Communications in Contemporary Mathematics

In this article, we study the existence and multiplicity of solutions of the singular N-Laplacian equation:

In this paper, we prove sharp inequalities between the volume and the integral of the kth mean curvature for (k + 1)-convex domains in the Euclidean space for all k. We generalize the recent results of Chang and Wang [Some higher order isoperimetric inequalities via the method of optimal transport, preprint (2013); arXiv:1305.3004], where they prove the case k = 1, 2. The idea is the same as [S.-Y. A. Chang and Y. Wang, Some higher order isoperimetric inequalities via the method of optimal transport, preprint (2013); arXiv:1305.3004], but calculations are involved.

Consider the Liouville–Gelfand type problems with nonlinear Neumann boundary conditions

In this paper, we present the global phase portraits in the Poincaré disc of the planar quadratic polynomial systems which admit invariant straight lines with total multiplicity two and Darboux invariants.

We consider a nonlinear logistic type equation. For all big values of the parameter, we show that the problem admits nontrivial solutions of constant sign and in fact we establish the existence of extremal constant sign solutions. Using these extremal solutions, we produce a nodal (sign-changing) solution. We also investigate the uniqueness and continuous dependence on the parameter of positive solutions. Finally, we study the degenerate p-logistic equation.

We study addition operations between convex sets. We show that, under a short list of natural assumptions, one has polynomiality of volume only for the Minkowski addition. We also give two other characterization theorems. For the first theorem we define the induced homothety of an addition operation, and characterize additions by this homothety. The second theorem characterizes all additions which satisfy a short list of natural conditions.

In this paper, we are concerned with the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation with typical discontinuous initial data (u(0, x), ∂_tu(0, x)) = (0, φ(x)), where m ∈ ℕ, x = (x₁,…,x_n), n ≥ 2, and f(t, x, u) is C^∞ smooth on its arguments. When the initial data φ(x) is homogeneous of degree zero or piecewise smooth along the hyperplane {t = x₁ = 0}, it is shown that the local solution u(t, x) ∈ L^∞([0, T] × ℝⁿ) exists and is C^∞ away from the forward cuspidal conic surface or the cuspidal wedge-shaped surfaces respectively. On the other hand, for n = 2 and piecewise smooth initial data φ(x) along the two straight lines {t = x₁ = 0} and {t = x₂ = 0}, we establish the local existence of a solution and further show that in general due to the degenerate character of the equation under study, where . This is an essential difference to the well-known result for solution to the two-dimensional semilinear wave equation with (v(0, x), ∂_tv(0, x)) = (0, φ(x)), where Σ₀ = {t = |x|}, and .

We consider the nonlinear Schrödinger equation with L²-critical exponent and an inhomogeneous damping term. By using the tools developed by Merle and Raphael, we prove the existence of blowup phenomena in the energy space H¹(ℝ).

Let Ω ⊂ ℝ^N be a bounded regular domain of ℝ^N and 1 < p < ∞. The paper is divided into two main parts. In the first part, we prove the following improved Hardy inequality for convex domains. Namely, for all , we have

Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in H^s with . The equation is L²-critical, and hence a derivative gap is left between their result and the scaling prediction. In this paper, we consider initial data in the Fourier–Lebesgue space for 1 < p ≤ 2 which coincides with H^s when p = 2 but scales like lower regularity Sobolev spaces for 1 < p < 2. In particular, we will see that as p → 1⁺, the critical exponent , in which case is the critical space. We shall prove almost optimal local well-posedness to the space-time monopole equation in Lorenz gauge with initial data in the aforementioned spaces that correspond to p close to 1.