In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P) \begin{cases} -\Delta_{H,p}u =\frac{\lambda f(x)}{u^{q(x)}}+g(u) \ \text{ in }\Omega,\\ u > 0 \text{ in }\Omega,\ \ \ \ u=0\text{ on }\partial\Omega, \end{cases} \end{equation*} under the assumption $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with $p,N\geq 2$, $\lambda>0$ and $0 < q \in C(\bar \Omega)$. For the purely singular case that is $g\equiv 0$, we proved existence and uniqueness of solution. We also demonstrate the existence of multiple solution to $(P)$ provided $f\equiv 1$ and $g(u)=u^r$ for $r\in (p-1,p^*-1)$.

We present a blow-up rate estimate for a solution to the parabolic Gross-Pitaevskii and related systems on entire space with Sobolev subcritical nonlinearity. We extend the results of [Y. Giga, S. Matsui, and S. Sasayama, Indiana Univ. Math. J., {53} (2004), 483--514] to the parabolic systems.

This paper is concerned with the problem of prescribing the Paneitz curvature on the standard $n$-sphere, $n\geq 5$. We employ the critical points theory of A. Bahri [3] to establish an index-counting criteria for existence of solutions when the prescribed function is flat near its critical points for an order $\beta\in (1,n]$.

Let $X_{1},...,X_{m}$ be a family of real smooth vector fields defined in $\mathbb{R}^{n}$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying Hörmander's rank condition at $0$ (and therefore at every point of $\mathbb{R}^{n}$). The vector fields are not assumed to be translation invariant with respect to any Lie group structure. Let us consider the nonvariational evolution operator \[ \mathcal{H}:=\sum_{i,j=1}^{m}a_{i,j}(t,x)X_{i}X_{j}-\partial_{t} , \] where $(a_{i,j}(t,x))_{i,j=1}^{m}$ is a symmetric uniformly positive $m\times m$ matrix and the entries $a_{ij}$ are bounded Hölder continuous functions on $\mathbb{R}^{1+n}$, with respect to the ``parabolic'' distance induced by the vector fields. We prove the existence of a global heat kernel $\Gamma(\cdot;s,y)\in C_{X,\mathrm{loc}}^{2,\alpha}(\mathbb{R}^{1+n} \setminus\{(s,y)\})$ for $\mathcal{H}$, such that $\Gamma$ satisfies two-sided Gaussian bounds and $\partial_{t}\Gamma, X_{i}\Gamma,X_{i}X_{j}\Gamma$ satisfy upper Gaussian bounds on every strip $[0,T]\times\mathbb{R}^{n}$. We also prove a scale-invariant parabolic Harnack inequality for $\mathcal{H} $, and a standard Harnack inequality for the corresponding stationary operator \[ \mathcal{L}:=\sum_{i,j=1}^{m}a_{i,j}(x)X_{i}X_{j} \] with Hölder continuous coefficients.