Related papers: Boundary value problems for semilinear Schr\"oding…

We study boundary value problems for semilinear elliptic equations of the form $-\Delta u+g\circ u=\mu$ in a smooth bounded domain $\Omega\subset R^N$. Let $\{\mu_n\}$ and $\{\tau_n\}$ be sequences of measure in $\Omega$ and $\partial…

We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_\mu u+u^q=0$ in a bounded smooth domain $\Omega\subset \mathbb R^N$. Here $q>1$, $L_\mu=\Delta+\frac{\mu}{\delta_\Omega^2}$ and…

Consider operators $L_{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega\subset \mathbb{R}^N$. Assume that $V\in C^\alpha(\Omega)$ satisfies $|V(x)| \leq \bar a\,\mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and that $L_V$ has a…

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $F \subset \partial \Omega$ be a $C^2$ submanifold of dimension $0 \leq k \leq N-2$. Put $\delta_F(x)=dist(x,F)$, $V=\delta_F^{-2}$ in $\Omega$ and $L_{\gamma…

Let $\Omega \subset \mathbb{R}^N$ ($N>2$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in $\Omega…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in…

We study the boundary value problem with Radon measures for nonnegative solutions of $-\Delta u+Vu=0$ in a bounded smooth domain $\Gw$, when $V$ is a locally bounded nonnegative function. Introducing some specific capacity, we give…

We study the boundary value problem with Radon measures for nonnegative solutions of $L_Vu:=-\Delta u+Vu=0$ in a bounded smooth domain $\Gw$, when $V$ is a locally bounded nonnegative function. Introducing some specific capacity, we give…

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…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ is a $C^2$ compact boundaryless submanifold in $\mathbb{R}^N$ of dimension $k$, $0\leq k < N-2$. For $\mu\leq (\frac{N-k-2}{2})^2$, put…

We study the problem of finding a function u verifying --$\Delta$u = 0 in $\Omega$ under the boundary condition $\partial$u $\partial$n + g(u) = $\mu$ on $\partial$$\Omega$ where $\Omega$ $\subset$ R N is a smooth domain, n the normal unit…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ and $\delta(x)=\text{dist}\,(x,\partial \Omega)$. Assume $\mu>0$, $\nu$ is a nonnegative finite measure on $\partial \Omega$ and $g \in C(\Omega \times \mathbb{R}_+)$. We study…

We study existence and uniqueness of solutions of (E 1) --$\Delta$u + $\mu$ |x| ^{-2} u + g(u) = $\nu$ in $\Omega$, u = $\lambda$ on $\partial$$\Omega$, where $\Omega$ $\subset$ R N + is a bounded smooth domain such that 0 $\in$…

We consider the steady fractional Schr\"odinger equation $L u + V u = f$ posed on a bounded domain $\Omega$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-\Delta)^s$; $V\ge 0$ is a potential…

Let $\Omega\subset\BBR^N$ be a bounded $C^2$ domain and $\CL_\gk=-\Gd-\frac{\gk}{d^2}$ the Hardy operator where $d=\dist (.,\prt\Gw)$ and $0<\gk\leq\frac{1}{4}$. Let $\ga_{\pm}=1\pm\sqrt{1-4\gk}$ be the two Hardy exponents, $\gl_\gk$ the…

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^N$. We study positive solutions of equation (E) $-L_\mu u+ u^q = 0$ in $\Omega$ where $L_\mu=\Delta + \frac{\mu}{\delta^2}$, $0<\mu$, $q>1$ and…

The boundary value problem is examined for the system of elliptic equations of from $-\Delta u + A(x)u = 0 \quad\text{in} \Omega,$ where $A(x)$ is positive semidefinite matrix on $\mathbb{R}^{{k}\times{k}},$ and $\frac{\partial u}{\partial…

We prove the existence of a solution of (--$\Delta$) s u + f (u) = 0 in a smooth bounded domain $\Omega$ with a prescribed boundary value $\mu$ in the class of positive Radon measures for a large class of continuous functions f satisfying a…

We study the generalized boundary value problem for nonnegative solutions of of $-\Delta u+g(u)=0$ in a bounded Lipschitz domain $\Omega$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Omega$, we define a trace…

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 $t=0$,…