Related papers: Overdetermined boundary value problems for the $\i…

Let $1<p<+\infty$ and let $\Omega\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the…

We study the family of operators $\{\mathcal{R}_a\}_{a\in [0,+\infty)}$ associated to the Robin-type problems in a bounded domain $\Omega\subset\mathbb{R}^2$ $$ \begin{cases} -\Delta u = f & \text{in } \Omega, \\ 2 \bar \nu \partial_{\bar…

In this paper, we study the boundary value problem of the classical semilinear parabolic equations $$ u_t-\Delta u=|u|^{p-1}u, \ \ in \ \ \Omega\times (0,T) $$ and $u=0$ on the boundary $\partial\Omega\times [0,T)$ and $u=\phi$ at $t=0$,…

In this paper we study a non-homogeneous Neumann problem, where the $p(x)$-Laplacian is involved and $p=\infty$ in a subdomain. By considering a suitable sequence $p_k$ of bounded variable exponents such that $p_k \to p$ and replacing $p$…

We prove a sharp $L^p$ weighted Hardy inequality involving boundary distance $\delta$ for any domain $\Omega\subsetneq \mathbb R^n$. The inequality may be improved substantially under the additional assumption that $-\log \delta$ is…

To our knowledge, this paper is the first attempt to consider the existence issue for fractional $p$-Laplacian equation: $(-\Delta)_p^s u= \lambda f(u),\; u> 0 ~\text{in}~\Omega;\; u=0\;\text{in}~ \mathbb{R}^N\setminus\Omega$, where $p>1$,…

We study some Dirichlet problem for a $p$--Laplacian type operator in the setting of Orlicz--Zygmund space $L^q\log^{-\alpha}L(\Omega,\mathbb R^N)$, $q >1$ and $\alpha>0$. More precisely, our aim is to establish which assuptions on the…

Let $N\geq 1$ and $s\in (0,1)$. In the present work we characterize bounded open sets $\Omega$ with $ C^2$ boundary (\textit{not necessarily connected}) for which the following overdetermined problem \begin{equation*} ( -\Delta)^s u = f(u)…

Given a bounded $(\epsilon,\delta)$-domain $\Omega\subseteq\mathbb{R\!}^N$ ($N\geq2$) whose boundary $\Gamma:=\partial\Omega$ is a $d$-set for $d\in(N-p,N)$, we investigate a generalized quasi-linear elliptic boundary value problem governed…

We study a class of boundary value problems with $\varphi$-Laplacian (e.g., the prescribed mean curvature equation, in which $\varphi(s)=\frac{s}{\sqrt{1+s^2}}$) \begin{center} $-\left(\varphi(u')\right)'=\lambda f(u)\; \text{ on }(-L,…

This paper deals with the obstacle problem for the infinity Laplacian. The main results are a characterization of the solution through comparison with cones that lie above the obstacle and the sharp $C^{1,1/3}$--regularity at the free…

In this paper, we study eigenvalues of the poly-Laplacian with arbitrary order on a bounded domain in an $n$-dimensional Euclidean space and obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine…

We investigate the multiplicity of solutions for a generalized poly-Laplacian system on weighted finite graphs and a generalized poly-Laplacian system with Dirichlet boundary value on weighted locally finite graphs, respectively, via the…

Let $\Omega$ be a bounded open interval, let $p>1$ and $\gamma>0$, and let $m:\Omega\rightarrow\mathbb{R}$ be a function that may change sign in $\Omega $. In this article we study the existence and nonexistence of positive solutions for…

In this paper, we study a partially overdetermined mixed boundary value problem in a half ball. We prove that a domain in which this partially overdetermined problem admits a solution if and only if the domain is a spherical cap…

We deal with eigenvalue problems for the Laplacian on noncompact Riemannian manifolds $M$ of finite volume. Sharp conditions ensuring $L^q(M)$ and $L^\infty (M)$ bounds for eigenfunctions are exhibited in terms of either the isoperimetric…

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain and let $\lambda_1, \lambda_2, \dots$ denote the sequence of eigenvalues of the Laplacian subject to Dirichlet boundary conditions. We consider inequalities for $\lambda_n$ that are…

We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contains local elliptic boundary…

We study a class of fractional $p$-Laplacian problems with weights which are possibly singular on the boundary of the domain. We provide existence and multiplicity results as well as characterizations of critical groups and related…

Assume that $p > 1$ and $p - 1 \le \alpha \le p$ are real numbers and $\Omega$ is a non-empty open subset of ${\mathbb R}^n$, $n \ge 2$. We consider the inequality $$ {\rm div} \, A (x, D u) + b (x) |D u|^\alpha \ge 0, $$ where $D =…