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Given a smooth compact k-dimensional manifold \Lambda embedded in $\mathbb {R}^m$, with m\geq 2 and 1\leq k\leq m-1, and given \epsilon>0, we define B_\epsilon (\Lambda) to be the geodesic tubular neighborhood of radius \epsilon about…

In this paper, we investigate the existence and uniqueness of solutions for the following model problem, involving singularities and inhomogeneous Robin boundary conditions \begin{equation*} \left\{ \begin{array}{ll}…

We have considered the following semi linear elliptic problem on the unit disk $B$ $-\Delta u = \lambda_1 u+e^u+f $ in $B$ with the Dirichlet boundary condition and $f$ satisfying the following condition : $f\in L^r(B)$, for some $r>2$ and…

We study existence and non-existence of solutions for singular elliptic boundary value problems as \begin{equation}\label{eintro}\begin{cases}\tag{1} \displaystyle -\Delta_p u+ \frac{a(x)}{u^{\gamma}}=\mu f(x) \ &\text{ in }\Omega, \newline…

We consider the following elliptic system with Neumann boundary: \begin{equation} \begin{cases} -\Delta u + \mu u=v^p, &\hbox{in } \Omega, \\-\Delta v + \mu v=u^q, &\hbox{in } \Omega, \\\frac{\partial u}{\partial n} = \frac{\partial…

This article establishes the boundary H\"{o}lder continuity of stable solutions to semilinear elliptic problems in the optimal range of dimensions $n \leq 9$, for $C^{1,1}$ domains. We consider equations $- L u = f(u)$ in a bounded…

This paper is devoted to the study of semi-stable radial solutions $u\in H^1(B_1)$ of $-\Delta u=g(u) {in} B_1\setminus \{0\}$, where $g\in C^1(R)$ is a general nonlinearity and $B_1$ is the unit ball of $R^N$. We establish sharp pointwise…

We will investigate the asymptotic behavior of positive solutions of the elliptic equation \Delta u+|x|^{l_{1}}u^{p}+|x|^{l_{2}}u^{q}=0 {in} R^{n}. We establish that for $n\geq 3$ and $q>p>1$, any positive radial solution of (0.1) has the…

We consider semilinear elliptic problems of the form \[ -\Delta u + \lambda u = f(x,u), \quad u\in H^1_0(A), \] where $A\subset\mathbb{R}^N$, $N\geq3$, is either a bounded or unbounded annulus, and $\lambda \geq0$. We study a broad class of…

In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem \begin{align*} \begin{split} (-\Delta)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+\mu|u|^{-\gamma-1}u,~\text{in}~\mathbb{R},…

Let $\Omega \subset\mathbb{R}^N$ ($N\geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \partial\Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$, $0\leq k \leq N-1$. We assume that $\Sigma = \{0\}$ if $k = 0$ and…

In this paper we prove radial symmetry for solutions to a free boundary problem with a singular right hand side, in both elliptic and parabolic regime. More exactly, in the unit ball $B_1$ we consider a solution to the fully nonlinear…

In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^{\alpha}, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n…

In this paper we are mainly concerned with nontrivial positive solutions to the Dirichlet problem for the degenerate elliptic equation \begin{gather} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial…

We study semilinear elliptic equations \begin{equation*} \begin{cases} -\Delta u = f(u) & \text{in } \Omega, \\ \partial_\nu u = 0 & \text{on } \partial\Omega, \end{cases} \end{equation*} with homogeneous Neumann boundary conditions in…

We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem $\Delta u=u^{\gamma-1}$, with $\gamma\in(0,1)$. Our main results imply that, once free boundaries are $C^{1,\alpha}$, then they are $C^\infty$. In…

We study the fully nonlocal semilinear equation $\partial_t^\alpha u+(-\Delta)^\beta u=|u|^{p-1}u$, $p\ge1$, where $\partial_t^\alpha$ stands for the Caputo derivative of order $\alpha\in (0,1)$ and $(-\Delta)^\beta$, $\beta\in(0,1]$, is…

We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form $Lu=\text{div}(A \nabla u)-u_t=0$ in Lip$(1,1/2)$ time-varying cylinders, where the coefficient matrix $A = \left[…

In this paper, we study the existence and the summability of solutions to a Robin boundary value problem whose prototype is the following: $$ \begin{cases} -\text{div}(b(|u|)\nabla u)=f &\text{in }\Omega,\\[.2cm] \displaystyle\frac{\partial…

We consider positive solutions of the following elliptic Hamiltonian systems \begin{equation} \left\{ \begin{aligned} -\Delta u+u&=a(x)v^{p-1}~~~\text{in}~~A_R\\ -\Delta v+v&=b(x)u^{q-1}~~~\text{in}~~A_R~~~~~~~~~~~~~~~~~(0.1)\\ u,…