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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.

For a connected graph F of order at least three, we say that a graph G is F-free if G does not contain an induced subgraph isomorphic to F. We call a connected graph G a split graph if the vertex set of G can be partitioned into a clique and an independent set. Motivated by a hamiltonicity result due to Renjith and Sadagopan (arXiv:1610.00855v3), involving K1,3-free split graphs, we study the hamiltonian properties of K1,r-free split graphs. In our first main result, we show that a K1,3-free split graph G is pancyclic if and only if G is 2-connected, which improves a result of Renjith and Sadagopan. Also, we prove that a K1,4-free split graph G is hamiltonian if G is 3-connected. Further, we give a conjecture as following: Let G be a K1,r+1-free split graph with at least three vertices. If G is r-connected, then G is hamiltonian.

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 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] .

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.

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 study global bifurcation phenomena for the following Kirchhoff type problem { - M ( ? ? | ? u ( x ) | 2 d x ) Δ u = λ f ( x , u ) in ? , u = 0 on ? ? , where M is a continuous function. Under some natural hypotheses, we show that ( λ 1 ( a ) M ( 0 ) , 0 ) is a bifurcation point and there is a global continuum C emanating from ( λ 1 ( a ) M ( 0 ) , 0 ) , where λ 1 ( a ) denotes the first eigenvalue of the above problem with f ( x , s ) = a ( x ) s . As an application of the above result, we study the existence of positive solution for this problem with asymptotically linear nonlinearity.

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.

In this paper, we shall establish the unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity of a class of nonlinear Sturm–Liouville problems with nondifferentiable nonlinearity, respectively. As applications of the above results, we shall determine the interval of r, in which there exist nodal solutions for the following problem { u ″ ( t ) + r a ( t ) f ( u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where a ( t ) is a positive continuous function on [ 0 , 1 ] , f ∈ C ( R , R ) but is not necessarily differentiable at the origin and infinity. Moreover, as a special case of the above problem, we shall prove more details about the existence of nodal solutions for a class of half-linear eigenvalue problems.

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 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 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 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.

In this paper we establish a nonsmooth version of Fountain theorem by adopting the framework of nonsmooth analysis theory, which is a generalization of Theorem 3.6 of Willem (1996) [2] . Then we present an application of this theorem to a Dirichlet-type differential inclusion problem.

This paper deals with an eigenvalue problem for hemivariational inequalities on domains of the type ω × R ( ω is a bounded open subset of R N − 1 , N ≥ 2 ) and it involves concave–convex nonlinearities. Under suitable conditions on the nonlinearities, two nontrivial solutions are obtained which belong to a special closed convex cone of W 0 1 , p ( ω × R ) whenever the eigenvalues are of certain range. Our approach is variational based on the theories of non-smooth analysis. Our results are a generalization of the case of Laplacian from A. Kristaly, et al. to the case of p -Laplacian.

In this paper, we consider a Dirichlet problem involving the p ( x ) -Laplacian of the type { − 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.

In this paper we consider differential inclusion problem involving the p ( x ) -Laplacian of the type { − div ( | ∇ u | p ( x ) − 2 ∇ u ) ∈ λ ∂ F ( x , u ) in Ω , u = 0 on ∂ Ω . Applying a version of the non-smooth three-critical-points theorem we obtain the existence of three solutions of the problem in W 0 1 , p ( x ) ( Ω ) .

This paper deals with a perturbed p(x)-Laplacian equation involving oscillatory terms: -div|@?u|^p^(^x^)^-^[email protected]?u+|u|^p^(^x^)^-^2u=Q(x)f(u)[email protected](u)[email protected]?R^N,u>=0,u(x)->0as|x|->~. By using variational methods and the non-smooth version symmetric criticality principle, we establish (a) the unperturbed problem P"0 has infinitely many distinct solutions; (b) the number of distinct solutions for P"@e becomes greater and greater whenever @e is smaller and smaller. Our results are a generalization of the case of Laplacian from Kristaly to the case of p(x)-Laplacian.

This paper is concerned with the existence of infinitely many positive solutions to a class of p(x)-Kirchhoff-type problem. By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, we establish the existence of infinitely many distinct positive solutions whose W1,p(x)(Ω)-norms and L∞-norms tend to zero under suitable hypotheses about nonlinearity.

This paper is concerned with the existence and multiplicity of solutions to a class of p(x)-Kirchhoff-type problem with Dirichlet boundary data. By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence and multiplicity of solutions for the problem.

In this paper we consider a differential inclusion system involving the ( p ( x ) , q ( x ) ) -Laplacian in R N . Applying a version of the nonsmooth three critical points theorem we obtain the existence of three pairs of radially symmetric solutions of the problem in W 1 , p ( x ) ( R N ) × W 1 , q ( x ) ( R N ) .

In this paper we consider the differential inclusion problem in R N involving the p ( x ) -Laplacian of the type { − div ( | ∇ u | p ( x ) − 2 ∇ u ) + | u | p ( x ) − 2 u ∈ ∂ F ( x , u ) in R N , u ∈ W 1 , p ( x ) ( R N ) . We prove the existence of infinitely many radially symmetric solutions of the problem under suitable hypotheses by applying a nonsmooth variational principle due to Marano and Motreanu and the theory of the variable exponent Sobolev spaces.

This paper deals with a p ( x ) -Laplacian equation in R N . By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, we establish the existence of infinitely many distinct homoclinic radially symmetric solutions whose W 1 , p ( x ) ( R N ) -norms tend to zero (to infinity, respectively) under weaker hypotheses about nonlinearity at zero (at infinity, respectively).

This paper deals with a hemivariational inequality involving the p ( x ) -Laplacian. By using nonsmooth variational methods and the theory of the variable exponent Sobolev spaces, we establish the existence of infinitely many distinct non-negative solutions of the problem in both cases—one at the origin and the other at infinity under suitable hypotheses.

In this paper we consider Neumann-type differential inclusion problems involving the p ( x ) -Laplacian of the type { − div ( | ∇ u | p ( x ) − 2 ∇ u ) + | u | p ( x ) − 2 u ∈ λ ∂ F ( x , u ) in Ω , ∂ u ∂ γ = 0 on ∂ Ω . Applying a version of the non-smooth three-critical-points theorem we obtain the existence of three solutions of the problem in W 0 1 , p ( x ) ( Ω ) .

In this paper we consider a differential inclusion problem involving the p ( x ) -Laplacian of the type { − div ( | ∇ u | p ( x ) − 2 ∇ u ) + λ ( x ) | u | p ( x ) − 2 u ∈ ∂ F ( x , u ) + ∂ G ( x , u ) in Ω , ∂ u ∂ γ = 0 on ∂ Ω . We prove the existence of infinitely many solutions of this problem under suitable hypotheses by applying a non-smooth Ricceri-type variational principle and the theory of the variable exponent Sobolev spaces.

A novel ion-pair reversed phase electrospray ionization (IP-RP-ESI) liquid chromatography–mass spectrometry (LC–MS) method has been developed for identification and quantification of Bcl-2 antisense phosphorothioate oligonucleotides G3139 and metabolites in plasma. This method utilized solid phase extraction for desalting and matrix removal and detection by an ion trap mass spectrometer. Resolution was accomplished on a micro C18 column eluted with a mobile phase consisting of hexafluoro-2-propanol and triethylamine in methanol at 50 °C. Five G3139 metabolites were identified in plasma and urine from treated patients and rats. A cassette HPLC–MS/MS quantification method for G3139 and three metabolites was developed and validated with a limit of quantification (LOQ) of 17.6 nM in human and rat plasma with acceptable precision and accuracy. Plasma pharmacokinetics of G3139 and metabolites in these species were described.

In this note, using the moving plane method, we establish the properties of symmetric and monotonic solutions of a nonlocal diffusive logistic equation under the assumption of ( f 1 ). Thus, we give a positive answer for the conjecture by Sun, Shi and Wang.