We propose a level-set method for a mean curvature flow whose boundary is prescribed by interpreting the boundary as an obstacle. Since the corresponding obstacle problem is globally solvable, our method gives a global-in-time level-set mean curvature flow under a prescribed boundary with no restriction of the profile of an initial hyper-surface. We show that our solution agrees with a classical mean curvature flow under the Dirichlet condition.We moreover prove that our solution agrees with a level-set flow under the Dirichlet condition constructed by P.~Sternberg and W.~P.~Ziemer (1994), where the initial hyper-surface is contained in a strictly mean-convex domain and the prescribed boundary is on the boundary of the domain.

In this paper, we consider theChoquard-Kirchhoff involving fractional $p$-Laplace equation of the form: \begin{align*}M \Big (\varepsilon^{sp-N}[u]^p_{s,p}+\varepsilon^{-N}\int_{\mathbb{R}^{N}}V(x)|u|^pdx \Big )&\big(\varepsilon^{sp}(-\Delta)_p^su+V(x)|u|^{p-2}u)\big)\\& = \varepsilon^{\mu -N}(\frac {1}{|x|^\mu}*F(u))f(u) \end{align*} in $\mathbb{R}^N,$ where $\varepsilon > 0$ is a parameter, $s\in (0 , 1)$,$0 < \mu < ps$, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $M$ represents Kirchhoff function, $V$ is a continuous potential function and $f$ is a continuous function involvingsubcritical growth.With the help of well-known penalization methods and Ljusternik-Schnirelmann theory, we obtain the existence, multiplicity and concentration of solutions forthe above equations.

This paper deals with the following fractional Brezis-Nirenberg problem\begin{equation*}\begin{cases}(-\Delta)^s u=|u|^{2^{*}_s-2}u+\lambda u,\quad & \text{in}\ \Omega,\\u=0, & \text{in}\ \mathbb R^N\backslash\Omega,\end{cases}\end{equation*}where$2^{*}_s=\frac{2N}{N-2s}$, $s\in (0,\,1)$,$\Omega$ is a bounded smooth open connected set in$\mathbb R^N$,$0 < \lambda < \lambda_{1}$ and $\lambda_1$ is the first eigenvalue of fractional Laplacian$(-\Delta)^s$ under the condition $u=0$ in$\mathbb R^N\backslash\Omega$. We establish the existence of the least energy sign-changing solutions for$N\geq 5s$ to the above problem, which includes the lower dimensional case. Our results extend and improve the recent works on theexistence of sign-changing solutions established by Cora et al. in [9]and Guo et al. in [13] respectively.

We investigate existence of solutions to the general boundary problem forthe transport of particles (neutrons, photons, molecules of gas, etc.) in aplane parallel domain in Musielak-Orlicz type spaces avoiding growthrestrictions. The growth of the nonlinearities are controlled by ageneralized nonhomogeneous and anisotropic convex $j$-function. The approach does not require any particular type of growth condition of $j$ or itsconjugate $j^*$ (neither $\Delta_2)$. Thus, we extend the analysis ofboundary problems for the transport equation beyond the spaces $L^p$ inwhich the problems are well understood. We develop the study of a convexmodular and provide new extensions of Krasnoselskii fixed point theorem.