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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...

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 study the following displacement problem in nonlinear elastostatics − I + ∇ u ∇ 2 u = λ f ( u ) in Ω , c u = 0 on ∂ Ω . We obtain some global and local bifurcation results, which are related to an open problem.

Belief Propagation (BP), a distributed, message-passing algorithm, has been widely used in different disciplines including information theory, artificial intelligence, statistics and optimization problems in graphical models such as Bayesian networks and Markov random fields. Despite BP algorithm has a great success in many application fields and many progress about BP algorithm has been made, the rigorous analysis about the correctness and convergence of BP algorithm are known in only a few cases for arbitrary graph. In this paper, we will investigate the correctness and convergence of BP algorithm for determining the optimal solutions of the Chinese postman problem in both undirected and directed graphs. As a main result, we prove that BP algorithm converges to the optimal solution of the undirected Chinese postman problem within O(n) iterations where n represents the number of vertices, provided that the optimal solution is unique. For the directed case, we consider the directed Chinese postman problem with capacity and show that BP algorithm also converges to its optimal solution after O(n) iterations, as long as its corresponding linear programming relaxation has the unique optimal solution.

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 ].

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 global structure of admissible solutions for the following k -Hessian equation S k D 2 u = λ k f ( − u ) . By bifurcation and topological methods, we determine the interval of λ for the existence of admissible solution for this problem.

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] .

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.

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.

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.

Consider the boundary value problem u ′ ′ + u = α u − + p ( t ) , u ( 0 ) = 0 = u ( π ) , α ≤ 0 . We show that a necessary and sufficient condition for the problem to be solvable is that ∫ 0 π p ( t ) sin t d t ≥ 0 . We thus answer positively a counter part of a long-standing open problem posed by S. Fucik. A more general problem is also considered and the sufficient condition for the existence of solution is obtained.

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.

In this paper, we introduce the notion of a strict lower/upper solution to nonlinear Sturm-Liouville boundary value problems. Based on the maximum principles, we establish a result of Leray-Schauder degree on the ordered intervals induced by the pairs of strict lower and upper solutions. Applying the result and the fixed point index theory in cones, we obtain the global existence results of positive solutions for sub-superlinear Sturm-Liouville problems.

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.

In this paper, we study unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity for nonlinear Sturm--Liouville problems of the form \begin{equation} \left\{ \begin{array}{l} -\left(pu'\right)'+qu=\lambda au+af\left(x,u,u',\lambda\right)+g\left(x,u,u',\lambda\right)\,\,\text{for}\,\, x\in(0,1),\\ b_0u(0)+c_0u'(0)=0,\\ b_1u(1)+c_1u'(1)=0, \end{array} \right.\nonumber \end{equation} where $a\in C([0, 1], [0,+\infty))$ and $a(x)\not\equiv 0$ on any subinterval of $[0, 1]$, $f,g\in C([0,1]\times\mathbb{R}^3,\mathbb{R})$. Suppose that $f$ and $g$ satisfy \begin{equation} \vert f(x,\xi,\eta,\lambda)\vert\leq M_0\vert \xi\vert+M_1\vert \eta\vert,\,\, \forall x\in [0,1]\,\,\text{and}\,\,\lambda \in\mathbb{R}, \nonumber \end{equation} \begin{equation} g(x,\xi,\eta,\lambda)=o(\vert \xi\vert+\vert \eta\vert),\,\, \text{uniformly in}\,\, x\in [0,1]\,\,\text{and}\,\,\lambda \in \Lambda,\nonumber \end{equation} as either $\vert \xi\vert+\vert \eta\vert\rightarrow 0$ or $\vert \xi\vert+\vert \eta\vert\rightarrow +\infty$, for some constants $M_0$, $M_1$, and any bounded interval $\Lambda$.

In this paper, we shall establish the unilateral global bifurcation result from an interval for a class of high-dimensional p -Laplacian problems. As an application of the above result, we shall prove the existence of the principal half-eigenvalues for a class of half-quasilinear eigenvalue problems. Our result partially answers the open problem proposed by Berestycki. Moreover, we shall investigate the existence of one-sign solutions for a class of high-dimensional p -Laplacian problems with non- p − 1-homogeneous nonlinearities.

We are concerned with determining values of γ, for which there exist radial nodal solutions of the boundary-value problem where B is the unit open ball in ℝ N with N ≥ 1. The proofs of our main results are based upon bifurcation techniques. In particular, we establish the global bifurcation theory for a second-order weighted p-Laplacian eigenvalue problem in B.

In this paper, we shall establish a Dancer-type unilateral global bifurcation result for a class of quasilinear elliptic problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that is a bifurcation point of the above problems and there are two distinct unbounded continua, and , consisting of the bifurcation branch from , where is the th positive or negative eigenvalue of the linear problem corresponding to the above problems, . As applications of the above unilateral global bifurcation result, we study the existence of nodal solutions for a class of quasilinear elliptic problems with sign-changing weight. Moreover, based on the bifurcation result of Drabek and Huang (1997) Dai G, Ma R. Unilateral global bifurcation phenomena and nodal solutions for -Laplacian. J. Differ. Equ. 2012;252:2448–2468., we study the existence of one-sign solutions for a class of high-dimensional quasilinear elliptic problems with sign-changing weight.

In this paper, we shall establish unilateral global bifurcation result for a class of fourth-order eigenvalue problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that @m"k^@n,0 is a bifurcation point of the above problems and there are two distinct unbounded continua, C"k^@n^+ and C"k^@n^-, consisting of the bifurcation branch C"k^@n from @m"k^@n,0, where @m"k^@n is the kth positive or negative eigenvalue of the linear problem corresponding to the above problems, @[email protected]?{+,-}. As the applications of the above result, we study the existence of nodal solutions for a class of fourth-order eigenvalue problems with sign-changing weight. Moreover, we also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight.

In this article, we consider a nonlocal problem involving the p(x)-Laplacian of the type Applying a three critical points theorem from Bonanno [A critical points theorem and nonlinear differential problems, J. Global Optim. 28 (2004), pp. 249–258], we obtain the existence of two intervals of positive real parameters λ for which the above problem admits three weak solutions in , whose norms are uniformly bounded with respect to λ belonging to one of the two open intervals. In particular, we also prove some properties of p(x)-Kirchhoff–Laplace operator.

In this paper, we shall establish a unilateral global bifurcation theorem from infinity for a class of p -Laplacian problems. As an application of the above result, we shall study the global behavior of the components of nodal solutions of the following problem { ( φ p ( u ′ ) ) ′ + λ a ( t ) f ( u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where φ p ( s ) = | s | p − 2 s , a ∈ C ( [ 0 , 1 ] , [ 0 , + ∞ ) ) with a ≢ 0 on any subinterval of [ 0 , 1 ] ; f : R → R is continuous, and there exist two constants s 2 0 s 1 such that f ( s 2 ) = f ( s 1 ) = f ( 0 ) = 0 , f ( s ) s > 0 for s ∈ R ∖ { s 2 , 0 , s 1 } . Moreover, we give the intervals for the parameter λ which ensure the existence of multiple nodal solutions for the problem if f 0 ∈ ( 0 , + ∞ ) and f ∞ ∈ ( 0 , + ∞ ) , where f ( s ) / φ p ( s ) approaches f 0 and f ∞ as s approaches 0 and ∞ , respectively. We use topological methods and nonlinear analysis techniques to prove our main results.

In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight. By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic $p$-Laplacian with the sign-changing weight. In particular, we show that there exist two simple, isolated, principal eigenvalues $\lambda_0^+$ and $\lambda_0^-$. Furthermore, under some natural hypotheses on perturbation function, we show that $(\lambda_0^\nu,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $C_\nu^{+}$ and $C_\nu^{-}$, consisting of the continuum $C_\nu$ emanating from $(\lambda_0^\nu, 0)$, where $\nu\in\{+,-\}$. As an application of the above result, we study the existence of one-sign solutions for a class of quasilinear periodic boundary problems with the sign-changing weight. Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter $\lambda$ are also studied.

In this work, we study the existence of nodal solutions for the following problem: { ( φ p ( u ′ ) ) ′ + λ a ( t ) f ( u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where φ p ( s ) = | s | p − 2 s , a ∈ C ( [ 0 , 1 ] , [ 0 , + ∞ ) ) with a ≢ 0 on any subinterval of [ 0 , 1 ] and f : R → R is continuous with f ( s ) s > 0 for s ≠ 0 . We give the intervals for the parameter λ which ensure the existence of single or multiple nodal solutions for the problem if f 0 ∉ ( 0 , + ∞ ) or f ∞ ∉ ( 0 , + ∞ ) , where f ( s ) / φ p ( s ) approaches f 0 and f ∞ as s approaches 0 and ∞ , respectively. We use bifurcation techniques to prove our main results.

In this paper we correct a gap of Whyburn type topological lemma and establish two superior limit theorems. As the applications of our Whyburn type topological theorems, we study the following Monge–Ampere equation $$\begin{aligned} \left\{ \begin{array}{lll} \det \left( D^2u\right) =\lambda ^N a(x)f(-u)\,\, &{}\quad \text {in}\,\, \Omega ,\\ u=0~~~~~~~\,\,&{}\quad \text {on}\,\, \partial \Omega . \end{array} \right. \end{aligned}$$ We establish global bifurcation results for the problem. We find intervals of \(\lambda \) for the existence, multiplicity and nonexistence of strictly convex solutions for this problem.

Using bifurcation method, we investigate the existence, nonexistence and multiplicity of positive solutions for the following Dirichlet problem involving mean curvature operator in Minkowski space $$\begin{aligned} \left\{ \begin{array}{lll} -\text {div}\left( \frac{\nabla v}{\sqrt{1-\vert \nabla v\vert ^2}}\right) = \lambda f(\vert x\vert ,v) &{}\quad \text {in}\,\, B_R(0),\\ v=0&{}\quad \text {on}\,\, \partial B_R(0). \end{array} \right. \end{aligned}$$ We managed to determine the intervals of the parameter \(\lambda \) in which the above problem has zero, one or two positive radial solutions corresponding to sublinear, linear, and superlinear nonlinearities f at zero respectively. We also studied the asymptotic behaviors of positive radial solutions as \(\lambda \rightarrow +\infty \).

In this paper, we study a class of nonlocal semilinear elliptic problems with inhomogeneous strong Allee effect. By means of variational approach, we prove that the problem has at least two positive solutions for large $\lambda$ under suitable hypotheses about nonlinearity. We also prove some nonexistence results. In particular, we give a positive answer to the conjecture of Liu-Wang-Shi.

In this paper, we establish a Dancer-type unilateral global bifurcation result for one-dimensional p-Laplacian problem { − ( φ p ( u ′ ) ) ′ = μ m ( t ) φ p ( u ) + g ( t , u ; μ ) , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . Under some natural hypotheses on the perturbation function g : ( 0 , 1 ) × R × R → R , we show that μ k ( p ) is a bifurcation point of the above problem and there are two distinct unbounded continua, C k + and C k − , consisting of the bifurcation branch C k from ( μ k ( p ) , 0 ), where μ k ( p ) is the k-th eigenvalue of the linear problem corresponding to the above problem. As the applications of the above result, we study the existence of nodal solutions for the following problem { ( φ p ( u ′ ) ) ′ + f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . Moreover, based on the bifurcation result of Girg and Takac (2008) [13] , we prove that there exist at least a positive solution and a negative one for the following problem { − div ( φ p ( ∇ u ) ) = f ( x , u ) , in Ω , u = 0 , on ∂ Ω .

In this paper we study the existence of periodic solutions with prescribed wavelength for two classes of nonlocal fourth-order nonautonomous differential equations. Existence of nontrivial solutions for the first equation is proved using Clark’s theorem. Existence of nontrivial solutions for the second equation is proved using the symmetric mountain-pass theorem.

This paper is concerned with the existence and multiplicity of solutions to a class of p ( x ) -Kirchhoff type problem with Neumann boundary data of the following form { − M ( ∫ Ω 1 p ( x ) ( | ∇ u | p ( x ) + | u | p ( x ) ) d x ) ( div ( | ∇ u | p ( x ) − 2 ∇ u ) − | u | p ( x ) − 2 u ) = f ( x , u ) in Ω , ∂ u ∂ υ = 0 on ∂ Ω . By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on f and M , we obtain a number of results on the existence and multiplicity of solutions for the problem. In particular, we also obtain some results which can be considered as extensions of the classical result named “combined effects of concave and convex nonlinearities”. Moreover, the positive solutions and the regularity of weak solutions of the problem are considered.

In this paper, we consider the Dirichlet problem involving the p ( x ) -Kirchhoff-type { − ( a + b ∫ Ω 1 p ( x ) | ∇ u | p ( x ) d x ) div ( | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) in Ω , u = 0 on ∂ Ω . We prove the existence of infinitely many non-negative solutions of the problem by applying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.