相关论文: Classification of Positive Radial Solutions to A W…

This paper is devoted to radial solutions of the following weighted fourth-order equation \begin{equation*} \mathrm{div}(|x|^{\alpha}\nabla(\mathrm{div}(|x|^\alpha\nabla u)))=u^{2^{**}_{\alpha}-1},\quad u>0\quad \mbox{in}\quad \mathbb{R}^N,…

For $n \geq 5$, we consider positive solutions $u$ of the biharmonic equation \[ \Delta^2 u = u^\frac{n+4}{n-4} \qquad \text{on}\ \mathbb R^n \setminus \{0\} \] with a non-removable singularity at the origin. We show that…

By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*}…

The biharmonic supercritical equation $\Delta^2u=|u|^{p-1}u$, where $n>4$ and $p>(n+4)/(n-4)$, is studied in the whole space $\mathbb{R}^n$ as well as in a modified form with $\lambda(1+u)^p$ as right-hand-side with an additional eigenvalue…

We prove the existence of a pair of positive radial solutions for the Neumann boundary value problem \begin{equation*} \begin{cases} \, \mathrm{div}\,\Biggl{(} \dfrac{\nabla u}{\sqrt{1- | \nabla u |^{2}}}\Biggr{)} + \lambda a(|x|)u^p = 0, &…

We consider positive solutions $u$ of the semilinear biharmonic equation $\Delta^2 u = |x|^{-\frac{n+4}{2}} g(|x|^\frac{n-4}{2} u)$ in $\mathbb R^n \setminus \{0\}$ with non-removable singularities at the origin. Under natural assumptions…

We consider the biharmonic equation $\Delta^2 u = u^\alpha$ in $\mathbf R^n$ with $n \geqslant 1$. It was proved that this equation has a positive classical solution if, and only if, either $\alpha \leqslant 1$ with $n \geqslant 1$ or…

In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation $$\begin{cases} -\Delta u-\displaystyle\frac \gamma{|x|^2}u=\displaystyle\frac{1}{|x|^s}|u|^{p_s-2}u & \text{ in }…

We prove new results on the existence of positive radial solutions of the elliptic equation $-\Delta u= \lambda h(|x|,u)$ in an annular domain in $\mathbb{R}^{N}, N\geq 2$. Existence of positive radial solutions are determined under the…

In this paper, we first classify all radially symmetry solutions of the following weighted fourth-order equation \begin{equation*} \Delta(|x|^{-\gamma}\Delta u)=|x|^\gamma u^{\frac{N+4+3\gamma}{N-4-\gamma}},\quad u\geq 0 \quad…

We study the regularity of the extremal solution of the semilinear biharmonic equation $\beta \Delta^2 u-\tau \Delta u=\frac{\lambda}{(1-u)^2}$ on a ball $B \subset \R^N$, under Navier boundary conditions $u=\Delta u=0$ on $\partial B$,…

The aim of this paper is to prove radial symmetry results for positive weak solutions with finite energy to the following quasilinear doubly critical system \begin{equation} \begin{cases} -\Delta_p u\,=\gamma \frac{u^{p-1}}{|x|^p} +…

The necessary and sufficient conditions for a regular positive entire solution $u$ of the biharmonic equation: \begin{equation} \label{0.1} -\Delta^2 u=u^{-p} \;\; \mbox{in $\R^N \; (N \geq 3)$}, \;\; p>1 \end{equation} to be a radially…

We study the problem $-\Delta u=\lambda u-u^{-1}$ with a Neumann boundary condition; the peculiarity being the presence of the singular term $-u^{-1}$. We point out that the minus sign in front of the negative power of $u$ is particularly…

We are concerned with positive radial solutions of the inhomogeneous elliptic equation $\Delta u+K(|x|)u^p+\mu f(|x|)=0$ on $\mathbb{R}^N$, where $N\ge 3$, $\mu>0$ and $K$ and $f$ are nonnegative nontrivial functions. If $K(r)\sim…

Many existence and nonexistence results are known for nonnegative radial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left|x\right| ^{-\alpha }dx)$ to the equation \[ -\triangle u+\dfrac{A}{\left| x\right| ^{\alpha…

In this work we study the nonnegative solutions of the elliptic system \Delta u=|x|^{a}v^{\delta}, \Delta v=|x|^{b}u^{\mu} in the superlinear case \mu \delta>1, which blow up near the boundary of a domain of R^{N}, or at one isolated point.…

We consider the H\'enon problem \begin{equation*} \left\{ \begin{array} - - \Delta u = |x|^{\alpha} u^{\frac{N+2+2\alpha}{N-2}-\varepsilon} & \ \ \text{in} \ B_1, \\ u > 0 & \ \ \text{in} \ B_1, \\ u=0 & \ \ \text{on} \ \partial B_1,…

We study the critical Neumann problem \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu}=0 &\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} in the unbounded…

In this paper, we consider the following nonlinear elliptic equation with gradient term: \[ \left\{ \begin{gathered} - \Delta u - \frac{1}{2}(x \cdot \nabla u) + (\lambda a(x)+b(x))u = \beta u^q +u^{2^*-1}, \hfill 0<u \in…