相关论文: Asymptotic behavior of extremals for fractional So…

Let $\Omega$ be a bounded domain of $\mathbf{R}^{N},$ $N\geq2.$ Let, for $p>N,$ \[ \Lambda_{p}(\Omega):=\inf\left\{ \left\Vert \nabla u\right\Vert _{p}^{p}:u\in W_{0}^{1,p}(\Omega)\quad and\quad\left\Vert u\right\Vert _{\infty}=1\right\} .…

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $p>N$ and $1\leq q(p)<\infty$ set \[ \lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in W_{0}^{1,p}(\Omega)\text{ \ and…

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain (open and connected) in $\mathbb{R}^n$. Given $u_0\in L^2(\Omega)$, $g\in L^\infty(\Omega)$ and $\lambda \in \mathbb{R}$, our purpose is to describe the asymptotic behavior of…

We study a minimizing problem associated with the singular problem \[ \left\{ \begin{array} [c]{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =\lambda u^{-1} & \mathrm{in\ }\Omega\\ u>0 & \mathrm{in\…

\[ \left\{ \begin{array} [c]{lll} -\left( \Delta_{p}+\Delta_{q(p)}\right) u=\lambda_{p}\left\vert u(x_{u})\right\vert ^{p-2}u(x_{u})\delta_{x_{u}} & \mathrm{in} & \Omega\\ u=0 & \mathrm{on} & \partial\Omega, \end{array} \right. \] where…

Let $\Omega\subset\mathbb{R}^\nu$, $\nu\ge 2$, be a $C^{1,1}$ domain whose boundary $\partial\Omega$ is either compact or behaves suitably at infinity. For $p\in(1,\infty)$ and $\alpha>0$, define \[…

We consider families $u_p$ of solutions to the problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-\Delta u= u^p & \mbox{ in }\Omega\\ u>0 & \mbox{ in }\Omega\\ u=0 & \mbox{ on }\partial \Omega…

In this paper, we investigate the asymptotic behavior, as $\beta \to 0$, of positive solutions to the semilinear elliptic Robin problem \begin{equation*} \begin{cases} -\Delta u = u^p, & \text{in } \Omega,\\ u > 0, & \text{in } \Omega,\\…

In this paper, we study the asymptotic behavior of radial extremal functions to an inequality involving Hardy potential and critical Sobolev exponent. Based on the asymptotic behavior at the origin and the infinity, we shall deduce a strict…

In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish…

We study the asymptotic behavior, as $\gamma$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated to singular semilinear elliptic equations whose model is $$ -\Delta u=\frac{f(x)}{u^\gamma}\,\text{ in }\Omega,…

We study the behavior as $p\rightarrow\infty$ of $u_{p},$ a positive least energy solution of the problem \[ \left\{\begin{array} [c]{lll} \left[ \left( -\Delta_{p}\right) ^{\alpha}+\left( -\Delta_{q(p)}\right) ^{\beta}\right]…

Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations…

We study the nodal solutions of the Lane Emden Dirichlet problem $-\Delta u = |u|^{p-1}u with DBC on a smooth bounded domain $\Omega$ in $\IR^2$ and where $p>1$. We consider solutions $u_p$ satisfying $p \int_{\Omega}\abs{\nabla u_p}^2\to…

We consider the Lane-Emden Dirichlet problem -\Delta u = \abs{u}^{p-1}u, in B, u =0, on \partial B, where $p>1$ and $B$ denotes the unit ball in $\IR^2$. We study the asymptotic behavior of the least energy nodal radial solution $u_p$, as…

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 study the limiting behavior as $|x|\rightarrow \infty$ of extremal functions $u$ for Morrey's inequality on $\mathbb{R}^n$. In particular, we compute the limit of $u(x)$ as $|x|\rightarrow \infty$ and show $|x||Du(x)|$ tends to $0$. To…

We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{ in $\Omega$}, \\ \quad u>0 &\text{ in $\Omega$}, \\ \quad u=0 & \text{ on $\partial \Omega$}, \end{cases} \] where $L(v)=-{\rm…

We consider, for $a,l\geq1,$ $b,s,\alpha>0,$ and $p>q\geq1,$ the homogeneous Dirichlet problem for the equation $-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left\vert \nabla u\right\vert ^{b}+mu^{l-1}e^{\alpha u^{s}}$ in a smooth bounded…

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $1<p<N$ and $0<q(p)<p^{\ast}:=\frac{Np}{N-p}$ let \[ \lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in…