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We consider a number of boundary value problems involving the $p$-Laplacian. The model case is $-\Delta_p u=V|u|^{p-2}u$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R}^n$. We derive necessary conditions for the existence of…

In this paper we study a partially overdetermined mixed boundary value problem for domains $\Omega$ contained in an unbounded set $\mathcal C$. We introduce the notion of Cheeger set relative to $\mathcal C$ and show that if a domain…

We study a fractional analogue of a plasma problem arising from physics. Specifically, for a fixed bounded domain $\Omega$ we study solutions to the eigenfunction equation \[ (- \Delta)^s u = \lambda(u- \gamma)_+ \] with $u \equiv 0$ on…

The aim of this paper is to obtain the existence of solution for the fractional p-Laplacian Dirichlet problem with mixed derivatives \begin{eqnarray*} &{_{t}}D_{T}^{\alpha}\left(|_{0}D_{t}^{\alpha}u(t))|^{p-2}{_{0}}D_{t}^{\alpha}u(t)\right)…

We study the existence of sign-changing solutions with multiple bubbles to the slightly subcritical problem $$-\Delta u=|u|^{2^*-2-\e}u \hbox{in}\Omega, \quad u=0 \hbox{on}\partial \Omega,$$ where $\Omega$ is a smooth bounded domain in…

We prove a result of existence of positive solutions of the Dirichlet problem for $-\Delta_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$, where $\Delta_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight…

We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equation \[ -\Delta_\infty u - \beta |Du| = f, \] subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions. Our arguments rely…

In this paper, we discuss the existence and uniqueness of solutions of a boundary value problem for a fractional differential equation of order $\alpha\in(2,3)$, involving a general form of fractional derivative. First, we prove an…

We are looking for an optimal convex domain on which the boundary value problem $$\left\{\begin{array}{cc}(-\Delta)^2 u_\gamma-\gamma\Delta u_\gamma = f,& \mbox{ in }\Omega\\ u_\gamma=\partial_\nu u_\gamma=0,& \mbox{ on…

Let $\Omega \subset \mathbb{R}^n$, for $n \geq 2$, be a bounded $C^2$ domain. Let $q \in L^1_{loc} (\Omega)$ with $q \geq 0$. We give necessary conditions and matching sufficient conditions, which differ only in the constants involved, for…

In this paper we prove existence and uniqueness results for nonlinear parabolic problems with Dirichlet boundary values whose model is \[ \left\{ \begin{aligned} &b(u)_t-\Delta_{p}u=\mu\;\mbox{in }(0,T)\times\Omega,\\…

We look for a solutions to a nonlinear, fractional Schr\"odinger equation $$(-\Delta)^{\alpha / 2}u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where potential $V$ is coercive or $V=V_{per} + V_{loc}$ is a sum of periodic…

Let $\Omega $ be a bounded domain in $\mathbb{R} ^N $, and let $u\in C^1 (\overline{\Omega }) $ be a weak solution of the following overdetermined BVP: $-\nabla (g(|\nabla u|)|\nabla u|^{-1} \nabla u )=f(|x|,u)$, $ u>0 $ in $\Omega $ and…

This paper deals with the Neumann boundary value problem for the system $$u_t=\nabla\cdot\left(D(u)\nabla u\right)-\nabla\cdot\left(S(u)\nabla v\right)+f(u) ,\quad x\in\Omega,\ t>0$$ $$v_t=\Delta v-v+u,\quad x\in\Omega,\ t>0$$ in a smooth…

In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation $\frac{\partial u}{\partial t} = \Delta\left(\Delta u\right)^{-3}$. This problem arises in the mathematical modeling…

In the unit ball B(0,1), let $u$ and $\Omega$ (a domain in $\R$) solve the following overdetermined problem: $$\Delta u =\chi_\Omega\quad \hbox{in} B(0,1), \qquad 0 \in \partial \Omega, \qquad u=|\nabla u |=0 \quad \hbox{in} B(0,1)\setminus…

Using Leray-Schauder degree or degree for $\alpha$-condensing maps we obtain the existence of at least one solution for the boundary value problem of the type \[ \left\{\begin{array}{lll} (\varphi(u' ))' = f(t,u,u') & & \\ u(T)=0=u'(0), & &…

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…

As the main problem, the bi-Laplace equation $\Delta^2u=0 (\Delta=D_x^2+D_y^2)$ in a bounded domain $\Omega \subset \re^2$, with inhomogeneous Dirichlet or Navier-type conditions on the smooth boundary $\partial \Omega$ is considered. In…

This paper is concerned with the numerical approximation of the $L^2$ Dirichlet eigenpairs of the operator $-\Delta + V$ on a simply connected $C^2$ bounded domain $\Omega \subset \mathbb{R}^2$ containing the origin, where $V$ is a radial…