Related papers: Optimal uniform bounds for competing variational e…

We consider an equation $$ L_{\alpha ,\beta ,\gamma} (u) \equiv u_{xx} + u_{yy} + u_{zz} + \displaystyle \frac{{2\alpha}}{x}u_x + \displaystyle \frac{{2\beta}}{y}u_y + \displaystyle \frac{{2\gamma}}{z}u_z = 0 $$ in a domain ${\bf R}_3^ +…

In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system {lll} -\Delta_{p(x)}u = a(x)|u|^{p(x)-2}u - b(x)|u|^{\alpha(x)}|v|^{\beta(x)} v + f(x) in \Omega, \Delta_{q(x)}v = c(x) |v|^{q(x)-2}v -…

We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain $\Omega$, where $\Omega$ is either in $\mathbb{R}^n$ or in a Riemannian manifold. For linear systems of equations arising from low-order…

Given a Lipschitz domain $\Omega $ in ${\mathbb R} ^N $ and a nonnegative potential $V$ in $\Omega $ such that $V(x)\, d(x,\partial \Omega)^2$ is bounded in $\Omega $ we study the fine regularity of boundary points with respect to the…

We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -\mathrm{div }(A\nabla u)+\mathbf{b} \cdot \nabla u+\lambda u=f+\mathrm{div } \mathbf{F}\quad \text{in }…

In this paper we consider positive supersolutions of the nonlinear elliptic equation \[- \Delta u = \rho(x) f(u)|\nabla u|^p, \qquad \hfill \mbox{ in } \Omega,\] where $0\le p<1$, $ \Omega$ is an arbitrary domain (bounded or unbounded) in $…

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline…

We study positive solutions of semilinear elliptic equations in a planar triangular domain under mixed boundary conditions, consisting of homogeneous Dirichlet boundary conditions on one side and homogeneous Neumann boundary conditions on…

Let $\Omega\subset \mathbb{R}^N$ be an open bounded domain and $m\in \mathbb{N}$. Given $k_1,\ldots,k_m\in \mathbb{N}$, we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, of the…

In this paper we investigate the existence of solutions to the following Schr\"{o}dinger system in the critical case \begin{equation*} -\Delta u_{i}+\lambda_{i}u_{i}=u_{i}\sum_{j = 1}^{d}\beta_{ij}u_{j}^{2} \text{ in } \Omega, \quad u_i=0…

We establish the existence and uniqueness of the solution to the Dirichlet problem for the variable exponent $p$-Laplacian on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$, where the boundary datum belongs to $W^{1,p}(\Omega)$.…

We investigate optimal control problems governed by the elliptic partial differential equation $-\Delta u=f$ subject to Dirichlet boundary conditions on a given domain $\Omega$. The control variable in this setting is the right-hand side…

Let $\textbf{A}$ be a symmetric convex quadratic form on $\mathbb{R}^{Nn}$ and $\Omega\Subset \mathbb{R}^n$ a bounded convex domain. We consider the problem of existence of solutions $u: \Omega \subset \mathbb{R}^n \longrightarrow…

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: \begin{align*} \begin{cases} -div(A(x,\nabla u)) &= \mu \quad \text{in} \ \ \Omega, \\ u &=0 \quad…

In this paper, we study a solvability result for the nonlinear problem $$ \mbox {div } \left ( \vert \nabla_\omega u\vert^{p-2}\nabla_\omega u \right )+v(x) u^{q-1}+\mu u^{\gamma-1}=0, \quad z\in \Omega, \quad u \Big \vert_{\partial…

For all functions on an arbitrary open set $\Omega\subset\R^3$ with zero boundary values, we prove the optimal bound \[ \sup_{\Omega}|u| \leq (2\pi)^{-1/2} \left(\int_{\Omega}|\nabla u|^2 \,dx\, \int_{\Omega}|\Delta u|^2 \,dx\right)^{1/4}.…

Let $\Omega\subset \mathbb{R}^{N}$ be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schr\"{o}dinger system with $d\geq 2$ equations \begin{equation*} -\Delta…

In this paper we investigate a class of variational reaction-diffusion systems with strong competition driven by beyond-pairwise interactions. The model involves $d$ nonnegative components interacting through $k$-wise terms, with $3 \leq k…

We prove the existence of positive solutions with optimal local regularity to homogeneous elliptic equations of Schr\"{o}dinger type, under only a form boundedness assumption on $\sigma \in D'(\Omega)$ and ellipticity assumption on…

A family of interior penalty $hp$-discontinuous Galerkin methods is developed and analyzed for the numerical solution of the quasilinear elliptic equation $-\nabla{} \cdot (\mathbf{A}(\nabla{u}) \nabla{u} = f$ posed on the open bounded…