Quasilinear Schrodinger-Poisson equations involving a nonlocal term and an integral constraint

In this paper,we consider a class of quasilinear Schrodinger-Poisson problems of the form∫-(a+b∫_(R)^(N)|■μ|^(2)dx)■μ+V(x)u+Фu-1/2u■(u^(2))-λ|u|^(p-2)u=0 in R^(N),-ΔФ=u^(2),u(x)→0,|x|→∞in R^(N),∫_(R)^(N)|u|^(p)dx=1,where a>0,b≥0,N≥3,λappears as a Lagrangian multiplier,and 4<p<2·2^(*)=4N/N-2.We deal with two different cases simultaneously,namely lim|x|→∞V(x)=1 and limjxj!1 V(x)=V1.By using the method of invariant sets of the descending flow combined with the genus theory,we prove the existence of infinitely many sign-changing solutions.Our results extend and improve some recent work.

Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.

Well-posedness for A Plate Equation with Nonlocal Source term

In this paper,we investigate the initial boundary value problem for a plate equation with nonlocal source term.The local,global existence and exponential decay result are established under certain conditions.Moreover,we also prove the blow-up in finite time and the lifespan of solution under certain conditions.

Lower Bound of Blow-Up Time for Solutions of a Class of Cross Coupled Porous Media Equations

In this paper,blow-up phenomena of solutions to a class of parabolic equations for porous media with nonlocal source terms cross-coupled under Dirichlet and Neumann boundary conditions are studied.The differential inequality techniques are used to obtain the lower bounds on the blow up time of the equation set under two different boundary conditions.

Lower Bound Estimate of Blow Up Time for the Porous Medium Equations under Dirichlet and Neumann Boundary Conditions

In this paper,we establish the lower bounds estimate of the blow up time for solutions to the nonlocal cross-coupled porous medium equations with nonlocal source terms under Dirichlet and Neumann boundary conditions.The results are obtained by using some differential inequality technique.