Related papers: Nonlocal Sublinear Elliptic Problems Involving Mea…

In this work we analyze the existence of solutions to the nonlinear elliptic system: \begin{equation*} \left\{ \begin{array}{rcll} -\Delta u & = & v^q+\a g & \text{in }\Omega , \\ -\Delta v& = &|\nabla u|^{p}+\l f &\text{in }\Omega , \\…

We give $L^p$ estimates for the second derivatives of weak solutions to the Dirichlet problem for equation $\Div(\mathbf{A}\nabla u) = f$ in $\Omega\subset \mathbb{R}^d$ with Sobolev coefficients. In particular, for $f\in L^2(\Omega)…

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \[ \mathcal{L}u = \sigma u^{q} + \mu \quad \text{in} \;\; \Omega, \] in the sublinear case $0<q<1$, with…

We study the homogenization of the equation $-A(\frac{\cdot}{\varepsilon}):D^2 u_{\varepsilon} = f$ posed in a bounded convex domain $\Omega\subset \mathbb{R}^n$ subject to a Dirichlet boundary condition and the numerical approximation of…

We study a class of fractional semilinear elliptic equations and formulate the corresponding Calder\'on problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order…

This paper examines the behavior of a positive solution $u\in C^{1,\alpha}(\Bar{\Omega})$ of the $(p,q)$ Laplace equation with a singular term and zero Dirichlet boundary condition. Specifically, we consider the equation: \begin{equation*}…

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.…

We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is…

We prove the monotonicity of positive solutions to the problem $-\Delta u = f(u)$ in $\mathbb{R}^N_+ := \{(x',x_N)\in\mathbb{R}^N \mid x_N>0 \}$ under zero Dirichlet boundary condition with a possible singular nonlinearity $f$. In some…

We consider elliptic equations of the form (E) $-Au=f(x,u)+\mu$, where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincreasing with respect to $u$ and $\mu$ is a Borel measure of…

In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of…

Let $M$ be a complete Riemannian manifold and $G$ a Lie subgroup of the isometry group of $M$ acting freely and properly on $M.$ We study the Dirichlet Problem \begin{align*} \operatorname{div}\left( \frac{a\left( \left\Vert \nabla…

We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: \[ -\triangle u +\mathrm{div}(u\mathbf{b}) =f \quad\text{ and }\quad -\triangle v -\mathbf{b} \cdot \nabla v =g \] in a bounded Lipschitz domain…

We consider the Lane-Emden Dirichlet problem \begin{equation}\tag{1} \left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }\Omega u=0\qquad\qquad\qquad\mbox{ on }\partial \Omega \end{array}\right. \end{equation} when $p>1$ and…

We consider the semilinear elliptic equation $-\Delta u =\lambda f(u)$ in a smooth bounded domain $\Omega$ of $R^{n}$ with Dirichielt boundary condition, where $f$ is a $C^{1}$ positive and nondeccreasing function in $[0,\infty)$ such that…

We study the periodic boundary value problem associated with the second order nonlinear equation \begin{equation*} u'' + ( \lambda a^{+}(t) - \mu a^{-}(t) ) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear…

We extend monotonicity-based inversion methods to an inverse coefficient problem for the isotropic nonlocal elliptic equation \[ (-\nabla \cdot \sigma \nabla)^s u = 0 \quad \text{in } \Omega \subset \mathbb{R}^n, \] where $0 < s < 1$, $n…

We derive a priori bounds for positive supersolutions of $ - \Delta_{p} u = \rho(x) f(u) $, where $p>1$ and $\Delta_{p}$ is the $p$-Laplace operator, in a smooth bounded domain of $R^{N}$ with zero Dirichlet boundary conditions. We apply…

We study the dependence of least nontrivial critical levels of the energy functional corresponding to the zero Dirichlet problem $-\Delta_p u = f(u)$ in a bounded domain $\Omega \subset \mathbb{R}^N$ upon domain perturbations. Assuming that…

In this work, we address the questions of existence, uniqueness, and boundary behavior of the positive weak-dual solution of equation $\mathbb{L}_\gamma^s u = \mathcal{F}(u)$, posed in a $C^2$ bounded domain $\Omega \subset \mathbb{R}^N$,…