Related papers: The Dirichlet problem for perturbed Stark operator…

Let $\Omega =\omega\times\mathbb R$ where $\omega\subset \mathbb R^2$ be a bounded domain, and $V : \Omega \to\mathbb R$ a bounded potential which is $2\pi$-periodic in the variable $x_{3}\in \mathbb R$. We study the inverse problem…

In this article, we study a model problem featuring a L\'evy process in a domain with semi-transparent boundary by considering the following perturbed fractional Laplacian operator \[\mathscr{L}_{b,q} := (-\Delta)^t +…

In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the problem given by \begin{equation*} \label{1} \left\{\begin{split} \mathcal{L}u\: &= \lambda u^{q} + u^{p}, \quad u>0 ~~ \text{in} ~\Omega,…

A Toeplitz operator $T_\varphi$, $\varphi \in L^\infty(\mathbb{T}^n)$, is a partial isometry if and only if there exist inner functions $\varphi_1, \varphi_2 \in H^\infty(\mathbb{D}^n)$ such that $\varphi_1$ and $\varphi_2$ depends on…

We consider the operator ${\mathcal A}_h=-\Delta+iV$ in the semi-classical $h\rightarrow 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the…

In this work, we study the two following minimization problems for $r \in \mathbb{N}^{*}$, \begin{equation*} \begin{array}{ccc} S_{0,r}(\varphi)=\displaystyle\inf_{u\in H_{0}^{r}(\Omega)\,|u+\varphi\|_{L^{2^{*r}}}=1}\|u\|_{r}^{2}&…

This paper studies a nonlinear Dirichlet problem for the $p$-Laplacian operator with nonlinearity consisting of power components. The problem under consideration can be though of as a perturbation of the Ambrosetti-Brezis-Cerami problem…

We consider divergence form elliptic operators L = - div A(x)\nabla, defined in the half space R^{n+1}_+, n \geq 2, where the coefficient matrix A(x) is bounded, measurable, uniformly elliptic, t-independent, and not necessarily symmetric.…

We consider the following class of fractional parametric problems \begin{equation*} \left\{ \begin{array}{ll} (-\Delta_{Dir})^{s} u= f(x, u)+t\varphi_{1}+h &\mbox{ in } \Omega\\ u=0 &\mbox{ on } \partial \Omega, \end{array} \right.…

After introducing the concept of functional dissipativity of the Dirichlet problem in a domain $\Omega\subset {\mathbb R}^N$ for systems of partial differential operators of the form $\partial_{h}({\mathscr A}^{hk}(x)\partial_{k})$…

We study the Dirichlet problem for an elliptic equation involving the $1$-Laplace operator and a reaction term, namely: $$ \left\{\begin{array}{ll} \displaystyle -\Delta_1 u =h(u)f(x)&\hbox{in }\Omega\,,\\ u=0&\hbox{on }\partial\Omega\,,…

In this article, we consider the spectral problem for the mixed local and nonlocal $p$-Laplace operator. We discuss the existence and regularity of eigenfunction of the associated Dirichlet $(p,q)$-eigenvalue problem in a bounded domain…

We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain $\Omega$ and on the domain $\phi(\Omega)$ resulting from $\Omega$ by means of a bi-Lipschitz…

In this paper we study the Cauchy problem for the semilinear damped wave equation for the sub-Laplacian on the Heisenberg group. In the case of the positive mass, we show the global in time well-posedness for small data for power like…

We consider a nonlinear parametric Dirichlet problem driven by the $p$-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carath\'eodory perturbation which is ($p-1$)-linear…

We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and…

We continue the study of the A-amplitude associated to a half-line Schrodinger operator, -d^2/dx^2+ q in L^2 ((0,b)), b <= infinity. A is related to the Weyl-Titchmarsh m-function via m(-\kappa^2) =-\kappa - \int_0^a A(\alpha)…

In this paper we study the following nonlocal Dirichlet equation of double phase type \begin{align*} -\psi \left [ \int_\Omega \left ( \frac{|\nabla u |^p}{p} + \mu(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x\right] \mathcal{G}(u) =…

In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $\textit{\textbf x}_0$ a fixed point in a bounded Lipschitz domain $\Omega$. Then there exists a constant $C > 0$ such that if $u$ is a harmonic function in…

For $s_1,s_2\in(0,1)$ and $p,q \in (1, \infty)$, we study the following nonlinear Dirichlet eigenvalue problem with parameters $\alpha, \beta \in \mathbb{R}$ driven by the sum of two nonlocal operators: \begin{equation*} (-\Delta)^{s_1}_p…