Related papers: Isolated Singularities of Polyharmonic Operator in…

We are concerned with the study of the twin non-local inequalities featuring non-homogeneous differential operators $$\displaystyle -\Delta^2 u + \lambda\Delta u \geq (K_{\alpha, \beta} * u^p)u^q \quad\text{ in } \mathbb{R}^N (N\geq 1),$$…

Let $\Omega$ be a bounded $C^2$ domain in $\mathbb{R}^n$ and $u\in C(\mathbb{R}^n)$ solves \begin{equation*} \begin{aligned} \Delta u + a Iu + C_0|Du| \geq -K\quad \text{in}\; \Omega, \quad \Delta u + a Iu - C_0|Du|\leq K \quad \text{in}\;…

Let us consider a semilinear boundary value problem $ - \Delta u= f(x,u),$ in $\Omega,$ with Dirichlet boundary conditions, where $ \Omega \subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions…

We study stable solutions to fractional semilinear equations $(-\Delta)^s u = f(u)$ in $\Omega \subset \mathbb{R}^n$, for convex nonlinearities $f$, and under the Dirichlet exterior condition $u=g$ in $\mathbb{R}^n \setminus \Omega$ with…

The aim of this paper is to prove the existence of weak solutions to the equation $\Delta u + u^p = 0$ which are positive in a domain $\Omega \subset {\Bbb R}^N$, vanish at the boundary, and have prescribed isolated singularities. The…

In this paper we show the uniqueness of the critical point for \emph{semi-stable} solutions of the problem $$\begin{cases} -\Delta u=f(u)&\text{in }\Omega\\ u>0&\text{in }\Omega\\ u=0&\text{on } \partial\Omega,\end{cases}$$ where…

The biharmonic supercritical equation $\Delta^2u=|u|^{p-1}u$, where $n>4$ and $p>(n+4)/(n-4)$, is studied in the whole space $\mathbb{R}^n$ as well as in a modified form with $\lambda(1+u)^p$ as right-hand-side with an additional eigenvalue…

We are concerned with Dirichlet problems of the form $${\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f(u)=0\ \mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial\Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1<p<n$…

In this article we prove the Pohozaev identity for the semilinear Dirichlet problem of the form $-\Delta u + a(-\Delta)^s u = f(u)$ in $\Omega$, and $u=0$ in $\Omega^c$, where $a$ is a non-negative constant and $\Omega$ is a bounded $C^2$…

In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity \begin{equation*} \quad (P_{t}^s) \left\{ \begin{split} \quad u_t + (-\Delta)^s u &= u^{-q} + f(x,u), \;u >0\;…

We consider the equation $-\epsilon^{2}\Delta u + u = u^ {p}$ in a bounded domain $\Omega\subset\R^{3}$ with edges. We impose Neumann boundary conditions, assuming $1<p<5$, and prove concentration of solutions at suitable points of…

We study the problem $(-\Delta)^su=\lambda e^u$ in a bounded domain $\Omega\subset\mathbb R^n$, where $\lambda$ is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result…

In 1978 E. De Giorgi formulated a conjecture concerning the one-dimensional symmetry of bounded solutions to the elliptic equation \Delta u=F'(u), which are monotone in some direction. In this paper we prove the analogous statement for the…

We study the interior regularity of solutions to the Dirichlet problem $Lu=g$ in $\Omega$, $u=0$ in $\R^n\setminus\Omega$, for anisotropic operators of fractional type $$ Lu(x)= \int_{0}^{+\infty}\,d\rho \int_{S^{n-1}}\,da(\omega)\, \frac{…

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by $$ -\Delta u= h(u){f} \ \ \text{in}\,\ \Omega, $$ where $f$ is an irregular datum,…

We study the uniqueness problem of $\sigma$-regular solution of the equation, $$-\Delta_p u+ \abs u^{q-1}u =h \quad on\quad \RN, $$ where $q>p-1>0.$ and $N> p.$ Other coercive type equations associated to more general differential operators…

The classical result of A. Ambrosetti and G. Prodi [1], in the form of M.S. Berger and E. Podolak [4], gives the exact number of solutions for the problem \[ \Delta u+g(u)= \mu \phi _1(x)+e(x) \;\; \mbox{in $D$} , \;\; u=0 \;\; \mbox{on…

In this paper, we prove the existence and uniqueness of $W^{2,p}$ ($n<p<\infty$) solutions of a double obstacle problem with $C^{1,1}$ obstacle functions. Moreover, we show the optimal regularity of the solution and the local $C^1$…

We investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right…

We consider the semilinear Dirichlet problem \[ \Delta u+kg(u)=\mu _1 \varphi _1+\cdots +\mu _n \varphi _n+e(x) \;\; \mbox{for $x \in \Omega$}, \;\; u=0 \;\; \mbox{on $\partial \Omega$}, \] where $\varphi _k$ is the $k$-th eigenfunction of…