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AbstractThe aim of this paper is to establish an Ambrosetti-Prodi type result for the problemwhere is a parameter, is continuous and satisfies is continuous. with . [ABSTRACT FROM AUTHOR]

We are concerned with semilinear biharmonic equations of the form where B denotes the unit ball in ℝN, N ≥ 1, λ > 0 is a parameter, ∂/∂ν is the outward normal derivative, f : ℝ → (0, +∞) is a continuous function that has super-linear growth at infinity. We use bifurcation theory, combined with an approximation scheme to establish the existence of an unbounded branch of positive radial solutions, which is bounded in positive λ–direction. If in addition, f satisfies certain subcritical condition, we show that the branch must bifurcate from infinity at λ = 0. [ABSTRACT FROM AUTHOR]

We show the existence and multiplicity of solutions for the fourth-order periodic boundary value problem where is a parameter, , and is an -Carathéodory function. Moreover, is sublinear at and nondecreasing with respect to the second variable. We obtain that if is sufficiently close to from the left or right, then the problem has at least one or two solutions, respectively. The proof of main results is based on bifurcation theory and the method of lower and upper solutions. [ABSTRACT FROM AUTHOR]

In this paper, we clarify the connection of the Fokas–Lenells (FL) equation to the Kadomtsev–Petviashvili (KP)–Toda hierarchy by using a set of bilinear equations as a bridge and confirm multidark soliton solution to the FL equation previously given by Matsuno (J. Phys. A 2012 45 (475202). We also show that the set of bilinear equations in the KP–Toda hierarchy can be generated from a single discrete KP equation via Miwa transformation. Based on this finding, we further deduce the multibreather and general rogue wave solutions to the FL equation. The dynamical behaviors and patterns for both the breather and rogue wave solutions are illustrated and analyzed. [ABSTRACT FROM AUTHOR]

We show the existence of nodal solutions of the second-order nonlinear boundary value problem - u ″ x = λ g u x + p x , u x , u ′ x , x ∈ 0 , 1 , u 0 = u 1 = 0 , (P) where λ > 0 is a parameter, p : [0, 1]×ℝ2 → ℝ and g : ℝ →ℝ are continuous functions, and g(0) = 0. For a nonnegative integer k, we say that a solution is nodal if it has only simple zeros in (0, 1) and has exactly k-1 such zeros. Under some suitable conditions, we obtain that there exists λ∗ > 0 (or λ∗ > 0) such that for fixed k ∈ {1, 2,...}, problem (P) has at least one nodal solution for λ ∈ (k2π2/g∞, λ∗) (or λ ∈ (λ∗, k2π2/g∞)), where g∞ = lim|s|→∞g(s)/s. The proof of our main results relies on the bifurcation technique. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with elliptic problems { − Δ u = f (u) + g (| x | , u , x | x | ⋅ ∇ u) , x ∈ Ω , u | ∂ Ω = 0 , where Ω = { x ∈ R N : R 1 < | x | < R 2 } is an annular domain, N > 2 , 0 < R 1 < R 2 < ∞ , 4 (R 2 − R 1) (N − 1) ≤ R 1 . f : R → R is continuous and satisfies lim | ξ | → ∞ f (ξ) ξ = ∞ . g : [ R 1 , R 2 ] × R 2 → R is continuous, | g (r , ξ 0 , ξ 1) | ≤ C + β | ξ 0 | for some C > 0 , 0 < β < 1 4 (R 2 − R 1) 2 . We obtain infinitely many radial solutions with prescribed nodal properties using bifurcation techniques. [ABSTRACT FROM AUTHOR]

We are concerned with the Neumann problem with Minkowski-curvature operator (P) { − div (∇ u (x) 1 − | ∇ u (x) | 2 ) = g (u (x)) in Ω , u (x) > 0 in Ω , u → 0 as | x | → ∞ , ∂ u ∂ ν = − σ on ∂ Ω , where Ω is an exterior domain in R N and ∂ ∂ ν denotes the normal interior derivative on ∂ Ω. We have shown the existence of radial positive solutions of (P) when g satisfies some suitable assumptions. The proof of our main result is based upon the shooting method. [ABSTRACT FROM AUTHOR]

We are concerned with the global structure of the radial positive solution set of the radially symmetric prescribed curvature problem with the Neumann boundary condition in some Friedmann–Lemaître–Robertson–Walker spacetimes. The proof of our main result is based on bifurcation techniques. [ABSTRACT FROM AUTHOR]

We investigate the following second-order problem - u ′ ′ (t) + a (t) u (t) = λ b (t) f (u (t)) , t ∈ R , (P) lim | t | → ∞ u (t) = 0 , where λ > 0 is a parameter, a ∈ C (R , (0 , ∞)) , b ∈ C (R , [ 0 , ∞)) such that lim | t | → ∞ b (t) a (t) = 0 , f : R → R is a continuous function with s f (s) > 0 for s ≠ 0 . For the linear case, i.e., f (u) = u , we investigate the existence of principal eigenvalue of (P). For the nonlinear case, depending on the behavior of f near 0 and ∞ , we obtain asymptotic behavior and the existence of homoclinic solutions of (P). The proof of our main results is based upon bifurcation technique. [ABSTRACT FROM AUTHOR]

[Display omitted] • Seventy Flavors Pearl Pill (SFPP) could ameliorate the severity of stroke rats. • SFPP could protect nerve cells against ischemia, hypoxia and hyperoxia injury in vitro. • SFPP exerts neuroprotective effects by regulating OXPHOS pathway. Ischemic stroke has high mortality and morbidity rates and is the second leading cause of death in the world, but there is no definitive medicine. Seventy Flavors Pearl Pill (SFPP) is a classic formula in Tibetan Medicine. Clinical practice has shown the attenuation effect of SFPP on blood pressure disorders, strokes and their sequelae and other neurological symptoms, but its mechanism remains to be elucidated. In this study, we established three animal models in vivo and three cell models to evaluate the anti-hypoxia, anti-ischemia, and reperfusion injury prevention effects of SFPP. Quantitative proteomics revealed that oxidative phosphorylation (OXPHOS) is essential for SFPP's efficacy. Then, cysteine-activity based protein profiling technology, which reflects redox stress at the proteome level, was employed to illustrate that SFPP brought functional differences of critical proteins in OXPHOS. In addition, quantitative metabolomics revealed that SFPP affects whole energy metabolism with OXPHOS as the core. Finally, we performed a compositional identification of SFPP to initially explore the components of potential interventions in OXPHOS. These results provide new perspectives and tools to explore the mechanism of herbal medicine. The study suggests that OXPHOS could be a potential target for further research and intervention of ischemic stroke treatment. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with one of the generalized short wave equations proposed by Hone et al (2018 Lett. Math. Phys 108 927). We show that the derivative form of this equation can be viewed as a short wave limit of the Novikov (sw-Novikov) equation. Furthermore, this generalized short wave equation and its derivative form are found to be connected to period 3 reduction of two-dimensional CKP(BKP)-Toda hierarchy, same as the short wave limit of the Depasperisâ€"Procesi (sw-DP) equation. We propose a two-component short wave equation which contain the sw-Novikov equation and sw-DP equation as two special cases. As a main result, we construct two types of integrable semi-discretizations via Hirota’s bilinear method and provide multi-soliton solution to the semi-discrete sw-Novikov equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear eigenvalue problems u ′ ′ ′ ′ = λ h (t) f (u) , 0 < t < 1 , u (0) = u (1) = u ′ (0) = u ′ (1) = 0 , where h ∈ C ([ 0 , 1 ] , (0 , ∞)) ; f ∈ C (R , R) and s f (s) > 0 for s ≠ 0 , and f 0 = f ∞ = 0 , f 0 = lim | s | → 0 f (s) / s , f ∞ = lim | s | → ∞ f (s) / s . We investigate the global structure of one-sign solutions by using bifurcation techniques. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global structure of positive solutions of where is a parameter, , and for . We show that the problem has three positive solutions suggesting suitable conditions on the nonlinearity. Furthermore, we also establish the existence of infinitely many positive solutions. The proof is based on the bifurcation method. [ABSTRACT FROM AUTHOR]

Sijunzi decoction (SJZD), a traditional Chinese medicine formula, is commonly used in clinical practice for the treatment of gastric precancerous lesions (GPL). However, the mechanism of gastric protection is not fully understood. The purpose of this study was to systematically evaluate the efficacy of SJZD in blocking the development of GPL and to reveal the underlying mechanism. First, we established a rat model of GPL, which was induced by N-methyl-N′-nitro-N-nitrosoguanidine (MNNG) combined with an irregular diet and 40% ethanol. The efficacy of SJZD was evaluated based on pathological sections and serum biochemical indices. Then, the pharmacodynamic mechanism of SJZD was revealed by quantitative proteomics based on stable isotope dimethyl labeling. At the same time, the pharmacodynamic mechanism was verified by quantitative metabolomics. In addition, the anti-gastritis effect of SJZD was confirmed by a serum pharmacology method in a cell model, and the functional mechanism was further verified. We demonstrated that SJZD could block the development of GPL in the animal model. Proteomics and metabolomics revealed that SJZD blocks GPL development by regulating oxidative phosphorylation (OXPHOS). In addition, the serum pharmacology results showed that SJZD-containing serum (SJZD-CS) could inhibit apoptosis in MNNG-induced GES-1 cells. OXPHOS inhibitors could significantly reduce the protective effect of SJZD-CS. SJZD effectively ameliorates GPL, and proteomics and metabolomics revealed that its protective effects are closely related to OXPHOS. [Display omitted] [ABSTRACT FROM AUTHOR]

A single‐source precursor route is presented to prepare small‐sized Ni particles (∼5 nm) uniformly embedded into SiO2 sheet (Ni/SiO2). The synthetic process involved the coordination of APTES (3‐aminopropyltriethoxysilane) with Ni2+ to form Ni−Si precursor in high yield, followed by pyrolysis under NH3 atmosphere. During the heating, the Si component in the Ni−Si precursor was transformed into SiO2, which could restrict the in‐situ generated Ni from aggregation, resulting in the formation of the small‐sized Ni particles. Small size is conducive to promoting the catalytic performance, and meanwhile SiO2 protection plays a vital role in improving stability. During aromatic nitro reduction, Ni/SiO2 shows excellent catalytic performance, which can be separated conveniently by an external magnet after use. The route is indicative of designing other catalysts protected by SiO2 due to the versatile coordination ability of APTES with other metal ions (Fe, Co, Cu, etc). [ABSTRACT FROM AUTHOR]

We study the local and global bifurcation of nonnegative nonconstant solutions of a discrete general Brusselator model. We generalize the linear u in the standard Brusselator model to the nonlinear f u . Assume that f ∈ C 0 , ∞ , 0 , ∞ is a strictly increasing function, and f ′ f − 1 a ∈ 0 , ∞ . Taking b as the bifurcation parameter, we obtain that the solution set of the problem constitutes a constant solution curve and a nonconstant solution curve in a small neighborhood of the bifurcation point b 0 j , f − 1 a , a b / f − 1 a 2 . Moreover, via the Rabinowitz bifurcation theorem, we obtain the global structure of the set of nonconstant solutions under the condition that f s / s 2 is nonincreasing in 0 , ∞ . In this process, we also make a priori estimation for the nonnegative nonconstant solutions of the problem. [ABSTRACT FROM AUTHOR]

We show the global structure of the set of positive solutions of a discrete Dirichlet problem involving the p-Laplacian difference operator suggesting suitable conditions on the weight function and nonlinearity. We obtain existence and multiplicity of positive solutions for λ lying in various intervals in R by using the directions of a bifurcation and the Picone-type identity for discrete p-Laplacian operators. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear eigenvalue problem u ′′′′ = λ h (t) f (u) , 0 < t < 1 , u (0) = u (1) = u ″ (0) = u ″ (1) = 0 , where h ∈ C ([ 0 , 1 ] , (0 , ∞)) ; f ∈ C (R , R) and s f (s) > 0 for s ≠ 0 , and f 0 = f ∞ = 0 , f 0 = lim | s | → 0 f (s) / s , f ∞ = lim | s | → ∞ f (s) / s . We investigate the global structure of one-sign solutions by using bifurcation techniques. [ABSTRACT FROM AUTHOR]

Let T > 1 be an integer, and let T = { 1 , 2 , ... , T } . We show the existence of positive solutions of the Dirichlet boundary value problem with second-order difference operator { − △ 2 u (j − 1) = λ f (j , u (j)) , j ∈ T , u (0) = u (T + 1) = 0 , where λ > 0 is a parameter, and f : T × R + → R is a continuous function satisfying f (j , 0) < 0 for all j ∈ T . The proofs of the main results are based upon topological degree and global bifurcation techniques. [ABSTRACT FROM AUTHOR]

We show the existence of three radially symmetric k-admissible solutions for k-Hessian equation with 0-Dirichlet boundary condition under suitable conditions on the nonlinearity. Proofs are mainly based on the bifurcation technique. [ABSTRACT FROM AUTHOR]

We establish the global structure of positive solutions of fourth-order periodic boundary-value problems u⁗(t) + Mu(t) = λf (t, u(t)), t ∈ [0, T], uk(0) = u(k)(T), k = 0, 1, 2, 3, with M ∈ (0, 4(2πnM4/T)4) and u(4)(t) − Mu(t) + λg(t, u(t)) = 0, t ∈ [0, T], uk(0) = u(k)(T), k = 0, 1, 2, 3, with M ∈ (0, (2πM4/T)4);here g, f ∈ C([0,T] × [0, ∞), [0, ∞)), M is constant, and λ> 0 is a real parameter. The main results are based on a global bifurcation theorem. [ABSTRACT FROM AUTHOR]

The effective catalysts for the liquid‐phase catalytic (reduction) reaction are constant pursued. Traditional powder catalysts usually suffered from the separation from reaction medium, the aggregation at higher temperature, and poor reused stability, which are not favorable for the practical application. Here, we have shown the cobalt‐nickel nitrides anchored on Nickel foam (NF) as low‐cost catalysts overcoming above disadvantages by taking the reduction of 4‐nitrophenol to 4‐aminophenol as model. The conversion can reach to 97.6% after the reaction for 5 minutes at 25 °C, being comparable to most non‐noble metal based catalysts. Notably, the catalyst can be manually separated from reaction medium benefited from anchoring nitrides on macroscopical NF substrate. The anchoring can endow the high stability and avoid loss during separation process, thus is especially favorable to give good recycled ability and to use at higher temperature. Indeed, the catalyst can be used at higher temperature (75 °C) to accelerate dramatically reaction (conversion close to 100% within 20 s). Moreover, the activity can be remained after sequential 20 times of catalytic reaction, which is superior to the catalysts ever reported. The catalyst is promising for the practical catalytic application. We have demonstrated that the cobalt‐nickel nitrides anchored on Nickel foam (NF) can be used as effective catalyst for the reduction of 4‐NP to 4‐AP with the characteristics of direct manual separation from reaction medium, high stability suitable using at higher temperature to accelerate reaction (conversion of 4‐NP close to 100% within 20s at 75 °C), and excellent reused ability (no obvious change in activity after sequential 20 times of catalytic reactions). [ABSTRACT FROM AUTHOR]

The transition metal nitrides are promising catalyst in catalytic fields. The decrease of particle size is favourable to improve the catalytic ability, but its synthesis remains a challenge due to high calcination temperature in the synthesis. Here, we have reported the designed synthesis of cluster‐like Co4N paticles (NPs) uniformly embedded in N‐doped carbon microsphere (Co4N@NC) by controllable nitridation of a spherical Co‐HQ−S composed of Co2+ and 8‐hydroxy quinoline (8‐HQ), and its used as efficient hydrogenation catalyst. The Co4N have less size about 3 nm benefited the confined effect of carbon matrix. The small size is favorable to develop the catalytic ability of Co4N and a protection by carbon can enhance its stability. The test indicated that Co4N/NC can act effective catalysts for the hydrogenation by taking the conversion of 4‐nitrophenol (4‐NP) to 4‐aminophenol (4‐AP) as model. The rate constant of 4‐NP conversion is 1.14 min−1 at 25 °C (5 mg of catalyst), much higher than many non‐noble metal catalyst and even comparable to noble metal catalysts. The Co4N/NC can be easily separated by a external magnet with good reuse performance. Importantly, the catalytic ability can be further enhanced by slight increase of the dose of catalyst, and/or reaction temperature. The combination of magnetic property and good catalytic ability makes the Co4N/NC large promising in catalytic fields. Small‐sized Co4N nanopaticles uniformly embedded in N‐doped carbon microsphere (Co4N@Carbon) have been synthesized by controllable nitridation of spherical Co‐HQ−S precursor composed of Co2+ ions coordinated with 8‐hydroxy quinoline (8‐HQ) in NH3 atmosphere. The Co4N/Carbon can act effective catalysts for the catalytic hydrogenation of 4‐NP to form 4‐AP with good activity and easy separated ability by a external magnet for reuse. [ABSTRACT FROM AUTHOR]

We consider the discrete boundary value problems with mean curvature operator in the Minkowski space where is a parameter, n>4 and q>1. Using upper and lower solutions, topological methods and Szulkin's critical point theory for convex, lower semicontinuous perturbations of -functionals, we show that there exists such that the above problem has zero, at least one or two positive solutions according to , or . Moreover, Λ is strictly decreasing with respect to n. [ABSTRACT FROM AUTHOR]

In this paper we investigate the global behavior of positive rotationally symmetric solutions for Dirichlet problem involving the k -th mean curvature operator in Minkowski space L n + 1 , which is not elliptic for k > 1 . The proofs of our main results are based upon bifurcation techniques. [ABSTRACT FROM AUTHOR]

Let T>2 be an integer and T={1,2,…,T}. We show the existence of the principal eigenvalues of linear periodic eigenvalue problem -Δ2u(j-1)+q(j)u(j)=λg(j)u(j),  j∈T, u(0)=u(T),  u(1)=u(T+1), and we determine the sign of the corresponding eigenfunctions, where λ is a parameter, q(j)≥0 and q(j)≢0 in T, and the weight function g changes its sign in T. As an application of our spectrum results, we use the global bifurcation theory to study the existence of positive solutions for the corresponding nonlinear problem. [ABSTRACT FROM AUTHOR]

The aim of this paper is to explore the method of lower and upper solutions in order to give some existence results for equations of the form with the Navier condition under the condition $k_{1}<0

This paper establishes the existence and multiplicity of solutions for the discrete Neumann-Steklov problem with singular ϕ-Laplacian by using the method of lower and upper solutions, a priori estimates and Brouwer degree theory. [ABSTRACT FROM AUTHOR]

We consider the spectrum of discrete left definite Sturm–Liouville problems whereis an integer,,r(t) changes its sign on. We obtain the existence of eigenvalues, the oscillation properties of the eigenfunctions, and the interlacing results of eigenvalues of this problem with the eigenvalues of the Dirichlet problem and the Neumann problem. [ABSTRACT FROM AUTHOR]

We investigate the existence of S-shaped connected component in the set of positive solutions of the fourth-order boundary value problem: u′′′′x=λhxfux, x∈(0,1),u(0)=u(1)=u′′0=u′′1=0, where λ>0 is a parameter, h∈C[0,1], and f∈C[0,∞) with f0≔lims→0⁡(f(s)/s)=∞. We develop a bifurcation approach to deal with this extreme situation by constructing a sequence of functions f[n] satisfying f[n]→f and (f[n])0→∞. By studying the auxiliary problems, we get a sequence of unbounded connected components C[n], and, then, we find an unbounded connected component C in the set of positive solutions of the fourth-order boundary value problem which satisfies 0,0∈C⊂lim⁡sup⁡C[n] and is S-shaped. [ABSTRACT FROM AUTHOR]

We investigate the spectrum structure of the eigenvalue problem u4x=λux,  x∈0,1;  u0=u1=u′0=u′1=0. As for the application of the spectrum structure, we show the existence of solutions of the fourth-order boundary value problem at resonance -u4x+λ1ux+gx,ux=hx,  x∈0,1;  u0=u1=u′0=u′1=0, which models a statically elastic beam with both end-points being cantilevered or fixed, where λ1 is the first eigenvalue of the corresponding eigenvalue problem and nonlinearity g may be unbounded. [ABSTRACT FROM AUTHOR]

We study the existence of positive increasing radial solutions for Neumann problems in the unit ball We do not impose any growth condition on the nonlinearity at infinity. Our result is optimal and extends one of the main results in Bonheure et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire. 2012;29:573–588]. The proof of our main result is based upon bifurcation techniques. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the existence of a reversed S-shaped component in the positive solutions set of the fourth-order boundary value problem where $\lambda>0$ is a parameter, $h\in C[0,1]$ and $f\in C[0,\infty )$ , $f(0)=0$ , $f(s)>0$ for all $s>0$ . By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to parameter λ, and especially, we obtain the existence of three distinct positive solutions for λ being in a certain interval. [ABSTRACT FROM AUTHOR]

Let B R be the ball of radius R in R N with N ≥ 2 . We consider the nonconstant radial positive solutions of elliptic systems of the form − Δ u + u = f ( u , v ) in B R , − Δ v + v = g ( u , v ) in B R , ∂ ν u = ∂ ν v = 0 on ∂ B R , where f and g are nondecreasing in each component. With few assumptions on the nonlinearities, we apply bifurcation theory to show the existence of at least one nonnegative, nonconstant and nondecreasing solution. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of an S-shaped component in the solution set of the following fourth-order boundary value problem: where $f\in C( [0,1]\times [0,\infty )\times (-\infty,0], [0,\infty))$ , and $\lambda >0$ is a parameter. By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to the parameter λ, and especially, we obtain the existence of at least three distinct positive solutions for λ being in a certain interval. [ABSTRACT FROM AUTHOR]

We consider the discrete right definite Sturm–Liouville problems − Δ ( p ( t − 1 ) Δ y ( t − 1 ) ) + q ( t ) y ( t ) = λ m ( t ) y ( t ) , t ∈ [ 1 , T ] Z , ( a 0 λ + b 0 ) y ( 0 ) = ( c 0 λ + d 0 ) Δ y ( 0 ) , ( a 1 λ + b 1 ) y ( T + 1 ) = ( c 1 λ + d 1 ) ∇ y ( T + 1 ) , where [ 1 , T ] Z = { 1 , 2 , ⋯ , T } , m ( t ) > 0 for t ∈ [ 1 , T ] Z , ( − 1 ) i δ i ≤ 0 , where δ i = a i d i − b i c i for i = 0 , 1 . We obtain the existence of the eigenvalues, the sign-changing times of the eigenfunctions and the interlacing results of the eigenvalues of the above problem, the Dirichlet problem and the Neumann problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with positive steady-state solutions of a class of cross-diffusion systems which arise in the study of the predator–prey systems with prey-taxis. Under homogeneous Dirichlet boundary conditions, we use the theory of fixed point index in positive cones to establish the existence of positive steady-state solutions. By analyzing two related eigenvalues, we further obtain the coexistence region with respect to the growth rates of two species and characterize the differences of coexistence region if different predator–prey interactions are adopted. Additionally, we investigate the limiting behavior of positive steady-state solutions as some parameter tends to infinity. Our results not only generalize the previously known one, but also present some new conclusions. [ABSTRACT FROM AUTHOR]

In the present paper, we attempt to construct both the breather and rogue wave solutions to the focusing complex short pulse (fCSP) equation via the KP–Toda reduction method. Following a procedure of reducing the bilinear equations satisfied by tau functions of Kadomtsev–Petviashvili (KP)–Toda hierarchy to the ones of the fCSP equation with nonzero boundary condition, we first deduce the general breather solution of the fCSP equation starting from a specially arranged tau-function of the KP–Toda hierarchy, then we construct and prove the N th order rogue wave solutions of the fCSP equation and express them in two different but equivalent forms of determinants. The dynamical behaviors of both the breather and rogue wave solutions are illustrated and analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global structure of radial positive solutions of boundary value problem div ( ϕ N ( ∇ v ) ) + λ f ( | x | , v ) = 0 in B ( R ) , v = 0 on ∂ B ( R ) , where ϕ N ( y ) = y 1 − | y | 2 , y ∈ R N , λ is a positive parameter, B ( R ) = { x ∈ R N : | x | < R } , and | ⋅ | denotes the Euclidean norm in R N . All results, depending on the behavior of nonlinear term f near 0, are obtained by using global bifurcation techniques. [ABSTRACT FROM AUTHOR]

We consider radial positive solutions of elliptic systems of the form { − Δ u + u = α ( | x | ) f ( u , v ) in B R , − Δ v + v = β ( | x | ) g ( u , v ) in B R , ∂ ν u = ∂ ν v = 0 on ∂ B R , where essentially α , β are assumed to be radially nondecreasing weights and f , g are nondecreasing in each component. We show the existence of at least one nondecreasing nontrivial radial solutions. Our result is sharp and is a complement for one of the main results in Bonheure et al. (2013) [2] . The proof of our main result is based upon bifurcation techniques. [ABSTRACT FROM AUTHOR]

In this paper, by using the quadrature method, we show how changes in the sign of f lead to multiple positive solutions for the semipositone Neumann problems when the parameter λ belongs to some intervals. [ABSTRACT FROM AUTHOR]

In this paper, we construct nonconstant lower and upper solutions for the periodic boundary value problem $x'+f(t,x)=e(t)$, $x(0)=x(T)$ and find their estimates. We prove the existence of positive solutions for the singular problem $x'+g(x)=e(t)$, $x(0)=x(T)$ by using these results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions of the discrete ϕ-Laplacian equation $\nabla[\phi(\Delta u_{k})]=\lambda f(k, u_{k}, \Delta u_{k})$, $k\in[2, n-1]_{\mathbb{Z}}$, with Dirichlet or mixed boundary conditions. Under general conditions, an explicit estimate of $\lambda_{0}$ is given such that the problem possesses a solution for any $|\lambda|<\lambda_{0}$. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions of the discrete ϕ-Laplacian equation $\nabla[\phi(\Delta u_{k})]=\lambda f(k, u_{k}, \Delta u_{k})$, $k\in[2, n-1]_{\mathbb{Z}}$, with Dirichlet or mixed boundary conditions. Under general conditions, an explicit estimate of $\lambda_{0}$ is given such that the problem possesses a solution for any $|\lambda|<\lambda_{0}$. [ABSTRACT FROM AUTHOR]

In this work, an elliptic system with nonlinear boundary conditions describing two subpopulations of the same species competing for resources is studied. Under some natural assumptions, we find the sufficient conditions for the existence of positive solutions of the system by using bifurcation techniques. [ABSTRACT FROM AUTHOR]

We establish several criteria for the existence and multiplicity of convex solutions for a boundary value problem arising from the study of the existence of convex radial solutions for the Monge–Ampère equations. The results are sharp. The proof of the main result is based upon the bifurcation techniques. [ABSTRACT FROM AUTHOR]

By virtue of the eigenvalues of discrete linear second-order Neumann eigenvalue problems, we study the eigenvalues of discrete linear second-order periodic and antiperiodic eigenvalue problems with sign-changing weight. We find that these two problems have T real eigenvalues (including the multiplicity) respectively. Furthermore, the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function. Furthermore, these eigenvalues, including the eigenvalues of Neumann problem, satisfy the order relation. [ABSTRACT FROM AUTHOR]

We study the existence of positive solutions for an elliptic system modeling two subpopulations of the same species competing for resources. Under some suitable assumptions, we find the necessary and sufficient conditions for the existence of positive solutions of the system by using bifurcation techniques. Moreover, an open problem proposed by Bouguima et al. (2008) is answered. [ABSTRACT FROM AUTHOR]

This paper is concerned with the traveling wave solutions and the spreading speeds for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, which is motivated by an age-structured population model with time delay. We first prove the existence of traveling wave solution with critical wave speed c = c*. By introducing two auxiliary monotone birth functions and using a fluctuation method, we further show that the number c = c* is also the spreading speed of the corresponding initial value problem with compact support. Then, the nonexistence of traveling wave solutions for c < c* is established. Finally, by means of the (technical) weighted energy method, we prove that the traveling wave with large speed is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm. [ABSTRACT FROM AUTHOR]