On sign-changing and multiple solutions of the p-Laplacian

In this paper, we construct the pseudo-gradient vector field in , by which we obtain the positive and negative cones of are both invariant sets of the descent flow of the corresponding functional. Then we use differential equations theory in Banach spaces and dynamics theory to study p-Laplacian boundary value problems with “jumping” nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems of p-Laplacian.     

On strong convergence to 3D steady vortex sheets

In this paper, we first establish a strong convergence criterion of approximate solutions for the 3D steady incompressible Euler equations. For axisymmetric flows, under the assumption that the vorticity is of one sign and uniformly bounded in L¹ space, we obtain a sufficient and necessary condition for the strong convergence in of approximate solutions. Furthermore, for one-sign and L¹-bounded vorticity, it is shown that if a sequence of approximate solutions concentrates at an isolated point in (r,z)-plane, then the concentration point can appear neither in the region near the axis (including the symmetry axis itself) nor in the region far away from the axis. Finally, we present an example of approximates solutions which converge strongly in by using Hill's spherical vortex.     

On a resolvent estimate of the interface problem for the Stokes system in a bounded domain

Obtained is the L_p estimate of solutions to the resolvent problem for the Stokes system with interface condition in a bounded domain in . It is the first step to consider the free boundary value problem.     

“Bubble-tower” radial solutions in the slightly supercritical Brezis-Nirenberg problem

In this paper, we consider the Brezis-Nirenberg problem in dimension N?4, in the supercritical case. We prove that if the exponent gets close to and if, simultaneously, the bifurcation parameter tends to zero at the appropriate rate, then there are radial solutions which behave like a superposition of bubbles, namely solutions of the form

     

Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms

In this paper, we study the boundary behavior of solutions to boundary blow-up elliptic problems , where Ω is a bounded domain with smooth boundary in R^N, q>0, , which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary, and f is rapidly varying or normalized regularly varying at infinity.     

Bubble towers for supercritical semilinear elliptic equations

We construct positive solutions of the semilinear elliptic problem with Dirichet boundary conditions, in a bounded smooth domain Ω⊂R^N(N?4), when the exponent p is supercritical and close enough to and the parameter λ∈R is small enough. As , the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino et al. (J. Differential Equations 193(2) (2003) 280) when Ω is a ball and the solutions are radially symmetric.     

Superlinear elliptic equations with singular coefficients on the boundary

In this paper, a superlinear elliptic equation whose coefficient diverges on the boundary is studied in any bounded domain Ω under the zero Dirichlet boundary condition. Although the equation has a singularity on the boundary, a solution is smooth on the closure of the domain. Indeed, it is proved that the problem has a positive solution and infinitely many solutions without positivity, which belong to or . Moreover, it is proved that a positive solution has a higher order regularity up to .     

Stability to the global large solutions of 3-D Navier-Stokes equations

In this paper, we consider the stability to the global large solutions of 3-D incompressible Navier-Stokes equations in the anisotropic Sobolev spaces. In particular, we proved that for any , given a global large solution v∈C([0,∞);H0,s₀(R³)∩L³(R³)) of (1.1) with and a divergence free vector satisfying for some sufficiently small constant depending on , v, and , (1.1) supplemented with initial data v(0)+w₀ has a unique global solution in u∈C([0,∞);H0,s₀(R³)) with ∇u∈L²(R⁺,H0,s₀(R³)). Furthermore, uh is close enough to vh in C([0,∞);H0,s(R³)).     

Steady compressible Navier-Stokes-Fourier system for monoatomic gas and its generalizations

We consider steady compressible Navier-Stokes-Fourier system for a gas with pressure p and internal energy e related by the constitutive law p=(γ−1)?e, γ>1. We show that for any there exists a variational entropy solution (i.e. solution satisfying the weak formulation of balance of mass and momentum, entropy inequality and global balance of total energy). This result includes the model for monoatomic gas (). If , these solutions also fulfill the weak formulation of the pointwise total energy balance.     

Pointwise estimates and Lp convergence rates to diffusion waves for p-system with damping

By introducing a new approximate Green function, we obtain the pointwise estimates on the solutions of Euler equations with linear frictional damping, from which we can deduce the optimal convergence rates to the nonlinear diffusion waves. The pointwise estimates and convergence rates given in this paper are new.     

A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity

We consider positive solutions of on B₁ (n?5) where μ and K>0 are smooth functions on B₁. If K is very sub-harmonic at each critical point of K in B2/3 and the maximum of u in is comparable to its maximum over , then all positive solutions are uniformly bounded on . As an application, a priori estimate for solutions of equations defined on Sn is derived.     

A priori bounds for superlinear problems involving the N-Laplacian

In this paper we establish a priori bounds for positive solutions of the equation