The purpose of this paper is to investigate the nonexistence of positive solutions of the following doubly nonlinear degenerate parabolic equations: \begin{align*}\begin{cases}  {\dfrac{\partial u}{\partial t}=\nabla_{k} \cdot \left( {u^{m-1}\left| {\nabla_{k} u} \right|^{p-2}\nabla_{k} u} \right)+V(w)u^{m+p-2}},\qquad & {\mbox{in}\  \Omega \times (0,T),} \\  {u(w,0)=u_{0} (w)\geqslant 0}, & {\mbox{in}\  \Omega ,} \\  {u(w,t)=0}, & {\mbox{on}\  \partial \Omega \times (0,T),}  \end{cases} \end{align*} where $\Omega $ is a Carnot-Carath\'{e}odory metric ball in $\mathbb{R}^{2n+1}$ generated by Greiner vector fields, $V\in L_{loc}^{1} (\Omega )$, $k\in \mathbb{N}$, $m\in \mathbb{R}$, $1<p<2n+2k$ and $m+p-2>0$. The better lower bound $p^*$ for $m + p_{ }$ is found and the nonexistence results are proved for $p^*\leqslant  m+p<3$.

Denote  $$ {\cal K} _{\psi, \theta} (\Omega) =\left\{ v\in W^{1,p} (\Omega) : v\ge \psi, \mbox { a.e. and }  v-\theta \in W_0^{1,p} (\Omega) \right\}, $$ where $\psi$ is any function in $\Omega \subset \mathbb R^N$, $N\ge 2$, with values in $\mathbb R \cup \{\pm \infty\}$ and $\theta $ is a measurable function. This paper deals with global integrability for $u \in {\cal K}_{\psi, \theta}$ such that

$$ \int_\Omega \langle {\cal A} (x,\nabla u), \nabla (w-u) \rangle {\rm d}x \ge \int_\Omega \langle F, \nabla (w-u) \rangle {\rm d}x, \ \ \forall\ w \in {\cal K}_{\psi,\theta} (\Omega), $$ with $|{\cal A} (x,\xi)| \approx |\xi| ^{p-1}$, $1<p<N$. Some global integrability results are obtained.

Let $(X,d,\mu)$ be a metric space with a Borel-measure $\mu$, suppose $\mu$ satisfies the Ahlfors-regular condition, i.e. \begin{equation*} b_1r^s\leq\mu(B_r(x))\leq b_2r^s,\qquad \forall B_r(x)\subset X, \;\; \  r>0, \end{equation*} where $b_1$, $b_2$ are two positive constants and $s$ is the volume growth exponent. In this paper, we mainly study two things, one is to consider the best constant of the Moser-Trudinger inequality on such metric space under the condition that $s$ is not less than 2. The other is to study the generalized Moser-Trudinger inequality with a singular weight.

In this paper, we consider the Cauchy problem of 3D tropical climate model with zero thermal diffusion. Firstly, we establish the global regularity for this system with fractional diffusion $\alpha=\beta=5/4$. Secondly, by adding only a damp term, we obtain the global well-posedness for small initial data.

In this paper we prove that the Schrödinger-Boussinesq system with solution $(u,v,$  $(-\partial_{xx})^{-\frac12} v_t)$ is locally wellposed in   $  H^{s}\times H^{s}\times H^{s-1}$, $s\geqslant-{1}/{4}$. The local wellposedness is obtained by the transformation from the problem into a nonlinear Schrödinger type equation system and the contraction mapping theorem in a suitably modified Bourgain type space inspired by the work of Kishimoto, Tsugawa. This result improves the known local wellposedness in $ H^{s}\times H^{s}\times H^{s-1}$, $s>-{1}/{4}$ given by Farah.

In this paper, we are interested in the following nonlocal problem with critical exponent \begin{align*} \begin{cases} -\left(a-b\displaystyle\int_{\Omega}|\nabla u|^2{\rm d}x\right)\Delta u=\lambda |u|^{p-2}u+|u|^{4}u, &\quad x\in\Omega,\\ u=0,  &\quad x\in\partial\Omega, \end{cases} \end{align*} where $a,b$ are positive constants, $2<p<6$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$ and $\lambda>0$ is a parameter. By variational methods, we prove that problem has a positive ground state solution $u_b$ for $\lambda>0$ sufficiently large. Moreover, we take $b$ as a parameter and study the asymptotic behavior of $u_b$ when $b\searrow0$.

In this paper, we study a class of sublinear operators and their commutators with a weighted BMO function. We first give the definition of a weighted Morrey space $L^{p,\kappa}_{\mu,\omega}(X)$ where $X$ is an RD-measure and $\omega$ is the weight function. The weighted Morrey spaces arise from studying the local behavior of solutions to certain partial differential equations. We will show that the aforementioned class of operators and their communtators with a weighted BMO function are bounded in the weighted Morrey space $L^{p,\kappa}_{\mu,\omega}(X)$ provided that the weight function $\omega$ belongs to the $A_p(\mu)$-class and satisfies the reverse H\"{o}lder's condition.