu=limn[(a+bΩ|un|2dx)(Ω|un|2dxΩunudx)+Ωun(unu)dxΩQ(x)|un|5(unu)dxλΩP(x)|un|q1(unu)dx]=(a+bA2)(A2Ω|u|2dx). $

    Then, we obtain that $ u_n\rightarrow u $ in $ H^1(\Omega) $. The proof is complete.

    As well known, the function

    $ Uε,y(x)=(3ε2)14(ε2+|xy|2)12,foranyε>0, $

    satisfies

    $ ΔUε,y=U5ε,yinR3, $

    and

    $ R3|Uε,y|2dx=R3|Uε,y|6dx=S320. $

    Let $ \phi\in C^1(\mathbb{R}^3) $ such that $ \phi(x) = 1 $ on $ B(x_M, \frac{R}{2}) $, $ \phi(x) = 0 $ on $ \mathbb{R}^3-B(x_M, R) $ and $ 0\leq\phi(x)\leq 1 $ on $ \mathbb{R}^3 $, we set $ v_\varepsilon(x) = \phi(x)U_{\varepsilon, x_M}(x) $. We may assume that $ Q(x) > 0 $ on $ B(x_M, R) $ for some $ R > 0 $ such that $ B(x_M, R)\subset \Omega $. From [4], we have

    $ {vε22=S320+O(ε),vε66=S320+O(ε3),vε22=O(ε),vε2=aS320+O(ε). $ (2.9)

    Moreover, by [28], we get

    $ {vε42S30+O(ε),vε82S60+O(ε),vε122S90+O(ε). $ (2.10)

    Then we have the following Lemma.

    Lemma 2.3. Suppose that $ 1 < q < 2 $, $ 3-q < \beta < \frac{6-q}{2} $, $ Q_M > 4Q_m $, $ (Q_1) $ and $ (Q_2) $, then $ \sup_{t\geq0}I_{\lambda}(tv_\varepsilon) < \Theta_1-D\lambda^{\frac{2}{2-q}}. $

    Proof. By Lemma 2.1, one has $ I_{\lambda}(tv_{\varepsilon})\rightarrow-\infty $ as $ t\rightarrow\infty $ and $ I_{\lambda}(tv_{\varepsilon}) < 0 $ as $ t\rightarrow 0 $, then there exists $ t_{\varepsilon} > 0 $ such that $ I_{\lambda}(t_{\varepsilon} v_{\varepsilon}) = \sup_{t > 0}I_{\lambda}(tv_{\varepsilon})\geq r > 0 $. We can assume that there exist positive constants $ t_1, t_2 > 0 $ and $ 0 < t_1 < t_{\varepsilon} < t_2 < +\infty $. Let $ I_{\lambda}(t_{\varepsilon}v_{\varepsilon}) = \beta(t_{\varepsilon}v_{\varepsilon})-\lambda\psi(t_{\varepsilon}v_{\varepsilon}) $, where

    $ β(tεvε)=t2ε2vε2+bt4ε4vε42t6ε6ΩQ(x)|vε|6dx, $

    and

    $ ψ(tεvε)=tqεqΩP(x)|vε|qdx. $

    Now, we set

    $ h(t)=t22vε2+bt44vε42t66ΩQ(x)|vε|6dx. $

    It is clear that $ \lim_{t\rightarrow0}h(t) = 0 $ and $ \lim_{t\rightarrow\infty}h(t) = -\infty $. Therefore there exists $ T_{1} > 0 $ such that $ h(T_{1}) = \max_{t\geq0}h(t) $, that is

    $ h(t)|T1=T1vε2+bT31vε42T51ΩQ(x)|vε|6dx=0, $

    from which we have

    $ vε2+bT21vε42=T41ΩQ(x)|vε|6dx. $ (2.11)

    By (2.11) we obtain

    $ T21=bvε42+b2vε82+4vε2ΩQ(x)|vε|6dx2ΩQ(x)|vε|6dx. $

    In addition, by $ (Q_2) $, for all $ \eta > 0 $, there exists $ \rho > 0 $ such that $ |Q(x)-Q_M| < \eta|x-x_M| $ for $ 0 < |x-x_M| < \rho $, for $ \varepsilon > 0 $ small enough, we have

    $ |ΩQ(x)v6εdxΩQMv6εdx|Ω|Q(x)QM|v6εdx<B(xM,ρ)η|xxM|(3ε2)32(ε2+|xxM|2)3dx+CΩB(xM,ρ)(3ε2)32(ε2+|xxM|2)3dxCηε3ρ0r3(ε2+r2)3dr+Cε3Rρr2(ε2+r2)3drCηερ/ε0t3(1+t2)3dt+CR/ερ/εt2(1+t2)3dtC1ηε+C2ε3, $

    where $ C_1, C_2 > 0 $ (independent of $ \eta $, $ \varepsilon $). From this we derive that

    $ lim supε0|ΩQ(x)v6εdxΩQMv6εdx|εC1η. $ (2.12)

    Then from the arbitrariness of $ \eta > 0 $, by (2.9) and (2.12), one has

    $ ΩQ(x)|vε|6dx=QMΩ|vε|6dx+o(ε)=QMS320+o(ε). $ (2.13)

    Hence, it follows from (2.9), (2.10) and (2.13) that

    $ β(tεvε)h(T1)=T21(13vε2+bT2112vε42)=bvε42vε24ΩQ(x)|vε|6dx+b3vε12224(ΩQ(x)|vε|6dx)2+vε2b2vε82+4vε2ΩQ(x)|vε|6dx6ΩQ(x)|vε|6dx+b2vε82b2vε82+4vε2ΩQ(x)|vε|6dx24(ΩQ(x)|vε|6dx)2b(S30+O(ε))(aS320+O(ε))4(QMS320+o(ε))+b3(S90+O(ε))24(QMS320+o(ε))2+(aS320+O(ε))b2(S60+O(ε))+4(aS320+O(ε))(QMS320+o(ε))6(QMS320+o(ε))+b2(S60+O(ε))b2(S60+O(ε))+4(aS320+O(ε))(QMS320+o(ε))24(QMS320+o(ε))2abS304QM+b3S6024Q2M+aS0b2S40+4aS0QM6QM+b2S40b2S40+4aS0QM24Q2M+C3ε=Θ1+C3ε, $

    where the constant $ C_3 > 0 $. According to the definition of $ v_\varepsilon $, from [29], for $ \frac{R}{2} > \varepsilon > 0 $, there holds

    $ ψ(tεvε)1q3q4tq1B(xM,R2)σεq2(ε2+|xxM|2)q2|xxM|βdxCεq2R/20r2(ε2+r2)q2rβdr=Cε6q2βR/2ε0t2(1+t2)q2tβdtCε6q2β10t2βdt=C4ε6q2β, $ (2.14)

    where $ C_4 > 0 $ (independent of $ \varepsilon, \lambda $). Consequently, from the above information, we obtain

    $ Iλ(tεvε)=β(tεvε)λψ(tεvε)Θ1+C3εC4λε6q2β<Θ1Dλ22q. $

    Here we have used the fact that $ \beta > 3-q $ and let $ \varepsilon = \lambda^{\frac{2}{2-q}} $, $ 0 < \lambda < \Lambda_1 = \min\{1, (\frac{C_3+D}{C_4})^{\frac{2-q}{6-2q-2\beta}}\} $, then

    $ C3εC4λε6q2β=C3λ22qC4λ82q2β2q=λ22q(C3C4λ62q2β2q)<Dλ22q. $ (2.15)

    The proof is complete.

    We assume that $ 0\in\partial\Omega $ and $ Q_m = Q(0) $. Let $ \varphi\in C^1(\mathbb{R}^3) $ such that $ \varphi(x) = 1 $ on $ B(0, \frac{R}{2}) $, $ \varphi(x) = 0 $ on $ \mathbb{R}^3-B(0, R) $ and $ 0\leq\varphi(x)\leq 1 $ on $ \mathbb{R}^3 $, we set $ u_\varepsilon(x) = \varphi(x)U_\varepsilon(x) $, the radius $ R $ is chosen so that $ Q(x) > 0 $ on $ B(0, R)\cap \Omega $. If $ H(0) $ denotes the mean curvature of the boundary at $ 0 $, then the following estimates hold (see [6] or [26])

    $ {uε22=O(ε),uε22uε26S0223A3H(0)εlog1ε+O(ε), $ (2.16)

    where $ A_3 > 0 $ is a constant. Then we have the following lemma.

    Lemma 2.4. Suppose that $ 1 < q < 2 $, $ 3-q < \beta < \frac{6-q}{2} $, $ Q_M\leq4Q_m $, $ H(0) > 0 $, Q is positive somewhere on $ \partial\Omega $, $ (Q_1) $ and $ (Q_3) $, then $ \sup_{t\geq0}I_{\lambda}(tu_\varepsilon) < \Theta_2-D\lambda^{\frac{2}{2-q}}. $

    Proof. Similar to the proof of Lemma 2.3, we also have by Lemma 2.1, there exists $ t_{\varepsilon} > 0 $ such that $ I_{\lambda}(t_{\varepsilon} u_{\varepsilon}) = \sup_{t > 0}I_{\lambda}(tu_{\varepsilon})\geq r > 0 $. We can assume that there exist positive constants $ t_1, t_2 > 0 $ such that $ 0 < t_1 < t_{\varepsilon} < t_2 < +\infty $. Let $ I_{\lambda}(t_{\varepsilon}u_{\varepsilon}) = A(t_{\varepsilon}u_{\varepsilon})-\lambda B(t_{\varepsilon}u_{\varepsilon}) $, where

    $ A(tεuε)=t2ε2uε2+bt4ε4uε42t6ε6ΩQ(x)|uε|6dx, $

    and

    $ B(tεuε)=tqεqΩP(x)|uε|qdx. $

    Now, we set

    $ f(t)=t22uε2+bt44uε42t66ΩQ(x)|uε|6dx. $

    Therefore, it is easy to see that there exists $ T_{2} > 0 $ such that $ f(T_{2}) = \max_{f\geq0}f(t) $, that is

    $ f(t)|T2=T2uε2+bT32uε42T52ΩQ(x)|uε|6dx=0. $ (2.17)

    From (2.17) we obtain

    $ T22=buε42+b2uε82+4uε2ΩQ(x)|uε|6dx2ΩQ(x)|uε|6dx. $

    By the assumption $ (Q_3) $, we have the expansion formula

    $ ΩQ(x)|uε|6dx=QmΩ|uε|6dx+o(ε). $ (2.18)

    Hence, combining (2.16) and (2.18), there exists $ C_5 > 0 $, such that

    $ A(tεuε)f(T2)=T22(13uε2+bT2212uε42)=buε42uε24ΩQ(x)|uε|6dx+b3uε12224(ΩQ(x)|uε|6dx)2+uε2b2uε82+4uε2ΩQ(x)|uε|6dx6ΩQ(x)|uε|6dx+b2uε82b2uε82+4uε2ΩQ(x)|uε|6dx24(ΩQ(x)|uε|6dx)2ab4Qm(uε62Ω|uε|6dx+O(ε))+b324Q2m(uε122(Ω|uε|6dx)2+O(ε))+a6Qm(uε22(Ω|uε|6dx)13b2uε82(Ω|uε|6dx)43+4aQmuε22(Ω|uε|6dx)13+O(ε))+b224Q2m(uε82(Ω|uε|6dx)43b2uε82(Ω|uε|6dx)43+4aQmuε22(Ω|uε|6dx)13+O(ε))abS3016Qm+b3S60384Q2m+aS0b2S40+16aS0Qm24Qm+b2S40b2S40+16aS0Qm384Q2m+C5ε=Θ2+C5ε. $

    Consequently, by (2.14) and (2.15), similarly, there exists $ \Lambda_2 > 0 $ such that $ 0 < \lambda < \Lambda_2 $, we get

    $ Iλ(tεuε)=A(tεuε)λB(tεuε)Θ2+C5εC6λε6q2β<Θ2Dλ22q. $

    where $ C_6 > 0 $ (independent of $ \varepsilon, \lambda $). The proof is complete.

    Theorem 2.5. Assume that $ 0 < \lambda < \Lambda_0 $ ($ \Lambda_0 $ is as in Lemma 2.1) and $ 1 < q < 2 $. Then problem $(1.1)$ has a nontrivial solution $ u_\lambda $ with $ I_\lambda(u_\lambda) < 0 $.

    Proof. It follows from Lemma 2.1 that

    $ minfu¯Bρ(0)Iλ(u)<0. $

    By the Ekeland variational principle [7], there exists a minimizing sequence $ \{u_n\}\subset {\overline{B_\rho(0)}} $ such that

    $ Iλ(un)infu¯Bρ(0)Iλ(u)+1n,Iλ(v)Iλ(un)1nvun,v¯Bρ(0). $

    Therefore, there holds $ I_\lambda(u_n)\rightarrow m $ and $ I_\lambda'(u_n)\rightarrow 0 $. Since $ \{u_n\} $ is a bounded sequence and $ {\overline{B_\rho(0)}} $ is a closed convex set, we may assume up to a subsequence, still denoted by $ \{u_n\} $, there exists $ u_{\lambda}\in{\overline{B_\rho(0)}}\subset H^{1}(\Omega) $ such that

    $ {unuλ,weaklyinH1(Ω),unuλ,stronglyinLp(Ω),1p<6,un(x)uλ(x),a.e.inΩ. $

    By the lower semi-continuity of the norm with respect to weak convergence, one has

    $ mlim infn[Iλ(un)16Iλ(un),un]=lim infn[13Ω(a|un|2+u2n)dx+b12(Ω|un|2dx)2+λ(161q)ΩP(x)|un|qdx]13Ω(a|uλ|2+u2λ)dx+b12(Ω|uλ|2dx)2+λ(161q)ΩP(x)|uλ|qdx=Iλ(uλ)16Iλ(uλ),uλ=Iλ(uλ)=m. $

    Thus $ I_{\lambda}(u_\lambda) = m < 0 $, by $ m < 0 < c_\lambda $ and Lemma 2.2, we can see that $ \nabla u_n\rightarrow \nabla u_\lambda $ in $ L^2(\Omega) $ and $ u_\lambda\not\equiv0 $. Therefore, we obtain that $ u_\lambda $ is a weak solution of problem (1.1). Since $ I_\lambda(|u_\lambda|) = I_\lambda(u_\lambda) $, which suggests that $ u_\lambda\geq0 $, then $ u_\lambda $ is a nontrivial solution to problem (1.1). That is, the proof of Theorem 1.1 is complete.

    Theorem 2.6. Assume that $ 0 < \lambda < \Lambda_{*} $$ (\Lambda_{*} = \min\{\Lambda_0, \Lambda_1, \Lambda_2\}) $, $ 1 < q < 2 $ and $ 3-q < \beta < \frac{6-q}{2} $. Then the problem (1.1) has a nontrivial solution $ u_{1}\in H^1(\Omega) $ such that $ I_{\lambda}(u_{1}) > 0 $.

    Proof. Applying the mountain pass lemma [3] and Lemma 2.2, there exists a sequence $ \{u_n\}\subset H^1(\Omega) $ such that

    $ Iλ(un)cλ>0andIλ(un)0asn, $

    where

    $ c_\lambda = \inf\limits_{\gamma\in\Gamma}\max\limits_{t\in[0, 1]}I_{\lambda}(\gamma(t)), $

    and

    $ \Gamma = \left\{\gamma\in C([0, 1], H^{1}(\Omega)): \gamma(0) = 0, \gamma(1) = e\right\}. $

    According to Lemma 2.2, we know that $ \{u_n\}\subset H^1(\Omega) $ has a convergent subsequence, still denoted by $ \{u_n\} $, such that $ u_n\rightarrow u_{1} $ in $ H^1(\Omega) $ as $ n\rightarrow\infty $,

    $ Iλ(u1)=limnIλ(un)=cλ>r>0, $

    which implies that $ u_{1}\not\equiv0 $. Therefore, from the continuity of $ I'_\lambda $, we obtain that $ u_{1} $ is a nontrivial solution of problem (1.1) with $ I_{\lambda}(u_{1}) > 0 $. Combining the above facts with Theorem 2.5 the proof of Theorem 1.2 is complete.

    In this paper, we consider a class of Kirchhoff type equations with Neumann conditions and critical growth. Under suitable assumptions on $ Q(x) $ and $ P(x) $, using the variational method and the concentration compactness principle, we proved the existence and multiplicity of nontrivial solutions.

    This research was supported by the National Natural Science Foundation of China (Grant Nos. 11661021 and 11861021). Authors are grateful to the referees for their very constructive comments and valuable suggestions.

    The authors declare no conflict of interest in this paper.

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