This paper concerns the existence and multiplicity of solutions for some semilinear elliptic equations with critical Sobolev exponent, Hardy term and the sublinear nonlinearity at origin. By using Ekeland's variational principle, we conclude the existence of nontrivial solution for this problem, the Clark's critical point theorem is used to prove the existence of infinitely many solutions for this problem with odd nonlinearity.

In this paper we obtain some new results on Landau-Lifshitz equation by two explicit transforms.

Consider the blow up results for local smooth solutions of a quasilinear Schrödinger equation iu_t + Δu + β|u|^{p-2}u + θ(Δ|u|²)u = 0, u|_{t=0} = u_0(x), x ∈ \mathbb{R}^N in nonisotropic space.

In this parer, by using the polar coordinates for the generalized Baouendi- Grushin operator L_α = \sum^n_{i=1}\frac{∂²}{∂x²_i} + \sum^m_{j=1}|x|^{2α} \frac{∂²}{∂y²_j}, where x = (x_1, x_2, …, x_n) ∈ \mathbb{R}^n, y = (y_1, y_2, …, y_m) ∈ \mathbb{R}^m, α › 0, we obtain the volume of the ball associated to L_α and prove the nonexistence for a second order evolution inequality which is relative to L_α.

In this paper, an extended loop algebra is constructed from which an isospectral problem established. It follows that the integrable couplings of the Tu hierarchy and M-AKNS-KN hierarchy are obtained, and their Hamilton structures are presented by the quadratic-form identity. Moreover, we guarantee that the expanding model we obtained are also Liouville integrable.

In this paper the existence and nonexistence of non-trivial solution for the Cauchy problem of the form {ut = div(|∇u|^{p-2}∇u) - \frac{∂}{∂x_i}b_i(u) - u^q, \qquad(x, t) ∈ S_T = R^N × (0, T), u(x, 0) = 0, \qquad \qquad x ∈ R^N\ {0} are studied. We assume that |b^'_i(s)| ≤ Ms^{m-1}, and proved that if p > 2, 0 < q < p-1+ \frac{p}{N}, 0 ≤ m < p-1+ \frac{p}{N}, then the problem has a solution; if p > 2, q > p-1+ \frac{p}{N}, 0 ≤ m ≤ \frac{q(p+Np-N-1)}{p+Np-N} , then the problem has no solution; if p > 2,p-1 < q < p-1+ \frac{p}{N}, 0 ≤ m < q, then the problem has a very singular solution; if p > 2, q > p-1 + \frac{p}{N}, 0 < m < q - \frac{p}{2N}, then the problem has no very singular solution. We use P.D.E. methods such as regularization, Moser iteration and Imbedding Theorem.

In this paper we study the mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems in the domain D = {(t, x) | t ≥ 0, x ≥ 0}. Under the assumption that the source term satisfies the matching condition, a sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.