This paper is concerned with the existences of positive solutions of the following Dirichlet problem for p-mean curvature operator with supercritical potential: {-div((1 + |∇u|²) ^{\frac{p-2}{2}} ∇u) = λu^{r-1} + μ\fran{u^{q-1}}{|x|^s}, u > 0 ∈ Ω, u = 0\qquad\qquad x ∈ ∂Ω where u Ω W^{1, p}_0 (Ω), Ω is a bounded domain in R^N(N > p > 1) with smooth boundary ∂Ω and 0 ∈ Ω, 0 < q < p, 0 ≤ s < \frac{N}{p} (p - q) + q, p ≤ r < p∗, p∗ = \frac{Np}{N-p}, μ > 0. It reaches the conclusion where this problem has two positive solutions in the different cases . It discusses the existences of positive solutions of the Dirichlet problem for the p-mean curvature operator with supercritical potential firstly. Meanwhile, it extends some results of the p-Laplace operator to that of p-mean curvature operator for p ≥ 2 .

The long time behavior of the solutions of the generalized long-short wave equations with dissipation term is studied. The existence of global attractor of the initial periodic boundary value is proved by means of a uniform a priori estimate for time. And also the dimensions of the global attractor are estimated.

By use of the fibering method introduced by Pohozaev S I, the existence of positive solutions for homogeneous Dirichlet problem of a class of quasilinear elliptic systems with indefinite weights is obtained.

We study the nonlinear instability of plane wave solutions to a Ginzburg- Landau equation with derivatives. We show that, under some condition in coefficient of the equation, these waves are unstable.

This paper deals with the very weak solutions of A-harmonic equation divA(x, ∇u(x)) = 0 (∗) where the operator A satisfies the monotonicity inequality, the controllable growth condition and the homogeneity condition. The extremum principle for very weak solutions of A-harmonic equation is derived by using the stability result of Iwaniec-Hodge decomposition: There exists an integrable exponent r_1 = r_1\left(p, n, \frac{β}{α}\right) = \frac{1}{2} \bigg[p - \frac{α}{100n²β} + \sqrt{\left(p + \frac{α}{100n²β}\right)² - \frac{4α}{100n²β}}\bigg] such that if u(x) ∈ W^{1, r}(Ω) is a very weak solution of the A-harmonic equation (∗), and m ≤ u(x) ≤ M on ∂Ω in the Sobolev sense, then m ≤ u(x) ≤ M almost everywhere in Ω, provided that r > r1. As a corollary, we prove that the 0-Dirichlet boundary value problem  {divA(x, ∇u(x)) = 0 u ∈ W^{1, r}_0 (Ω) of the A-harmonic equation has only zero solution if r > r_1.

We study here a class of pseudodifferential operators with weighted symbols of Shubin type. First, we develop the basic elements of the pseudodifferential calculus for these operators, proving in particular a result of L^p-boundedness. Then we derive regularity results in the frame of suitably defined functional spaces of Sobolev type.

Consider the Cauchy problem for a wave equation on R²: u_{tt} - Δu = |u|^{p-1}u. In 1981 Glassey gave a guess to a critical value p(2) = \frac{1}{2}(3 + \sqrt{17}): when p > p(2) there may exist a global solution and when 1 < p < p(2) the solution may blow up. By our main result in this paper a counter example to the guess is given that the solution may also blow up in finite time even if p(2) < p < 5.

In this paper, we prove the existence and nonuniqueness of the weak solutions of the initial and boundary value problem for a nonlinear degenerate parabolic equation not in divergence form. Localization property of weak solutions will be also discussed.

The motivation of this paper is the study of the existence of weak global attractor for mixed initial boundary value problem for some multidimensional Ginzberg-Landau equations.