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
[1] | Lulu Tao, Rui He, Sihua Liang, Rui Niu . Existence and multiplicity of solutions for critical Choquard-Kirchhoff type equations with variable growth. AIMS Mathematics, 2023, 8(2): 3026-3048. doi: 10.3934/math.2023156 |
[2] | Zehra Yucedag . Variational approach for a Steklov problem involving nonstandard growth conditions. AIMS Mathematics, 2023, 8(3): 5352-5368. doi: 10.3934/math.2023269 |
[3] | Yun-Ho Kim . Multiple solutions to Kirchhoff-Schrödinger equations involving the $ p(\cdot) $-Laplace-type operator. AIMS Mathematics, 2023, 8(4): 9461-9482. doi: 10.3934/math.2023477 |
[4] | Deke Wu, Hongmin Suo, Linyan Peng, Guaiqi Tian, Changmu Chu . Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity and critical exponents. AIMS Mathematics, 2022, 7(5): 7909-7935. doi: 10.3934/math.2022443 |
[5] | Wangjin Yao . Variational approach to non-instantaneous impulsive differential equations with $ p $-Laplacian operator. AIMS Mathematics, 2022, 7(9): 17269-17285. doi: 10.3934/math.2022951 |
[6] | Zhenluo Lou, Jian Zhang . On general Kirchhoff type equations with steep potential well and critical growth in $ \mathbb{R}^2 $. AIMS Mathematics, 2024, 9(8): 21433-21454. doi: 10.3934/math.20241041 |
[7] | Fugeng Zeng, Peng Shi, Min Jiang . Global existence and finite time blow-up for a class of fractional $ p $-Laplacian Kirchhoff type equations with logarithmic nonlinearity. AIMS Mathematics, 2021, 6(3): 2559-2578. doi: 10.3934/math.2021155 |
[8] | Najla Alghamdi, Abdeljabbar Ghanmi . Multiple solutions for a singular fractional Kirchhoff problem with variable exponents. AIMS Mathematics, 2025, 10(1): 826-838. doi: 10.3934/math.2025039 |
[9] | Liuyang Shao, Haibo Chen, Yicheng Pang, Yingmin Wang . Multiplicity of nontrivial solutions for a class of fractional Kirchhoff equations. AIMS Mathematics, 2024, 9(2): 4135-4160. doi: 10.3934/math.2024203 |
[10] | Nanbo Chen, Honghong Liang, Xiaochun Liu . On Kirchhoff type problems with singular nonlinearity in closed manifolds. AIMS Mathematics, 2024, 9(8): 21397-21413. doi: 10.3934/math.20241039 |
We study the following Neumann problem of Kirchhoff type equation with critical growth
$ {−(a+b∫Ω|∇u|2dx)Δu+u=Q(x)|u|4u+λP(x)|u|q−2u,in Ω,∂u∂v=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+b∫Ω|∇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+b∫Ω|∇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+b∫R3|∇u|2dx)Δu+V(x)u=f(x,u),inR3,u>0,u∈H1(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|2∗−2u+λf(x,u),in Ω,∂u∂v=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+b∫Ω|∇u|2dx)Δu=u5+λuq−1|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+b∫Ω|∇u|2dx)Δu+u=λ|u|q−2u+f(x,u)+Q(x)u5,in Ω,∂u∂v=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)=12‖u‖2+b4(∫Ω|∇u|2dx)2−16∫ΩQ(x)|u|6dx−λq∫ΩP(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(a∇u∇φ+uφ)dx+b∫Ω|∇u|2dx∫Ω∇u∇φdx=∫ΩQ(x)|u|4uφdx+λ∫ΩP(x)|u|q−2uφdx. $ |
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=infu∈H1(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=infu∈D1,2(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|qdx≤P(x0)∫Ω|u|qdx≤P(x0)|Ω|6−q6S−q2‖u‖q, $ | (2.1) |
and there exists a constant $ C > 0 $, we get
$ |∫ΩQ(x)|u|6dx|≤C∫Ω|u|6dx≤CS−3‖u‖6. $ | (2.2) |
Hence, combining (2.1) and (2.2), we have the following estimate
$ Iλ(u)=12‖u‖2+b4(∫Ω|∇u|2dx)2−16∫ΩQ(x)|u|6dx−λq∫ΩP(x)|u|qdx≥12‖u‖2−C6∫Ω|u|6dx−λqP(x0)|Ω|6−q6S−q2‖u‖q≥‖u‖q(12‖u‖2−q−C6S−3‖u‖6−q−λqP(x0)|Ω|6−q6S−q2). $ |
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
$ limt→0+Iλ(tu)tq=−λq∫ΩP(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
$ m≜infu∈BρIλ(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)=t22‖v‖2+bt44(∫Ω|∇v|2dx)2−t66∫ΩQ(x)|v|6dx−λtqq∫ΩP(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+aS0√b2S40+4aS0QM6QM+b2S40√b2S40+4aS0QM24Q2M,Θ2=abS3016Qm+b3S60384Q2m+aS0√b2S40+16aS0Qm24Qm+b2S40√b2S40+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)→cλandI′λ(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)−16⟨I′λ(un),un⟩≥13‖un‖2+b12(∫Ω|∇un|2dx)2−λ(1q−16)P(x0)S−q2|Ω|6−q6‖un‖q≥13‖un‖2−λ(6−q)6qP(x0)S−q2|Ω|6−q6‖un‖q. $ |
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
$ {un⇀u,weaklyinH1(Ω),un→u,stronglyinLp(Ω)(1≤p<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|6dx⇀dν=|u|6dx+∑j∈Jνjδxj,|∇un|2dx⇀dμ≥|∇u|2dx+∑j∈Jμjδxj. $ |
Moreover, numbers $ \nu_{j} $ and $ \mu_{j} $ satisfy the following inequalities
$ S0ν13j≤μjifxj∈Ω,S0223ν13j≤μjifxj∈∂Ω. $ | (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 Ω∖B(xj,ε). $ |
There exists a constant $ C > 0 $ such that
$ limε→0limn→∞∫ΩP(x)|un|qϕε,jdx≤P(x0)limε→0limn→∞∫B(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
$ limε→0limn→∞(a+b∫Ω|∇un|2dx)∫Ω⟨∇un,∇ϕε,j⟩undx≤Climε→0limn→∞(∫Ω|∇un|2dx)12(∫Ω|un|2|∇ϕε,j|2dx)12≤Climε→0(∫B(xj,ε)|u|6dx)16(∫B(xj,ε)|∇ϕε,j|3dx)13≤Climε→0(∫B(xj,ε)|u|6dx)16(∫B(xj,ε)(2ε)3dx)13≤C1limε→0(∫B(xj,ε)|u|6dx)16=0, $ |
where $ C_1 > 0 $, and we also derive that
$ limε→0limn→∞∫Ω|∇un|2ϕε,jdx≥limε→0∫Ω|∇u|2ϕε,jdx+μj=μj, $ |
$ limε→0limn→∞∫ΩQ(x)|un|6ϕε,jdx=limε→0∫ΩQ(x)|u|6ϕε,jdx+Q(xj)νj=Q(xj)νj, $ |
$ limε→0limn→∞∫Ωu2nϕε,jdx=limε→0∫Ωu2ϕε,jdx≤limε→0∫B(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=limε→0limn→∞⟨I′λ(un),unϕε,j⟩=limε→0limn→∞{(a+b∫Ω|∇un|2dx)∫Ω⟨∇un,∇(unϕε,j)⟩dx+∫Ωu2nϕε,jdx−∫ΩQ(x)|un|6ϕε,jdx−λ∫ΩP(x)|un|qϕε,jdx}=limε→0limn→∞{(a+b∫Ω|∇un|2dx)∫Ω(|∇un|2ϕε,j+⟨∇un,∇ϕε,j⟩un)dx−∫ΩQ(x)|un|6ϕε,jdx}≥limε→0{(a+b∫Ω|∇u|2dx+bμj)(∫Ω|∇u|2ϕε,jdx+μj)−∫ΩQ(x)|u|6ϕε,jdx−Q(xj)νj}≥(a+bμj)μj−Q(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
$ ν13j≥bS20+√b2S40+4aS0QM2QMifxj∈Ω,ν13j≥bS20+√b2S40+16aS0Qm273Qmifxj∈∂Ω. $ | (2.6) |
From (2.5) and (2.6), we have
$ μj≥bS30+√b2S60+4aS30QM2QMifxj∈Ω,μj≥bS30+√b2S60+16aS30Qm8Qmifxj∈∂Ω. $ | (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)−16⟨I′λ(un),un⟩}=limn→∞{a3∫Ω|∇un|2dx+b12(∫Ω|∇un|2dx)2+13∫Ωu2ndx−λ6−q6q∫ΩP(x)|un|qdx}≥a3(∫Ω|∇u|2dx+∑j∈Jμj)+b12(∫Ω|∇u|2dx+∑j∈Jμj)2+13∫Ωu2dx−λ6−q6qP(x0)S−q2|Ω|6−q6‖u‖q≥a3μj0+b12μ2j0+13‖u‖2−λ6−q6qP(x0)S−q2|Ω|6−q6‖u‖q. $ |
Set
$ g(t)=13t2−λ6−q6qP(x0)S−q2|Ω|6−q6tq,t>0, $ |
then
$ g′(t)=23t−λ6−q6P(x0)S−q2|Ω|6−q6tq−1=0, $ |
we can deduce that $ \min_{t\geq0}g(t) $ attains at $ t_0 > 0 $ and
$ t0=(λ6−q4P(x0)S−q2|Ω|6−q6)12−q. $ |
Consequently, we obtain
$ cλ≥abS304QM+b3S6024Q2M+aS0√b2S40+4aS0QM6QM+b2S40√b2S40+4aS0QM24Q2M−Dλ22−q=Θ1−Dλ22−q, $ |
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
$ cλ≥abS3016Qm+b3S60384Q2m+aS0√b2S40+16aS0Qm24Qm+b2S40√b2S40+16aS0Qm384Q2m−Dλ22−q=Θ2−Dλ22−q. $ |
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|6dx→∫Ω|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=limn→∞⟨I′λ(un),un−u⟩=limn→∞[(a+b∫Ω|∇un|2dx)(∫Ω|∇un|2dx−∫Ω∇un∇udx)+∫Ωun(un−u)dx−∫ΩQ(x)|un|5(un−u)dx−λ∫ΩP(x)|un|q−1(un−u)dx]=(a+bA2)(A2−∫Ω|∇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+|x−y|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
$ {‖∇vε‖22=S320+O(ε),‖vε‖66=S320+O(ε3),‖vε‖22=O(ε),‖vε‖2=aS320+O(ε). $ | (2.9) |
Moreover, by [28], we get
$ {‖∇vε‖42≤S30+O(ε),‖∇vε‖82≤S60+O(ε),‖∇vε‖122≤S90+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)=t2ε2‖vε‖2+bt4ε4‖∇vε‖42−t6ε6∫ΩQ(x)|vε|6dx, $ |
and
$ ψ(tεvε)=tqεq∫ΩP(x)|vε|qdx. $ |
Now, we set
$ h(t)=t22‖vε‖2+bt44‖∇vε‖42−t66∫ΩQ(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=T1‖vε‖2+bT31‖∇vε‖42−T51∫ΩQ(x)|vε|6dx=0, $ |
from which we have
$ ‖vε‖2+bT21‖∇vε‖42=T41∫ΩQ(x)|vε|6dx. $ | (2.11) |
By (2.11) we obtain
$ T21=b‖∇vε‖42+√b2‖∇vε‖82+4‖vε‖2∫ΩQ(x)|vε|6dx2∫ΩQ(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)v6εdx−∫ΩQMv6εdx|≤∫Ω|Q(x)−QM|v6εdx<∫B(xM,ρ)η|x−xM|(3ε2)32(ε2+|x−xM|2)3dx+C∫Ω∖B(xM,ρ)(3ε2)32(ε2+|x−xM|2)3dx≤Cηε3∫ρ0r3(ε2+r2)3dr+Cε3∫Rρr2(ε2+r2)3dr≤Cηε∫ρ/ε0t3(1+t2)3dt+C∫R/ερ/εt2(1+t2)3dt≤C1ηε+C2ε3, $ |
where $ C_1, C_2 > 0 $ (independent of $ \eta $, $ \varepsilon $). From this we derive that
$ lim supε→0|∫ΩQ(x)v6εdx−∫ΩQMv6εdx|ε≤C1η. $ | (2.12) |
Then from the arbitrariness of $ \eta > 0 $, by (2.9) and (2.12), one has
$ ∫ΩQ(x)|vε|6dx=QM∫Ω|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(13‖vε‖2+bT2112‖∇vε‖42)=b‖∇vε‖42‖vε‖24∫ΩQ(x)|vε|6dx+b3‖∇vε‖12224(∫ΩQ(x)|vε|6dx)2+‖vε‖2√b2‖∇vε‖82+4‖vε‖2∫ΩQ(x)|vε|6dx6∫ΩQ(x)|vε|6dx+b2‖∇vε‖82√b2‖∇vε‖82+4‖vε‖2∫ΩQ(x)|vε|6dx24(∫ΩQ(x)|vε|6dx)2≤b(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))2≤abS304QM+b3S6024Q2M+aS0√b2S40+4aS0QM6QM+b2S40√b2S40+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)≥1q3q4tq1∫B(xM,R2)σεq2(ε2+|x−xM|2)q2|x−xM|βdx≥Cεq2∫R/20r2(ε2+r2)q2rβdr=Cε6−q2−β∫R/2ε0t2(1+t2)q2tβdt≥Cε6−q2−β∫10t2−βdt=C4ε6−q2−β, $ | (2.14) |
where $ C_4 > 0 $ (independent of $ \varepsilon, \lambda $). Consequently, from the above information, we obtain
$ Iλ(tεvε)=β(tεvε)−λψ(tεvε)≤Θ1+C3ε−C4λε6−q2−β<Θ1−Dλ22−q. $ |
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
$ C3ε−C4λε6−q2−β=C3λ22−q−C4λ8−2q−2β2−q=λ22−q(C3−C4λ6−2q−2β2−q)<−Dλ22−q. $ | (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])
$ {‖uε‖22=O(ε),‖∇uε‖22‖uε‖26≤S0223−A3H(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)=t2ε2‖uε‖2+bt4ε4‖∇uε‖42−t6ε6∫ΩQ(x)|uε|6dx, $ |
and
$ B(tεuε)=tqεq∫ΩP(x)|uε|qdx. $ |
Now, we set
$ f(t)=t22‖uε‖2+bt44‖∇uε‖42−t66∫ΩQ(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=T2‖uε‖2+bT32‖∇uε‖42−T52∫ΩQ(x)|uε|6dx=0. $ | (2.17) |
From (2.17) we obtain
$ T22=b‖∇uε‖42+√b2‖∇uε‖82+4‖uε‖2∫ΩQ(x)|uε|6dx2∫ΩQ(x)|uε|6dx. $ |
By the assumption $ (Q_3) $, we have the expansion formula
$ ∫ΩQ(x)|uε|6dx=Qm∫Ω|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(13‖uε‖2+bT2212‖∇uε‖42)=b‖∇uε‖42‖uε‖24∫ΩQ(x)|uε|6dx+b3‖∇uε‖12224(∫ΩQ(x)|uε|6dx)2+‖uε‖2√b2‖∇uε‖82+4‖uε‖2∫ΩQ(x)|uε|6dx6∫ΩQ(x)|uε|6dx+b2‖∇uε‖82√b2‖∇uε‖82+4‖uε‖2∫ΩQ(x)|uε|6dx24(∫ΩQ(x)|uε|6dx)2≤ab4Qm(‖∇uε‖62∫Ω|uε|6dx+O(ε))+b324Q2m(‖∇uε‖122(∫Ω|uε|6dx)2+O(ε))+a6Qm(‖∇uε‖22(∫Ω|uε|6dx)13√b2‖∇uε‖82(∫Ω|uε|6dx)43+4aQm‖∇uε‖22(∫Ω|uε|6dx)13+O(ε))+b224Q2m(‖∇uε‖82(∫Ω|uε|6dx)43√b2‖∇uε‖82(∫Ω|uε|6dx)43+4aQm‖∇uε‖22(∫Ω|uε|6dx)13+O(ε))≤abS3016Qm+b3S60384Q2m+aS0√b2S40+16aS0Qm24Qm+b2S40√b2S40+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+C5ε−C6λε6−q2−β<Θ2−Dλ22−q. $ |
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
$ m≜infu∈¯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)−1n‖v−un‖,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
$ {un⇀uλ,weaklyinH1(Ω),un→uλ,stronglyinLp(Ω),1≤p<6,un(x)→uλ(x),a.e.inΩ. $ |
By the lower semi-continuity of the norm with respect to weak convergence, one has
$ m≥lim infn→∞[Iλ(un)−16⟨I′λ(un),un⟩]=lim infn→∞[13∫Ω(a|∇un|2+u2n)dx+b12(∫Ω|∇un|2dx)2+λ(16−1q)∫ΩP(x)|un|qdx]≥13∫Ω(a|∇uλ|2+u2λ)dx+b12(∫Ω|∇uλ|2dx)2+λ(16−1q)∫ΩP(x)|uλ|qdx=Iλ(uλ)−16⟨I′λ(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>0andI′λ(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)=limn→∞Iλ(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.
[1] | Afifi TO, Cox BJ, Martens PJ, et al. (2010) The relationship between problem gambling and mental and physical health correlates among a nationally representative sample of Canadian women. Can J Public Health 2010: 171-175. |
[2] | Messerlian C, Derevensky JL, Gupta, R (2005) Youth gambling: A public health perspective. J Gambl Issues 20: 69-79. |
[3] | Newman SC, Thompson AH (2007) The association between pathological gambling and attempted suicide: findings from a national survey in Canada. Can J Psychiatry 52: 605-612. |
[4] | National Council on Problem Gambling. Problem Gambling. Available from: http://www ncpgambling org/help-treatment/faq/ 2014 |
[5] |
El-Guebaly N, Patten SB, Currie S, et al. (2006) Epidemiological associations between gambling behavior, substance use & mood and anxiety disorders. J Gambl Stud 22: 275-287. doi: 10.1007/s10899-006-9016-6
![]() |
[6] |
Desai RA, Maciejewski PK, Dausey DJ, et al. (2004) Health correlates of recreational gambling in older adults. Am J Psychiatry 161: 1672-1679. doi: 10.1176/appi.ajp.161.9.1672
![]() |
[7] |
Tse S, Hong SI, Wang CW, et al. (2012) Gambling behavior and problems among older adults: a systematic review of empirical studies. J Gerontol B Psychol Sci Soc Sci 67: 639-652. doi: 10.1093/geronb/gbs068
![]() |
[8] |
McNeilly DP, Burke WJ (2000) Late life gambling: The attitudes and behaviors of older adults. J Gambl Stud 16: 393-415. doi: 10.1023/A:1009432223369
![]() |
[9] | Volberg RA, McNeilly D (2003) Gambling and problem gambling among seniors in Florida. Maitland, FL: Florida Council on Compulsive Gambling. |
[10] |
Wiebe JM, Cox BJ (2005) Problem and probable pathological gambling among older adults assessed by the SOGS-R. J Gambl Stud 21: 205-221. doi: 10.1007/s10899-005-3032-9
![]() |
[11] |
Morasco BJ, Pietrzak RH, Blanco C, et al. (2006) Health problems and medical utilization associated with gambling disorders: results from the National Epidemiologic Survey on Alcohol and Related Conditions. Psychosom Med 68: 976-984. doi: 10.1097/01.psy.0000238466.76172.cd
![]() |
[12] |
Raylu N, Oei TP (2004) Role of culture in gambling and problem gambling. Clin Psychol Rev 23: 1087-1114. doi: 10.1016/j.cpr.2003.09.005
![]() |
[13] | Loo JM, Raylu N, Oei TP (2008) Gambling among the Chinese: A comprehensive review. ClinPsychol Rev 28:1152-1166. |
[14] |
Raylu N, Oei TP (2004) Role of culture in gambling and problem gambling. Clin Psychol Rev 23: 1087-1114. doi: 10.1016/j.cpr.2003.09.005
![]() |
[15] |
Tse S, Yu AC, Rossen F, et al. (2010) Examination of Chinese gambling problems through a socio-historical-cultural perspective. Scientific World Journal 10: 1694-704. doi: 10.1100/tsw.2010.167
![]() |
[16] | Victorian Casino and Gaming Authority (2000) The impact of gaming on specific cultural groups. Available from: https://assets.justice.vic.gov.au/vcglr/resources/b1bc0f57-c5d5-46f1-9e22-70050c99c86f/impactofgamingonspecificculturalgroups.pdf |
[17] | Sam Louie. Asian gambling addiction. 2014. Available from: http://www.psychologytoday.com/blog/minority-report/201407/asian-gambling-addiction |
[18] |
Papineau E (2005) Pathological gambling in Montreal's Chinese community: an anthropological perspective. J Gambl Stud 21: 157-178. doi: 10.1007/s10899-005-3030-y
![]() |
[19] |
Chen SX, Mak WW (2008) Seeking professional help: Etiology beliefs about mental illness across cultures. J Couns Psychol 55: 442-450. doi: 10.1037/a0012898
![]() |
[20] | UCLA Center for Civil Society. Asian gambling. 2007. Availabe from: http://civilsociety.ucla.edu/practitioners/profiles-in-engagement/asian-gambling |
[21] |
Dong X, Chen R, Chang E-S, et al. (2014) The prevalence of suicide attempts among community-dwelling US Chinese older adults GÇôfindings from the PINE study. Ethn Inequal Health Soc Care 7: 23-35. doi: 10.1108/EIHSC-10-2013-0030
![]() |
[22] |
Dong X, Chen R, Li C, et al. (2014) Understanding Depressive Symptoms Among Community-Dwelling Chinese Older Adults in the Greater Chicago Area. J Aging Health 26: 1155-1171. doi: 10.1177/0898264314527611
![]() |
[23] | Dong X, Chen R, Fulmer T, et al. (2014) Prevalence and correlates of elder mistreatment in a community-dwelling population of U.S. Chinese older adults. J Aging Health 26: 1209-1224. |
[24] | Dong X, Chen R, Wong E, et al. (2014) Suicidal ideation in an older US Chinese population. J Aging Health 267: 1189-1208. |
[25] |
Dong X (2014) The Population Study of Chinese Elderly in Chicago. J Aging Health 26: 1079-1084. doi: 10.1177/0898264314550581
![]() |
[26] | Dong X, Chang E, Wong E, et al. (2010) Assessing the health needs of Chinese older adults: Findings from a community-based participatory research study in Chicago's Chinatown. J Aging Res 2010: 124246. |
[27] |
Dong X, Chang E-S, Wong E, et al. (2011) Working with culture: lessons learned from a community-engaged project in a Chinese aging population. Aging Health 7: 529-537. doi: 10.2217/ahe.11.43
![]() |
[28] |
Dong X, Wong E, Simon MA (2014) Study design and implementation of the PINE study. J Aging Health 26: 1085-1099. doi: 10.1177/0898264314526620
![]() |
[29] | Ferris J, Wynne H (2001) The Canadian problem gambling index. Ottawa, ON: Canadian Centre on Substance Abuse. |
[30] |
Loo JM, Oei TP, Raylu N (2011) Psychometric evaluation of the problem gambling severity index-chinese version (PGSI-C). J Gambl Stud 27: 453-466. doi: 10.1007/s10899-010-9221-1
![]() |
[31] |
Philippe F, Vallerand RJ (2007) Prevalence rates of gambling problems in Montreal, Canada: A look at old adults and the role of passion. J Gambl Stud 23: 275-283. doi: 10.1007/s10899-006-9038-0
![]() |
[32] | Moore TL (2001) Older adults gambling in Oregon. Available from: http://problemgamblingprevention.org/older-adults/prevalence-older-adults.pdf |
[33] |
Lai DW (2006) Gambling and the older Chinese in Canada. J Gambl Stud 22: 121-141. doi: 10.1007/s10899-005-9006-0
![]() |
[34] |
Grant Stitt B, Giacopassi D, Nichols M (2003) Gambling among older adults: A comparative analysis. Exp Aging Res 29: 189-203. doi: 10.1080/03610730303713
![]() |
[35] |
Momper SL, Nandi V, Ompad DC, et al. (2009) The prevalence and types of gambling among undocumented Mexican immigrants in New York City. J Gambl Stud 25: 49-65. doi: 10.1007/s10899-008-9105-9
![]() |
[36] | Wiebe J, Single E, Falkowski-Ham A, Mun P, et al. (2004) Gambling and problem gambling among older adults in Ontario. Responsible Gambling Council. Available from: http://www.responsiblegambling.org/rg-news-research/rgc-centre/research-and-analysis/docs/research-reports/gambling-and-problem-gambling-among-older-adults-in-ontario |
[37] | Fong TW (2005) The vulnerable faces of pathological gambling. Psychiatry (Edgmont) 2: 34-42. |
[38] | Fong DKC, Ozorio B (2005) Gambling participation and prevalence estimates of pathological gamblingin far-east gambling city: Macao. UNLV Gaming Res Rev J 9: 15-28 |
1. | Ying Zhou, Jun Lei, Yue Wang, Zonghong Xiong, Positive solutions of a Kirchhoff–Schrödinger--Newton system with critical nonlocal term, 2022, 14173875, 1, 10.14232/ejqtde.2022.1.50 | |
2. | Deke Wu, Hongmin Suo, Jun Lei, Multiple Positive Solutions for Kirchhoff-Type Problems Involving Supercritical and Critical Terms, 2024, 23, 1575-5460, 10.1007/s12346-024-00999-w | |
3. | Jiaqing Hu, Anmin Mao, Multiple solutions to nonlocal Neumann boundary problem with sign‐changing coefficients, 2024, 0170-4214, 10.1002/mma.10419 | |
4. | A. Ahmed, Mohamed Saad Bouh Elemine Vall, MULTIPLICITY OF WEAK SOLUTIONS FOR A (P(X), Q(X))-KIRCHHOFF EQUATION WITH NEUMANN BOUNDARY CONDITIONS, 2024, 14, 2156-907X, 2441, 10.11948/20230449 | |
5. | Ahmed Ahmed, Mohamed Saad Bouh Elemine Vall, Weak solutions in anisotropic (α→(z),β→(z))-Laplacian Kirchhoff models, 2025, 1747-6933, 1, 10.1080/17476933.2025.2500342 |