In this paper,~we are concerned with the following discrete problem first $$\left\{ \begin{array}{ll} -\Delta^{2}u(t-1)=\lambda p(t)f(u(t)), &t\in[1,N-1]_{\mathbb{Z}},\\ \Delta u(0)=u(N)=0,\\ \end{array} \right. $$ where $N>2$~is an integer,~$\lambda>0$~is a parameter,~$p:[1,N-1]_{\mathbb{Z}}\rightarrow\mathbb{R}$~is sign-changing function,~$f:[0,+\infty)\rightarrow[0,+\infty)$~is continuous an...

We deal with a planar differential system of the form{u′=h(t,v),v′=−λa(t)g(u), where h is T-periodic in first variable and strictly increasing second variable, λ>0, sign-changing weight function g superlinear. Based on coincidence degree theory, dependence λ, we prove existence solutions (u,v) such that u(t)>0 for all t∈R. Our results generalize unify previous contributions about Butler's probl...

We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem. Such a problem is parameterized by a set Γ of rational functions, each of which assigns a weight to each variable assignment. Our dichotomy extends previous work in which the weight functions were restricted to being non-negative. We represent a weight functi...

where p > 0, c > 0, and λ > 0 are parameters and Ω is an open bounded region with boundary ∂Ω in class C2 in Rn for n ≥ 1. Here g :Ω→ R is a Cα function while h :Ω→ R is a nonnegative Cα function with ‖h‖∞ = 1. When p = 1, (1.1) arises in population dynamics where 1/λ is the diffusion coefficient and ch(x) represents the constant yield harvesting. In this case (p = 1), when g(x) is a positive c...

In this paper, the relationships between sign-changing critical point theorems and the linking type theorems of M. Schechter and the saddle point theorems of P. Rabinowitz are established. The abstract results are applied to the study of the existence of sign-changing solutions for the nonlinear Schrödinger equation −∆u+V (x)u = f(x, u), u ∈ H1(RN ), where f(x, u) is a Carathéodory function. Pr...

We prove the existence of infinitely many sign changing radial solutions for a \begin{document}$ p $\end{document}-Laplacian Dirichlet problem in ball. Our involves weight function that is positive at center unit ball and negative its boundary. Standard initial value problems-phase plane analysis argument...