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Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $i_0\in\mathbb{Z}^+$, $\Omega\subset\mathbb{R}^n$ be a smooth bounded domain, $a_1,a_2,\dots,a_{i_0}\in\Omega$, $\widehat{\Omega}=\Omega\setminus\{a_1,a_2,\dots,a_{i_0}\}$, $0\le f\in…

This paper is concerned with the existence and decay of solutions of the following Timoshenko system: $$ \left\|\begin{array}{cc} u"-\mu(t)\Delta u+\alpha_1 \displaystyle\sum_{i=1}^{n}\frac{\partial v}{\partial x_{i}}=0,\, \in \Omega\times…

In this paper we use the dynamical methods to establish the existence of nontrivial solution for a class of nonlocal problem of the type $$ \left\{\begin{array}{l} -a\left(x,\int_{\Omega}g(u)\,dx \right)\Delta u =f(u), \quad x \in \Omega \\…

In a smoothly bounded domain $\Omega \subset \mathbb{R}^N$ $(N\in \mathbb{N})$, a no-flux initial-boundary value problem for the degenerate chemotaxis system with volume-filling effects, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u -…

We consider the class of semi-stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain $\Omega$ of $R^n$ (with $\Omega$ convex in some results). This class includes all local minimizers, minimal, and extremal…

We investigate an overdetermined Torsion problem, with a non-constant positively homogeneous boundary constraint on the gradient. We interpret this problem as the Euler equation of a shape optimization problems, we prove existence and…

In this paper, we prove the existence of $k$ families of smooth unbounded domains $\Omega_s\subset\mathbb{R}^{N+1}$ with $N\geq1$, where \begin{equation} \Omega_s=\left\{(x,t)\in \mathbb{R}^N\times \mathbb{R}:\vert x\vert<1+s\cos…

We present a general $L_p$-solvability framework for both the classical and time-fractional heat equations in non-smooth domains under the zero Dirichlet boundary condition. We consider domains $\Omega$ admitting the Hardy inequality: There…

We consider a boundary value problem of a stationary advection equation with the homogeneous inflow boundary condition in a bounded domain with Lipschitz boundary, and consider its perturbation by $\epsilon \Delta$, where $\epsilon$ is a…

We prove a multiplicity result for non-constant weak solutions $u \in H^1(\Omega)$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) -…

We consider a hyperbolic ordinary differential equation perturbed by a nonlinearity which can be singular at a point and in particular this includes MEMS type equations. We first study qualitative properties of the solution to the…

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In this paper, we consider the following nonlocal parabolic equation \begin{equation*} u_{t}-\Delta u=\left( \int_{\Omega}\frac{|u(y,t)|^{2^{\ast}_{\mu}}}{|x-y|^{\mu}}dy\right) |u|^{2^{\ast}_{\mu}-2}u,\ \text{in}\ \Omega\times(0,\infty),…

We prove the existence of two smooth families of unbounded domains in $\mathbb{R}^{N+1}$ with $N\geq1$ such that \begin{equation} -\Delta u=\lambda u\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu…

We prove the existence of solutions $(\lambda, v)\in \mathbb{R}\times H^{1}(\Omega)$ of the elliptic problem \[ \begin{cases} -\Delta v+(V(x)+\lambda) v =v^{p}\ &\text{ in $ \Omega, $} \ v>0,\qquad \int_\Omega v^2\,dx =\rho. \end{cases} \]…

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We obtain a general sufficient condition on the geometry of possibly singular planar domains that guarantees global uniqueness for any weak solution to the Euler equations on them whose vorticity is bounded and initially constant near the…

We consider a nonlinear parabolic model that forces solutions to stay on a $L^2$-sphere through a nonlocal term in the equation. We study the local and global well-posedness on a bounded domain and the whole Euclidean space in the energy…

We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter…