Related papers: Positive solutions to a supercritical elliptic pro…

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 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 , \\…

Given an isoparametric function $f$ on the $n$-dimensional round sphere, we consider functions of the form $u=w\circ f$ to reduce the semilinear elliptic problem \[ -\Delta_{g_0}u+\lambda u=\lambda\ | u\ | ^{p-1}u\qquad\text{ on…

We study the asymptotic behavior of a sequence of positive solutions $(u_{\epsilon})_{\epsilon >0}$ as $\epsilon \to 0$ to the family of equations \begin{equation*} \left\{\begin{array}{ll} \Delta u_{\epsilon}+a(x)u_{\epsilon}=…

In this paper, we prove that there exists at most one positive radial weak solution to the following quasilinear elliptic equation with singular critical growth \[ \begin{cases} -\Delta_{p}u-{\displaystyle…

We show uniqueness for overdetermined elliptic problems defined on topological disks $\Omega$ with $C^2$ boundary, i.e., positive solutions $u$ to $\Delta u + f(u)=0$ in $\Omega \subset (M^2,g)$ so that $u = 0$ and $\frac{\partial…

The authors of this paper deal with the existence and regularities of weak solutions to the homogenous $\hbox{Dirichlet}$ boundary value problem for the equation $-\hbox{div}(|\nabla u|^{p-2}\nabla u)+|u|^{p-2}u=\frac{f(x)}{u^{\alpha}}$.…

Consider the nonlinear heat equation $v_t -\Delta v= |v|^{p-1} v$ in a bounded smooth domain $\Omega\subset \R^n$ with $n>2$ and Dirichlet boundary condition. Given $u_{p}$ a sign-changing stationary solution fulfilling suitable…

We study the semilinear elliptic problem \[ -\Delta u = Q_{\Omega} |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where \( Q_{\Omega} = \chi_{\Omega} - \chi_{\mathbb{R}^N \setminus \Omega} \) for a bounded smooth domain \( \Omega \subset…

In the present paper we investigate the following semilinear singular elliptic problem: \begin{equation*} (\rm P)\qquad \left \{\begin{array}{l} -\Delta u = \dfrac{p(x)}{u^{\alpha}}\quad \text{in} \Omega \\ u = 0\ \text{on} \Omega,\ u>0…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a bounded smooth domain and $\delta(x)=\text{dist}(x,\partial \Omega)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$…

Let $L$ be a second order elliptic operator $L$ with smooth coefficients defined on a domain $\Omega $ in $\mathbb{R}^d $, $d\geq3$, such that $L1\leq 0$. We study existence and properties of continuous solutions to the following problem…

We study the problem of the existence and nonexistence of positive solutions to a superlinear second-order divergence type elliptic equation with measurable coefficients $(*)$: $-\nabla\cdot a\cdot\nabla u=u^p$ in an unbounded cone--like…

In this article, we study the following Hamiltonian system: \begin{equation*} \begin{cases} \begin{aligned} &-\varepsilon^{2}\Delta_{g} u +u = |v|^{q-1}v, &-\varepsilon^{2}\Delta_{g} v +v = |u|^{p-1}u && \text{ in } \mathcal{M}, & \quad u,v…

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and let $m:\Omega\rightarrow\mathbb{R}$ be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem $-\Delta…

For an integer $n \ge 3$ and any positive number $\epsilon$ we establish the existence of smooth functions K on $R^n \setminus \{0 \}$ with $|K - 1| \le \epsilon$, such that the equation $\Delta u + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$…

The Lane-Emden system is written as \begin{equation*} \begin{cases} -\Delta u = v^p &\text{in } \Omega,\\ -\Delta v = u^q &\text{in } \Omega,\\ u, v > 0 &\text{in } \Omega,\\ u = v = 0 &\text{on } \partial \Omega \end{cases} \end{equation*}…

We consider a nonautonomous semilinear elliptic problem where the power nonlinearity is multiplied by a discontinuous coefficient that equals one inside a bounded open set $\Omega$ and it equals minus one in its complement. In the slightly…

We are concerned with Dirichlet problems of the form $${\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f(u)=0\ \mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial\Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1<p<n$…

We consider the following supercritical problem for the Lane-Emden system: \begin{equation}\label{eq00} \begin{cases} -\Delta u_1=|u_2|^{p-1}u_2\ &in\ D,\\ -\Delta u_2=|u_1|^{q-1}u_1 \ &in\ D,\\ u_1=u_2=0\ &on\ \partial D, \end{cases}…