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Background Relationship between polymorphisms in peroxisome proliferator-activated receptor gamma (PPARG) and progression of type 2 diabetes mellitus (T2DM) and coronary artery disease (CAD) remains to be clarified. Methods 635 subjects were divided into T2DM, CAD, T2DM complicated with CAD (T2DM/CAD) and control groups according to diagnostic criteria. The rs10865710 and rs3856806 polymorphisms were genotyped, and the severity of T2DM and CAD was evaluated for all subjects. Results In patients with T2DM, G allele carriers of rs10865710 polymorphism had significantly higher levels of glucose, triglycerides, apolipoprotein B (ApoB) and lipoprotein (a) (Lp(a)) than non-carriers, T allele carriers of rs3856806 polymorphism had significantly higher levels of glucose, low-density lipoprotein cholesterol (LDL-C), ApoB and Lp(a) than non-carriers. In patients with CAD, G allele carriers of rs10865710 polymorphism had significantly higher levels of total cholesterol (TC), ApoB and Lp(a) than non-carriers, T allele carriers of rs3856806 polymorphism had significantly higher levels of body mass index, blood pressure, TC, LDL-C and ApoB than non-carriers. Patients with one or two G alleles of rs10865710 polymorphism had significantly higher levels of Gensini scores and more diseased coronary branches than those patients without CAD. The rs3856806 polymorphism was not associated with CAD severity, but it was found to be significantly associated with T2DM/CAD, T allele frequency was significantly higher in T2DM/CAD group than that in T2DM/CAD-free group. Conclusions The rs10865710 and rs3856806 polymorphisms in PPARG are significantly associated with glucose levels in patients with T2DM. The rs10865710 polymorphism is significantly associated with the severity of CAD, which is possibly mediated by hyperlipidaemia and hyperglycaemia.

We investigate the existence of a family nontrivial exterior domain $\tilde{\Omega}\subset \mathbb{R}^{N}$ $\left(N\geq2, N\neq3\right)$, bifurcating from the complement of a ball such that \begin{equation} \Delta u=0\,\, \text{in}\,\,\tilde{\Omega}, \,\, u=u_0,\,\,\partial_\nu u=\gamma H+C_0\,\,\text{on}\,\,\partial\tilde{\Omega},\,\,\,\lim _{r \rightarrow+\infty} u=0\,\, \text{or}\,\,+\infty\nonumber \end{equation} has a positive solution, where the Neumann condition is non-constant with $H$ is mean curvature and $\gamma$, $C_0$ are constants. This result gives a negative answer to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems in the complement of the ball.

We discuss the following mean curvature equation $$\begin{aligned} -a\text {div} \left( \frac{\nabla u}{\sqrt{1-\vert \nabla u\vert ^2}}\right) +b\text {div} \left( \frac{\nabla u}{\sqrt{1+\vert \nabla u\vert ^2}}\right) =\lambda f(x,u) \end{aligned}$$ with 0-Dirichlet boundary condition on a bounded domain. We obtain the global gradient estimate of classical solutions. Furthermore, we investigate the existence and uniqueness of classical solution. By variational method, we also establish the multiplicity of strong solutions. Moreover, according to the behavior of f near 0, we obtain the global structure of positive solutions. Finally, we also investigate the symmetry of positive solutions when $$\Omega $$ is radially symmetric.

With suitable assumptions, uniqueness of nontrivial nonnegative solution is established for the following nonlinear integral equation φ(x)=∫DE(x,y)G(y,φ(y))dy. This result is applied to branching d...

This paper establishes global bifurcation and eigenvalue results for the following complex $ k $-Hessian equation $ \begin{equation} \left\{ \begin{array}{ll} S_k\left(u_{i\overline{j}}\right) = \lambda^k f(-u)\,\, &\text{in}\,\, B,\\ u = 0 &\text{on}\,\, \partial B. \end{array} \right.\nonumber \end{equation} $ The existence/nonexistence, uniqueness and multiplicity of radially symmetric negative solutions are investigated. Moreover, a-priori bound of radially symmetric negative solutions is also obtained.

In this paper, we shall study unilateral global bifurcation phenomenon for the following homogeneous Kirchhoff type problem \begin{equation*} \begin{cases} -\left(\int_0^1 \left\vert u'\right\vert^2\,dx\right)u''=\lambda u^3+h(x,u,\lambda)&\text{in}\,\, (0,1),\\ u(0)=u(1)=0. \end{cases} \end{equation*} As application of bifurcation result, we shall determine the interval of $\lambda$ in which there exist nodal solutions for the following homogeneous Kirchhoff type problem \begin{equation*} \begin{cases} -\left(\int_0^1 \left\vert u'\right\vert^2\,dx\right) u''=\lambda f(x,u)&\text{in}\,\, (0,1),\\ u(0)=u(1)=0, \end{cases} \end{equation*} where $f$ is asymptotically cubic at zero and infinity. To do this, we also establish a complete characterization of the spectrum of a homogeneous nonlocal eigenvalue problem.

We describe a two-dimensional biaxial apparatus that is used to conduct the experimental study of the jamming of granular media. The setup is designed based on the photoelastic imaging technique, which allows us to detect force-bearing contacts among particles, calculate the pressure on each particle according to the mean squared intensity gradient method, and compute contact forces on each particle [T. S. Majmudar and R. P. Behringer, Nature 435, 1079–1082 (2005)]. Particles float in a density-matched solution to avoid basal friction during experiments. We can compress (uniaxially or biaxially) or shear the granular system by an entangled comb geometry by moving the paired boundary walls independently. A novel design for the corner of each pair of perpendicular walls is described, which allows for independent motion. We control the system using a Raspberry Pi with Python code. Three typical experiments are described briefly. Furthermore, more complicated experiment protocols can be implemented to achieve specific granular materials research goals.

We establish the unilateral global bifurcation result for the following nonlinear operator equation $$u = L\left( \lambda \right)u + H\left( {\lambda,u} \right),\,\left( {\lambda,u} \right) \in \mathbb{R}{^m} \times X$$ where m is a positive integer, X is a Banach space, L(·) is a positively homogeneous completely continuous operator and H: ℝm·X → X is completely continuous with H = o (||u||) near u = 0 uniformly on bounded λ sets.

Relationship between polymorphisms in peroxisome proliferator-activated receptor gamma (635 subjects were divided into T2DM, CAD, T2DM complicated with CAD (T2DM/CAD) and control groups according to diagnostic criteria. The rs10865710 and rs3856806 polymorphisms were genotyped, and the severity of T2DM and CAD was evaluated for all subjects.In patients with T2DM, G allele carriers of rs10865710 polymorphism had significantly higher levels of glucose, triglycerides, apolipoprotein B (ApoB) and lipoprotein (a) (Lp(a)) than non-carriers, T allele carriers of rs3856806 polymorphism had significantly higher levels of glucose, low-density lipoprotein cholesterol (LDL-C), ApoB and Lp(a) than non-carriers. In patients with CAD, G allele carriers of rs10865710 polymorphism had significantly higher levels of total cholesterol (TC), ApoB and Lp(a) than non-carriers, T allele carriers of rs3856806 polymorphism had significantly higher levels of body mass index, blood pressure, TC, LDL-C and ApoB than non-carriers. Patients with one or two G alleles of rs10865710 polymorphism had significantly higher levels of Gensini scores and more diseased coronary branches than those patients without CAD. The rs3856806 polymorphism was not associated with CAD severity, but it was found to be significantly associated with T2DM/CAD, T allele frequency was significantly higher in T2DM/CAD group than that in T2DM/CAD-free group.The rs10865710 and rs3856806 polymorphisms in

In this paper we study quasilinear elliptic equations driven by the double phase operator and a right-hand side which has the combined effect of a singular and of a parametric term. Based on the fibering method by using the Nehari manifold we are going to prove the existence of at least two weak solutions for such problems when the parameter is sufficiently small.

In this paper, we consider the following coupled elliptic system \begin{document}$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u+\lambda_1 u = \mu_1 u^3+\beta uv^2-\gamma v &\text{in } \mathbb{R}^N, \\ -\Delta v+\lambda_2 v = \mu_2 v^3+\beta vu^2-\gamma u &\text{in } \mathbb{R}^N, \\ u(x), v(x)\rightarrow 0 \text{ as } \vert x\vert\rightarrow+\infty. \end{array} \right.\nonumber \end{equation} $\end{document} Under symmetric assumptions \begin{document}$ \lambda_1 = \lambda_2, \mu_1 = \mu_2 $\end{document} , we determine the number of \begin{document}$ \gamma $\end{document} -bifurcations for each \begin{document}$ \beta\in(-1, +\infty) $\end{document} , and study the behavior of global \begin{document}$ \gamma $\end{document} -bifurcation branches in \begin{document}$ [-1, 0]\times H_r^1\left( \mathbb{R} ^N\right)\times H_r^1\left( \mathbb{R} ^N\right) $\end{document} . Moreover, several results for \begin{document}$ \gamma = 0 $\end{document} , such as priori bounds, are of independent interests, which are improvements of corresponding theorems in [ 6 ] and [ 35 ].

We use bifurcation and topological methods to investigate the existence/nonexistence and the multiplicity of positive solutions of the following quasilinear Schrödinger equation$$\left\{ {\matrix{ {-\Delta u-\kappa \Delta \left( {u^2} \right)u = \beta u-\lambda \Phi \left( {u^2} \right)u{\mkern 1mu} {\mkern 1mu} } \hfill & {{\rm in}\;\Omega ,} \hfill \cr {u = 0} \hfill & {{\rm on}\;\partial \Omega } \hfill \cr } } \right.$$involving sublinear/linear/superlinear nonlinearities at zero or infinity with/without signum condition. In particular, we study the changes in the structure of positive solution withκas the varying parameter.

By variational method, we obtain various existence and multiplicity results for the following double phase problem { − div ( | ∇ u | p − 2 ∇ u + a ( x ) | ∇ u | q − 2 ∇ u ) = f ( x , u ) in Ω , u = 0 on ∂ Ω . In particular, we find a sign-changing ground state solution. Some properties of double phase operator are also obtained.

We establish a unilateral global bifurcation result for the following problem where Ω is a bounded domain in . Based on this global bifurcation result, we also studied the global structure of one-sign solutions according to different asymptotic behaviors (sublinear/linear/superlinear/jumping) nonlinearity near zero.

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz–Minkowski spacetime. Then, by using Rabinowitz's global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied.

In this paper, we investigate the global regularity of Leray–Hopf weak solutions to the 3D Navier–Stokes equations in terms of pressure. In particular, we establish three Serrin-type regularity criteria in framework of the anisotropic Lebesgue spaces. Our result can be understood as a generalization of the notable works of Y. Zhou (2006) [23] , C. Cao and E. Titi (2008) [5] , and Y. Zhou and M. Pokorný (2010) [24] .

The uniqueness obtained in [1] may not right if f ( s ) / s k is only decreasing for s > 0 . We correct this error by requiring f ( s ) / s k is “strictly” decreasing for s > 0 .

This paper is concerned with the following semilinear elliptic problem { − Δ u = λ m ( x ) f ( u ) in R N , u → 0 as | x | → + ∞ , where λ is a real parameter and m is a weight function which may be sign-changing. For the linear case, i.e., f ( u ) = u , we investigate the spectral structure. For the semilinear case, we study the existence and asymptotic behavior of one-sign and nodal solutions by bifurcation method.

We study the following eigenvalue problem of k-Hessian equation { S k ( D 2 u ) = λ k f ( − u ) in B , u = 0 on ∂ B . Global bifurcation result is established for this problem. As applications of the bifurcation result, we determine the intervals of λ for the existence, nonexistence, uniqueness and multiplicity of radially symmetric k-admissible solutions for this problem.

Using bifurcation analysis we investigate the structure of the set of positive solutions for the coupled nonlinear Schrodinger system \begin{equation*} \begin{cases} -\Delta u_1+ u_1= u_1^3+\beta u_1u_2^2 & \text{in } \mathbb{R}^N,\\ -\Delta u_2+\lambda u_2=\mu u_2^3+\beta u_2u_1^2 &\text{in } \mathbb{R}^N,\\ u_1(x),u_2(x)\rightarrow 0 &\text{as } \vert x\vert\rightarrow+\infty, \end{cases} \end{equation*} where $N=1,2,3$, $\mu$ is a positive constant, $\lambda$ and $\beta$ are positive real parameters. We prove the existence of two two-dimensional continua $\mathcal{S}_1$ and $\mathcal{S}_2$ emanating from the two sets of semi-positive solutions which cover some regions in term of $(\beta,\lambda)\in \mathbb{R}_+^2$. To do this, we establish a multi-parameter unilateral global bifurcation theorem.

We study the existence/nonexistence and multiplicity of spacelike graphs for the following mean curvature equation in a standard static spacetime div (a del u/root 1-a(2)vertical bar del u vertical bar(2)) + g (del u, del a)/root 1-a(2)vertical bar del u vertical bar(2) = lambda NH with 0-Dirichlet boundary condition on the unit ball. According to the behavior of H near 0, we obtain the global structure of one-sign radial spacelike graphs for this problem. Moreover, we also obtain the existence and multiplicity of entire spacelike graphs., National Natural Science Foundation of China (NSFC) 11871129, Xinghai Youqing funds from Dalian University of Technology, Spanish MINECO, European Union (EU) MTM2016-78807-C2-1-P MTM2017-82348-C2-1-P

We study the existence/nonexistence and multiplicity of radial solutions of the following mean curvature problem in Minkowski spacetime $$\begin{aligned} \left\{ \begin{array}{lll} -\text {div}\left( \frac{\nabla u}{\sqrt{1-\vert \nabla u\vert ^2}}\right) = \lambda f(x,u)\,\, &{}\quad \text {in}\,\, \mathbb {R}^N,\\ u\rightarrow 0&{}\quad \text {as}\,\, \vert x\vert \rightarrow +\infty . \end{array} \right. \end{aligned}$$ Moreover, we also obtain the rate of decay of solutions at \(\infty \).

In this paper we develop a non-autonomous reaction-diffusion model with the Robin boundary conditions to describe insect dispersal on an isotropically varying domain. We investigate the stability of the reaction-diffusion model. The stability results of the model describe either insect survival or vanishing.

In this paper, we establish a unilateral global bifurcation result from intervals for a class of semilinear problems on the unit ball. By applying the above result, we obtain the spectrum of a class of half-linear problems on the unit ball. Our results partially give a positive answer to the open problem proposed by Berestycki.

This paper is concerned with the existence and multiplicity of constant sign solutions for the following fully nonlinear equation \begin{document}$ \begin{equation*} \left\{ \begin{array}{l} -\mathcal{M}_\mathcal{C}^{\pm}(D^2u) = \mu f(u) \ \ \ \ \text{in} \ \ \Omega,\\ u = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial\Omega, \end{array} \right. \end{equation*} $\end{document} where \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} is a bounded regular domain with \begin{document}$ N\geq3 $\end{document}, \begin{document}$ \mathcal{M}_\mathcal{C}^{\pm} $\end{document} are general Hamilton-Jacobi-Bellman operators, \begin{document}$ \mu $\end{document} is a real parameter. By using bifurcation theory, we determine the range of parameter \begin{document}$ \mu $\end{document} of the above problem which has one or multiple constant sign solutions according to the behaviors of \begin{document}$ f $\end{document} at \begin{document}$ 0 $\end{document} and \begin{document}$ \infty $\end{document}, and whether \begin{document}$ f $\end{document} satisfies the signum condition \begin{document}$ f(s)s>0 $\end{document} for \begin{document}$ s\neq0 $\end{document}.

We study the following Monge-Ampere equation { ( det ⁡ ( D 2 u ) ) 1 N = λ f ( − u ) in Ω , u = 0 on ∂ Ω by bifurcation technique. We establish some results about the existence, nonexistence, uniqueness and multiplicity of convex solutions for this problem. Our results generalize and improve many important known results from previous literature.

We consider a double phase problem in RN with a q−1-superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Using variational methods based on the topological degree and critical point theory together with the Nehari manifold method and deformation lemma, we find three nontrivial solutions: among them, the first one is the positive ground state solution, the second one is the negative ground state solution, and the third one is the least energy sign-changing solution, which changes sign only once. We also discuss the existence of infinitely many solutions for even energy functional and that of radial solutions.

We study the global structure of positive solutions of the following mean curvature equation in the Minkowski space \[ -\div \bigg (\frac {\nabla u}{\sqrt {1-\vert \nabla u\vert ^2}}\bigg )= \lambda f(x,u), \] on an annular domain with the Robin boundary condition. According to the behavior of $f$ near $0$, we obtain the existence and multiplicity of positive solutions for this problem.

In this paper, we investigate the structure of the nontrivial solution set for the following nonlinear operator equation u = L ( λ ) u + H ( λ , u ) , ( λ , u ) ∈ R m × X , where m is a positive integer, X is a Banach space, L ( ⋅ ) : X → X is a (positively) homogeneous compact operator and H : R m × X → X is compact with H = o ( ‖ u ‖ ) near u = 0 uniformly on bounded λ sets. We obtain some results involving (unilateral) global bifurcation. Two examples of p-Laplacian problem with jumping nonlinearity and nonlocal boundary value problem are given to demonstrate how the theory can be applied.

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz-Minkowski spacetime. Then, by using Rabinowitz’s global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied

In this paper, we establish a unilateral global bifurcation result from interval for a class of $p$-Laplacian problems. By applying above result, we study the spectrum of a class of half-quasilinear problems. Moreover, we also investigate the existence of nodal solutions for a class of half-quasilinear eigenvalue problems.

In recent years, online social networks such as Twitter, have become a major source of information exchange and research on information diffusion in social networks has been accelerated. Partial differential equations are proposed to characterize temporal and spatial patterns of information diffusion over online social networks. The new modeling approach presents a new analytic framework towards quantifying information diffusion through the interplay of structural and topical influences. In this paper we develop a non-autonomous diffusive logistic model with indefinite weight and the Robin boundary condition to describe information diffusion in online social networks. It is validated with a real dataset from an online social network, Digg.com. The simulation shows that the logistic model with the Robin boundary condition is able to more accurately predict the density of influenced users. We study the bifurcation, stability of the diffusive logistic model with heterogeneity in distance. The bifurcation and stability results of the model information describe either information spreading or vanishing in online social networks.

We establish the existence of nontrivial nonnegative solution for the following 0-Dirichlet problem with mean curvature operator in the Minkowski space $$\begin{aligned} \left\{ \begin{array}{lll} -\text {div}\left( \frac{\nabla u}{\sqrt{1-\vert \nabla u\vert ^2}}\right) = \lambda f(x,u)\,\, &{}\text {in}\,\, \Omega ,\\ u=0&{}\text {on}\,\, \partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a general bounded domain of $$\mathbb {R}^N$$ . By bifurcation and topological methods, we determine the interval of parameter $$\lambda $$ in which the above problem has nontrivial nonnegative solution according to sublinear or linear nonlinearity at zero.

This paper is devoted to investigate the existence and multiplicity of radial nodal solutions for the following Dirichlet problem with mean curvature operator in Minkowski space \begin{eqnarray} \begin{cases} -\text{div} \Big (\frac{\nabla v}{\sqrt{1-\vert \nabla v\vert^2}} \Big ) = \lambda f(\vert x\vert,v)\,\, &\text{in}\,\, B_R(0),\\ v=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\, \partial B_R(0). \end{cases} \nonumber \end{eqnarray} By bifurcation approach, we determine the interval of parameter $\lambda$ in which the above problem has two or four radial nodal solutions which have exactly $n-1$ simple zeros in $(0,R)$ according to linear/sublinear/ superlinear nonlinearity at zero. The asymptotic behaviors of radial nodal solutions as $\lambda \to +\infty$ and $n \to +\infty$ are also studied.

In this paper, we establish the existence and non-existence of positive solutions for p-Kirchhoff type problems with a parameter on without assuming the usual compactness conditions. We show that the p-Kirchhoff type problems have at least one positive solution when the parameter is small, while the p-Kirchhoff type problems have no positive solutions when the parameter is large. Our argument is based on variational methods, monotonicity methods, cut-off functional techniques, and a priori estimates techniques. Copyright © 2014 John Wiley & Sons, Ltd.

We focus on the structure of the solution set for the nonlinear equation [Formula: see text] where [Formula: see text] and [Formula: see text] are continuous operators. Under certain hypotheses on [Formula: see text] and [Formula: see text], unilateral global bifurcations for eigenvalue problems are presented. Some applications are illustrated for nonlinear ordinary and partial differential equations. In particular, the existence and multiplicity of one-sign solutions for Monge–Ampère equation is discussed.

This paper studies the global bifurcation phenomenon for the following homogeneous nonlocal eigenvalue problem [Formula: see text] Under some natural hypotheses on [Formula: see text] and [Formula: see text], we show that [Formula: see text] is a bifurcation point of the nontrivial solution set of the above problem. As application of the above result, we determine the interval of [Formula: see text], in which there exist positive solutions for the following Kirchhoff type problem [Formula: see text] where [Formula: see text] is asymptotically 3-linear at zero and infinity. Our results provide a positive answer to an open problem. Moreover, we also study the spectral structure for a homogeneous nonlocal eigenvalue problem.