相关论文: Supercritical elliptic problems on the round spher…

Prescribing conformally the scalar curvature of a Riemannian manifold as a given function consists in solving an elliptic PDE involving the critical Sobolev exponent. One way of attacking this problem consist in using subcritical…

We consider the following elliptic system with Neumann boundary: \begin{equation} \begin{cases} -\Delta u + \mu u=v^p, &\hbox{in } \Omega, \\-\Delta v + \mu v=u^q, &\hbox{in } \Omega, \\\frac{\partial u}{\partial n} = \frac{\partial…

We consider in this paper the nonlinear elliptic equation with Neumann boundary condition \begin{align*} \begin{cases} \Delta u=a|u|^{m-1}u\,\,\mbox{ in }\,\,\rnp\\ \dfrac{\partial u}{\partial t}=b|u|^{\eta-1}u+f\,\,\mbox{ on…

In this paper we study the problem -\Delta u =\left(\frac{2+\alpha}{2}\right)^2\abs{x}^{\alpha}f(\lambda,u), & \hbox{in}B_1 \\ u > 0, & \hbox{in}B_1 u = 0, & \hbox{on} \partial B_1 where $B_1$ is the unit ball of $\R^2$, $f$ is a smooth…

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…

In this work, we study local minimizers of elliptic functionals with strong absorption terms and unbounded, sign-changing sources. These problems naturally interpolate between two classical free boundary problems: Bernoulli-type (cavity)…

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and $\delta(x)$ be the distance of a point $x\in \Omega$ to the boundary. We study the positive solutions of the problem $\Delta u +\frac{\mu}{\delta(x)^2}u=u^p$ in $\Omega$, where $p>0,…

We study the existence and the profile of sign-changing solutions to the slightly subcritical problem $$ -\De u=|u|^{2^*-2-\eps}u \hbox{in} \cB, \quad u=0 \hbox{on}\partial \cB, $$ where $\cB$ is the unit ball in $\rr^N$, $N\geq 3$,…

In this work we prove the existence of infinitely many nonradial solutions that change signal to the problem $-\Delta u=f(u)$ in $B$ with $u=0$ on $\partial B$, where $B$ is the unit ball in $\mathbb{R}^2$ and $f$ is a continuous and odd…

We deals with nonlinear elliptic Dirichlet problems of the form $${\rm div}(|D u|^{p-2}D u )+f(u)=0\quad\mbox{ in }\Omega,\qquad u\in H^{1,p}_0(\Omega) $$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\ge 2$, $p> 1$ and $f$ has…

In a recent paper, we established optimal Liouville-type theorems for conformally invariant second-order elliptic equations in the Euclidean space. In this work, we prove an optimal Liouville-type theorem for these equations in the…

We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with…

Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $\Delta u = f(u)$ on…

We prove the existence of solutions for the following critical Choquard type problem with a variable-order fractional Laplacian and a variable singular exponent \begin{align*} \begin{split} a(-\Delta)^{s(\cdot)}u+b(-\Delta)u&=\lambda…

We examine the H\'enon equation $ -\Delta u =|x|^\alpha u^p$ in $ \Omega \subset \mathbb{R}^N$ with $u=0$ on $ \partial \Omega$ where $ 0 < \alpha$. We show there exists a sequence $ \{p_k\}_k \subset [ \frac{N+2}{N-2}, p_{\alpha}(N)]$ with…

We consider positive solutions of the following elliptic Hamiltonian systems \begin{equation} \left\{ \begin{aligned} -\Delta u+u&=a(x)v^{p-1}~~~\text{in}~~A_R\\ -\Delta v+v&=b(x)u^{q-1}~~~\text{in}~~A_R~~~~~~~~~~~~~~~~~(0.1)\\ u,…

We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert-Einstein functional…

In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the…

In this paper we study the existence of solutions of a one-dimensional eigenvalue problem $-\left(|\phi_x|^{p-2}\phi_x\right)_x=\lambda \left(|\phi|^{q-2}\phi-f(\phi)\right)$ such that $\phi(0)=\phi(1)=0$, where $p,q>1$, $\lambda$ is a…

In these notes we study the Dirichlet problem for critical points of a convex functional of the form \[ F(u)=\int_{\Omega}\phi\left( \left\vert \nabla u\right\vert \right) , \] where $\Omega$ is a bounded domain of a complete Riemannian…