In this article, we apply local bifurcation theory to prove the existence of small-amplitude steady periodic water waves, which propagate over a flat bed with a specified fixed mean-depth, and where the underlying flow has a discontinuous vorticity distribution.

This work is devoted to prove a linear profile decomposition for the Airy equation in $\dot{H}_x^{s_k}(\mathbb R)$, where $s_k=(k-4)/2k$ and $k>4$. We also apply this decomposition to establish the existence of maximizers for a general class of Strichartz type inequalities associated to the Airy equation.

In this paper, we concern a weighted version of the Hardy inequality, which is a special case of the more general Caffarelli-Kohn-Nirenberg inequalities. We improve the inequality on the whole space or on a bounded domain by adding various remainder terms. On the whole space, we show the existence of a remainder term which has the form of ratio of two weighted integrals. Also we give a simple derivation of the remainder term involving a distance from the manifold of the “virtual extremals”. Finally, on a bounded domain, we prove the existence of remainder terms involving the gradient of functions.

We study the symmetry breaking phenomenon for an elliptic equation involving the fractional Laplacian in a large ball. Our main tool is an extension of the Strauss radial lemma involving the fractional Laplacian, which might be of independent interest; and from which we derive compact embedding theorems for a Sobolev-type space of radial functions with power weights.

For solutions of a class of divergence type quasilinear elliptic equations with $(p,q)$-growth conditions, we establish the condition for removability of singularity on manifolds.

We show the time local well-posedness in $H^s$ of the reduced NLS equation with third order dispersion (r3NLS) on $\mathbf{T}$ for $s > -1/6$. Our proof is based on the nonlinear smoothing effect, which is similar to that for mKdV. However, when (r3NLS) is considered in Sobolev spaces of negative indices, the unconditional uniqueness of solutions, that is, the uniqueness of solutions without auxiliary spaces breaks down in marked contrast to mKdV.

We derive a new condition for delay-independent stability of systems of linear neutral delay differential equations. The method applies ideas from linear semigroup theory, and involves renorming the underlying Hilbert space to obtain a dissipative inequality on the infinitesimal generator of the solution semigroup. The new stability condition is shown to either improve upon or be independent of existing stability conditions.

As was kindly pointed to one of us by an anonymous referee, there is a gap in the argument in [1] whose origin is in the statement and proof of Lemma 2.2. This error can be easily corrected, and after this correction all the main results in [1] remain valid, as explained below.