This paper is a sequel to a previous paper (Yang,Y.and Zhang,J.H.Existence of solutions for some fourth-order boundary value problems with parameters.Nonlinear Anal.69(2),1364–1375 (2008)) in which the nontrivial solutions to the fourthorder boundary value problems were studied.In the current work with the same conditions near infinity but different near zero,the positive,negative,and sign-changing solutions are obtained by the critical point theory,retracting property,and invariant sets.
In this paper, under some assumptions on the flow with a low Mach number, we study the nonexistence of a global nontrivial subsonic solution in an unbounded domain Ω which is one part of a 3D ramp. The flow is assumed to be steady, isentropic and irrotational, namely, the movement of the flow is described by the potential equation. By establishing a fundamental a priori estimate on the solution of a second order linear elliptic equation in Ω with Neumann boundary conditions on Ω and Dirichlet boundary value at some point of Ω, we show that there is no global nontrivial subsonic flow with a low Mach number in such a domain Ω.
In this paper, we study the existence and regularity of a solution to the initial datum problem of a semilinear generalized Tricomi equation in mixed-type domain. We suppose that an initial datum on the degenerate plane is smooth away from the origin, and has a conormal singularity at this point, then we show that in some mixed-type domain, the solution exists and is conormal with respect to the characteristic conic surface which is issued from the origin and has a cusp singularity.
We study blow-up,global existence and ground state solutions for the N-coupled focusing nonlinear Schr¨odinger equations.Firstly,using the Nehari manifold approach and some variational techniques,the existence of ground state solutions to the equations(CNLS) is established.Secondly,under certain conditions,finite time blow-up phenomena of the solutions is derived.Finally,by introducing a refined version of compactness lemma,the L2concentration for the blow-up solutions is obtained.
This paper is concerned with the problem on the global existence and stability of a subsonic flow in an infinitely long cylindrical nozzle for the 3D steady potential flow equation.Such a problem was indicated by Courant-Friedrichs in [8,p.377]:A flow through a duct should be considered as a steady,isentropic,irrotational flow with cylindrical symmetry and should be determined by solving the 3D potential flow equations with appropriate boundary conditions.By introducing some suitably weighted Hlder spaces and establishing a priori estimates,the authors prove the global existence and stability of a subsonic potential flow in a 3D nozzle when the state of subsonic flow at negative infinity is given.
In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schrdinger equation -ε2 Δuε + V(x)uε = K(x)|uε|p-2 uεeα0 |uε|γ,uε>0, uε∈H 1(R2),where 2 < p < ∞, α0 > 0, 0 < γ < 2. When the potential function V (x) decays at infinity like (1 + |x|)-α with 0 < α≤ 2 and K(x) > 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H1 (R2 )-solution uε exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schrdinger equation-Δu + V (ξ)u = K(ξ)|u| p-2 ue α0 |u|γ has local minimum points. Furthermore, the concentration property of uε is also established as ε tends to zero.
