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In this paper, using variational method, we study the existence of an infinite number of solutions (some are positive, some are negative, and others are sign-changing) for a non-homogeneous elliptic Kirchhoff equation with a nonlinear reaction term.

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 $ +\infty $. 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.

In this paper, we present a theorem for some impulsive boundary problems with derivative dependence by the upper and lower solutions method. Using the theorem obtained, we consider the existence of positive solutions of some class of singular impulsive boundary problems.

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)&gt, 0, as well as the monotonicity of the maximal solution on parameter λ. Thirdly, the uniqueness of the solution for K(x)&lt, 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 paper, we consider the existence of eigenvalues and relative eigenfunctions for Carrier equations and present spectral asymptotics and bifurcation concerning the eigenvalues of some related elliptic linear problem.

In this paper, we consider the singular Kirchhoff equation with two parameters $$\textstyle\begin{cases} -a ( \int_{\varOmega}|\nabla u(x)|^{2}\,dx )\triangle u(x)+K(x)g(u)=\lambda f(x,u)+\mu h(x) \quad \mbox{in } \varOmega, \\ u>0 \quad \mbox{in } \varOmega, \\ u=0 \quad \mbox{on } \partial\varOmega. \end{cases} $$ By using the sub-supersolution method together with the comparison principle for elliptic equations, we obtain several existence and nonexistence theorems. Our works improve the results in the previous literature.

In this paper, we consider the generalized logistic equation with nonlocal reaction term $$ -\Delta u=u \biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f(u) \biggr)\quad \text{in } \Omega,\qquad u>0 \quad \text{ in } \Omega,\qquad u=0 \quad \text{ on } \partial \Omega. $$ Using the bifurcation and sub-supersolution method, we obtain the non-existence, existence, and uniqueness of positive solutions for different parameters on the nonlocal terms. Our works about the nonlocal elliptic problem improve the results in the previous literature.

In this paper, using the theory of fixed point index, we prove some results on the multiplicity of positive solutions for a class of nonlocal elliptic problem.

In this paper, we consider inverse time-harmonic acoustic and electromagnetic scattering from locally perturbed rough surfaces in three dimensions. The scattering interface is supposed to be the graph of a Lipschitz continuous function with compact support. It is proved that an acoustically sound-soft or sound-hard surface can be uniquely determined by the far-field pattern of infinite number of incident plane waves with distinct directions. Moreover, a single point source or plane wave can be used to uniquely determine a scattering surface of polyhedral type. These uniqueness results apply to Maxwell equations with the perfectly conducting boundary condition. Our arguments rely on the mixed reciprocity relation in a half space and the reflection principle for Helmholtz and Maxwell equations.

In this paper, we consider a class of p-Kirchhoff type problems with a singularity in a bounded domain in $R^{N}$ . By using the variational method, the existence and multiplicity of positive solutions are obtained.

In this paper, using the fixed point index method, we present a result on the existence of sign-changing solutions for a class of nonlocal elliptic problem in an annulus.

In this paper, we prove the existence of a solution between a well-ordered subsolution and supersolution of a class of nonlocal elliptic problems and give some degree information. Using the method and bifurcation theory, we present the existence and multiplicity of positive solutions for the nonlocal problems with the changes of the parameter.

This paper discusses the nonexistence and the existence of positive solutions for second order singular three-point boundary value problems when the nonlinear term f(t,x,y) is sign-changing and may be singular at t=0, x=0, y=0.

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

The existence of multiple positive solutions for the singular Dirichlet boundary-value problem $$\begin{array}{*{20}c} {{x^{{\prime \prime }} + \Phi {\left( t \right)}f{\left( {t,x{\left( t \right)},x^{\prime } {\left( t \right)}} \right)} = 0,\,\,\,\,0 < t < 1,}} \\ {{x{\left( 0 \right)} = 0,x{\left( 1 \right)} = 0,}} \\ \end{array} $$ is presented by using the fixed point index; here f may be singular at x?=?0.

This paper discusses both the nonexistence of positive solutions for second-order three-point boundary value problems when the nonlinear term f(t, x, y) is superlinear in y at y = 0 and the existence of multiple positive solutions for second-order three-point boundary value problems when the nonlinear term f(t, x, y) is superlinear in χ at +∞.

The existence of at least one positive solution and the existence of multiple positive solutions are established for the singular second-order boundary value problem $$\left\{\begin{array}{l}{\frac{1}{p(t)}(p(t)x'(t))'}+\Phi(t)f(t,x,px')=0,\quad 0

The existence of at least one unbounded positive solution and the existence of multiple unbounded positive solutions are established for the singular second-order boundary value problem p(t)(-1)(p(t)x'(t))' + Phi(t)f(t,x,px') = 0, 0 < t < +infinity, x(0) = 0, lim(t ->+infinity) p(t)x'(t) = 0, using the fixed point index, where f may be singular at px' = 0.

The existence of multiple positive solutions is presented for the singular second-order boundary value problems $$\left\{\begin{array}{ll} x^{\prime\prime}+\Phi(t)f(t,x,x') =0, 0 < t < 1\\ x(0) = 0, x'(1) = 0 \end{array}\right.$$ using the fixed point index, where f may be singular at x = 0 and x′ = 0.

In this paper, we establish the existence of an unbounded solution to a second order boundary value problem on the half line. Also the existence of multiple unbounded positive solutions is discussed using the theory of fixed point index.

The existence of at least two positive solutions is presented for the singular second-order boundary value problems {1p(t)(p(t)x^'(t))^'+@F(t)f(t,x(t),p(t)x^'(t))=0,00p(t)x^'(t)=0,x(1)=0 using the fixed point index, where @!"0^11p(r)dr=+~ and f may be singular at x=0 and px^'=0. The solutions we construct may be unbounded.

Using the theory of fixed point index, this article discusses the existence of positive solutions for singular second-order boundary value problems, where f may be singular at x = 0 and x′=0, and f may change sign.

This paper presents a lower and upper solution technique for singular second-order boundary value problems on the half-line and discusses sufficient and necessary conditions for the existence of positive solutions. Also multiplicity is discussed using the fixed point index.

This paper presents a lower and upper solution approach for singular second order boundary value problems on the half line and establishes the existence of positive, unbounded and monotone solutions.

This paper presents some results on the existence of double solutions for singular impulsive boundary value problems on the half line.

This paper presents some results on the existence of multiple unbounded solutions for the boundary value problems for second order differential equations on half line and on the uniqueness of unbounded solution.

By constructing a special cone and using fixed point index theory in cone, this paper investigates the existence of multiple solutions of singular boundary value problems for differential systems.

In this paper using a fixed point theory on a cone we present some new results on the existence of multiple positive solutions for singular nonlocal boundary value problems involving integral conditions with derivative dependence.

In this paper, using fixed point index and the mixed monotone technique, we present some multiplicity and uniqueness results for the singular nonlocal boundary value problems involving nonlinear integral conditions. Our nonlinearity may be singular in its dependent variable and it is allowed to change sign.

In this paper, by proving a new comparison result, we present a result on the existence of extremal solutions for nonlinear impulsive delay differential equations.

This paper presents some results on the existence and uniqueness of solutions for the boundary value problems on the half-line with impulses and infinite delay. Moreover, an existence theorem of multiple solutions is obtained also. The problems may be singular at the boundary.

Existence results of multiple solutions are obtained under suitable conditions for impulsive integrodifferential boundary value problems on the half-line which may be singular at the boundary.

In this paper, using the theory of fixed point index on a cone and the Leray-Schauder fixed point theorem, we present the multiplicity of positive solutions for the singular nonlocal boundary-value problems involving nonlinear integral conditions and the existence of at least one positive solution for the singular nonlocal boundary-value problems with sign-changed nonlinearities.

This paper considers the existence of positive solutions of the boundary value problems ${v}^{\prime \prime }+\lambda \left({v}^{r}+{v}^{p}-{v}^{q}\right)=0$ and $v\left(-1\right)=v\left(1\right)=0$ , where $p>r>q>-1$ and $\lambda $ is a positive parameter. Using a time-map approach, we obtain the exact number of positive solutions in different cases.

Positive solutions to the singular initial-boundary value problems are obtained by applying the Schauder fixed-point theorem, where on and may be singular at and . As an application, an example is given to demonstrate our result.

The existence of multiple positive solutions is presented for the singular Dirichlet boundary value problems [GRAPHICS] using the fixed point index; here f may be singular at x = 0 and x' = 0.

In this paper, the nonlocal boundary value problems for a class of nonlinear functional convection reaction-diffusion equations with the singular reaction function are studied by using the method of upper and lower solutions and monotone iterative technique. Some of sufficient results on the existence and uniqueness of positive global solutions or positive solutions for the boundary value problems are presented, which are a generalization of some recent results in the area.

In this paper, using fixed-point index theory and approximation techniques, we consider the existence and multiplicity of fixed points of some nonlinear operators with singular perturbation. As an application we consider the existence and multiplicity of positive solutions of singular systems of multi-point boundary value problems, which improve the results in the literature.