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This paper considers a singular Kirchhoff equation with convection and a parameter. By defining new sub-supersolutions, we prove a new sub-supersolution theorem. Combining method of sub-supersolution with the comparison principle, for Kirchhoff equation with convection, we get the conclusion about positive solutions when nonlinear term is singular and sign-changing.

In this paper, we are concerned with the existence, uniqueness and long-time behavior of the solutions for a parabolic equation with nonlocal diffusion even if the reaction term is not Lipschitz-continuous at 0 and grows superlinearly or exponentially at +∞. First, we give a special sub-supersolution pair for some parabolic equations with nonlocal diffusion and establish the method of sub-supersolution. Second, using the sub-supersolution method, we prove the existence, uniqueness and long-time behavior of positive solutions. Finally, some one-dimensional numerical experiments are presented.

This paper deals with a nonlinear nonlocal parabolic system with nonlinear heat-loss boundary conditions, which arise in the thermal explosion model. Firstly, we prove a comparison principle for some kinds of parabolic systems under nonlinear boundary conditions. Using this, we improve a new theorem of the sub-and-super solution. Secondly, based on the new sub-and-super solution theorem, the sufficient conditions that the solution exists and blows up uniformly in finite time are presented. Then, we generalize some of the lemmas related to uniform blow-up solutions, which are used to introduce the uniform blow-up profiles of solutions. Finally, we give several numerical simulations to illustrate the existence and uniform blow-up of solutions.

Focusing on the physical context of the thermal explosion model, this paper investigates a semilinear parabolic equation ut=Δu+a∫Ωupdx,x,t∈QT,n·∇u+guu=0,x,t∈ST,ux,0=u0x,x∈Ω with nonlocal sources under nonlinear heat-loss boundary conditions, where a,p>0 is constant, QT=Ω×0,T, ST=∂Ω×0,T, and Ω is a bounded region in RN,N≥1 with a smooth boundary ∂Ω. First, we prove a comparison principle for some kinds of semilinear parabolic equations under nonlinear boundary conditions; using it, we show a new theorem of subsupersolutions. Secondly, based on the new method of subsupersolutions, the existence of global solutions and blow-up solutions is presented for different values of p. Finally, the blow-up rate for solutions is estimated also.

In this paper, we study the nonlocal equation ∫RN|u(x)|2dxγΔu=λu+μ|u|q−2u+|u|p−2u, xinRN having a prescribed mass ∫RN|u(x)|2dx=c2, where N≥3, μ, γ∈(0,+∞), q∈(2,2∗), c is a positive constant, p, q∈(2,2∗) with p≠q and 2∗=2NN−2. This research is meaningful from a physical point of view. Using variational methods, we present some results on the nonexistence and existence of solutions under different cases p and q which improve upon the previous ones via topological theory.

This article considers elliptic problems of Kirchhoff-type. We give some new definitions of lower and upper solutions for the problem and establish the method of lower and upper solutions when the upper and lower solutions are well ordered, i.e., the lower solution is less than the upper one, and we also consider the case when the upper and lower solutions have opposite ordering. In addition we use the relation between the topological degree and strict upper and lower solutions in both cases and using this we obtain multiplicity results for nonlinear Kirchhoff-type elliptic problems.

In this paper, a special class of boundary value problems, −▵u=λuq+ur,inΩ,u>0, inΩ,n·∇u+g(u)u=0,on∂Ω, where 00 is a parameter. The existence of the solution is verified via sub- and super-solutions method. In addition, the influences of parameters on the minimum solution are also discussed. The second positive solution is obtained by using the variational method.

In this paper, we consider the existence of multiple solutions for second-order equation with Stieltjes integral boundary condition using the three-critical-point theorem and variational method. Firstly, a novel space is established and proved to be Hilbert one. Secondly, based on the above work, we obtain the existence of multiple solutions for our problem. Finally, in order to illustrate the effectiveness of our problem better, the example is listed.

In this paper, the thermal explosion model described by a nonlinear boundary value problem is studied. Firstly, we prove the comparison principle under nonlinear boundary conditions. Secondly, using the sub-super solution theorem, we prove the existence of a positive solution for the case K(x)>0, as well as the monotonicity of the maximal solution on parameter λ. Thirdly, the uniqueness of the solution for K(x)<0 is proved, as well as the monotonicity of the solutions on parameter λ. Finally, we obtain some new results for the existence of solutions, and the dependence on the λ for the case K(x) is sign-changing.

In this article, using the sub-supersolution method and Rabinowitz-type global bifurcation theory, we prove some results on existence, uniqueness and multiplicity of positive solutions for some singular nonlocal elliptic problems.

In this article, we consider the singular elliptic boundary-value problem $$ -\Delta u+f(u)-u^{-\gamma} =\lambda u \text{ in } \Omega,\quad u>0\text{ in } \Omega,\quad u=0 \text{ on } \partial\Omega. $$ Using the upper-lower solution method, we show the existence and uniqueness of the solution. Also we study the boundary behavior and asymptotic behavior of the positive solutions.

In this article, we study the structure of global solutions to the boundary-value problem $$\displaylines{ -x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\cr \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\cr x(0)=x(1)=0, }$$ where $\lambda\neq0$, $\beta_1\geq\beta_{2}\geq0$, $\Delta x|_{t=1/2}=x(\frac{1}{2}+0)-x(\frac{1}{2})$, $\Delta x'|_{t=1/2}=x'(\frac{1}{2}+0)-x'(\frac{1}{2}-0)$, and $f:[0,1]\times\mathbb{R}\to\mathbb{R}$, $a:[0,1]\to(0,+\infty)$ are continuous. By a comparison principle and spectral properties of the corresponding linear equations, we prove the existence of solutions by using Rabinowitz-type global bifurcation theorems, and obtain results on the behavior of positive solutions for large $\lambda$ when $f(x)=x^{p+1}$.

This article presents global properties and existence of multiple solutions for a class of boundary value problems of impulsive differential equations. We first show that the spectral properties of the linearization of these problems are similar to the well-know properties of the standard Sturm-Liouville problems. These spectral properties are then used to prove two Rabinowitz-type global bifurcation theorems. Finally, we use the global bifurcation theorems to obtain multiple solutions for the above problems having specified nodal properties.

In this article, we study the existence and uniqueness of the positive solution for a second-order singular three-point boundary-value problem with sign-changing nonlinearities. Our main tool is a fixed-point theorem.

Using the theory of fixed point theorem in cone, this paper presents the existence of positive solutions for the singular $m$-point boundary value problem $$ \left\{\begin{array}{ll} x''(t)+a(t)f(t,x(t),x'(t))=0,0

Using the fixed point theorem in cones, this paper shows the existence of multiple positive solutions for the singular $m$-point boundary-value problem $$displaylines{ x''(t)+a(t)f(t,x(t),x'(t))=0,quad 0

Using the theory of fixed point index, this paper discusses the existence of at least one positive solution and the existence of multiple positive solutions for the singular three-point boundary value problem: $$displaylines{ y''(t)+a(t)f(t,y(t),y'(t))=0,quad 0

Using a fixed point theorem in cones, this paper shows the existence of positive solutions for the singular three-point boundary-value problem $$displaylines{ x''(t)+a(t)f(t,x(t),x'(t))=0,quad 0

In this paper, Krasnoselskii's theorem and the fixed point theorem of cone expansion and compression are improved. Using the results obtained, we establish the existence of multiple positive solutions for the singular second-order boundary-value problems with derivative dependance on finite and infinite intervals.

Using a specially constructed cone and the fixed point index theory, this paper shows the existence of multiple positive solutions for a class of nonresonant singular boundary-value problem of second-order differential equations. The nonexistence of positive solution is also studied.

This paper considers the following boundary value problem: ((-u'(t))n)'=ntn-1f(u(t)), 01 is odd. We establish the method of lower and upper solutions for some boundary value problems which generalizes the above equations and using this method we present a necessary and sufficient condition for the existence of positive solutions to the above boundary value problem and some sufficient conditions for the existence of positive solutions.

This paper considers the existence of positive solutions of the boundary value problems and , where and is a positive parameter. Using a time-map approach, we obtain the exact number of positive solutions in different cases.