Let $r>0, n\in\mathbb N, {\bf k}\in\mathbb N$. Consider the delay differential equation $$ x'(t)=g(x(t-d_1(Lx_t)),\ldots,x(t-d_{{\bf k}}(Lx_t))) $$ for $g:( \mathbb R^n)^{{\bf k}}\supset V\to\mathbb R^n$ continuously differentiable, $L$ a continuous linear map from $C([-r,0],\mathbb R^n)$ into a finite-dimensional vector-space $F$, each $d_k:F\supset W\to[0,r]$, $k=1,\ldots,{\bf k}$, continuously differentiable, and $x_t(s)=x(t+s)$. The solutions define a semi-flow of continuously differentiable solution operators on the sub-manifold $X_f\subset C^1([-r,0],\mathbb R^n)$ which is given by the compatibility condition $\phi'(0)=f(\phi)$ with $$f (\phi)=g(\phi(-d_1(L\phi)),\ldots,\phi(-d_{{\bf k}}(L\phi))). $$ We prove that $X_f$ has a finite atlas of at most $2^{{\bf k}}$ manifold charts, whose domains are almost graphs over $X_0$. The size of the atlas depends solely on the zero-sets of the delay functions $d_k$.

In this paper, we study the steady and purely azimuthal flow model representing the Antarctic Circumpolar Current, which is described as a semi-linear elliptic equation. By means of the Mercator projection, the mathematical model is transferred into a second order elliptic equation. We give sufficient conditions to guarantee the existence of solutions for this second elliptic equation with two forms of nonlinear ocean vorticity term.

This paper analyzes a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. This paper establishes the well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange (fractional differential) equations. This is achieved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which give rise to more general fractional differential equations.

We study the minimum problem for functionals of the form \begin{equation*} \mathcal{F}(u) = \int_{I} f(x, u(x), u^\prime(x))\,dx, \end{equation*} where the integrand $f:I\times\mathbb R^d\times\mathbb R^d\to \mathbb R$ is not convex in the last variable. We provide existence results in the Sobolev space $W^{1,1}(I,\mathbb R)$ analogous to the ones obtained in $W^{1,p}(I,\mathbb R^ d)$ ($p>1$) by a method inspired by integro-extremization and based on Euler equations. In addition, we treat functionals with nonsmooth Lagrangians and exhibit a comparison with a direct application of integro-extremality method to a class of functionals of sum type with a separate dependence on the components of the derivative $u^\prime$.

In this paper, we intend to study the following quasilinear Choquard equation involving the $p$-Laplacian operator and critical nonlinearities in $\mathbb{R}^N$: $$ -\big(\Delta_p u +\Delta_p (u^2)\big) = \lambda\big(I_\mu*|u|^q\big)|u|^{q-2}u + |u|^{2p^\ast-2}u \quad \mbox{in}\ \mathbb{R}^N, $$ where $\Delta_p u = \mathop{\rm div}(|\nabla u |^{p-2} \nabla u)$ is the $p$-Laplacian operator with $1 < p\leq N$, $N \geq 3$, $I_\mu(x)=|x|^{-\mu}$, $0 < \mu < N$, $\frac{p(N+\mu)}{N} \leq q < \frac{p(N+\mu)}{N-p}$, $p^*=\frac{Np}{N-p}$ is the critical exponent in the sense of Sobolev inequality, $\lambda$ is a positive parameter. The existence of nontrivial solutions are obtained by using the mountain pass theorem. The main features and novelty of the paper are the presence of the convolution terms as well as critical nonlinearities.