In this paper, we study isolated singularities of the following semilinear elliptic equation $-\Delta u+\frac12 x\cdot \nabla u+\frac{1}{q-1}u-u^q=0$ in $\Omega \setminus \{0\}$, where $n\ge 3$, $\Omega \subset \mathbb{R}^n$ is a domain, $0 \in \Omega$ and $q>1$. This equation arises in the study of blow-up profiles of semilinear heat equations. For $\frac{n}{n-2}<q<\frac{n+2}{n-2}$, we establish a complete classification of isolated singularities for nonnegative solutions and characterize the precise asymptotic behavior of singular solutions. Our results improve those of Guedda and Kirane (Trans. Amer. Math. Soc., 1995: 3595-3603), where analogous results were obtained only for radially symmetric positive solutions. In addition, we also derive the asymptotic behavior of solutions in the Serrin critical case $q=\frac{n}{n-2}$ and the supercritical case $q>\frac{n+2}{n-2}$.
We investigate positive radial solutions with an isolated nonremovable singularity for the semilinear elliptic equation \begin{align*} \Delta_{\mathbb{H}^N} u+\lambda u+u^p=0 \qquad\text{in }\mathbb{H}^N\setminus\{Q\}, \end{align*} where $N\geq 3$, $p>1$, $\lambda\le \frac{(N-1)^2}{4}$, and $Q\in\mathbb{H}^N$ is a pres...
This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $01$ and $...
We classify isolated singularities of classical solutions to the capillary equation with negative gravity, \[ \operatorname{div}\frac{Du}{\sqrt{1+|Du|^2}}=-u \qquad\text{in }B_R\setminus\{0\}\subset\mathbb R^n, \qquad n\ge2, \] without a priori assumptions on symmetry, sign, one-sided boundedness, or blow-up rate. Ever...
Bing Deng, Jia-Huan Li, Yi-Lu Liu et al.· 0 citations
In this paper, we study singular positive solutions to the semilinear elliptic equation $$ - \Delta u = f(u) ~~~~~~ \textmd{in} ~ \Omega \setminus \Gamma, $$ where $\Omega \subset \R^n$ is a bounded or unbounded domain, and $\Gamma \subset \Omega$ is a singular closed set with zero Newtonian capacity. When $\Omega = \R...
We investigate the Dirichlet problem for the variational integral $J[u] = \int_{\Omega} f(\nabla u) \, dx$ with density $f$ of linear growth satisfying appropriate ellipticity conditions. We show that the relaxed problem admits a unique solution $u$ in the space of functions of bounded variation, if the set $\Gamma_0$...
We study global regularity of solutions to Dirichlet or Neumann elliptic problems in spherical sectors $S_{D,R}$ of radius $R>0$ in $\mathbb{R}^N, N\ge 2$, where $D$ is the bounded domain on the unit sphere $\mathbb{S}^{N-1}$ which spans the spherical sector. One of the main results shows that boundedness of the gradie...
Carlo Alberto Antonini, F. Pacella, Camilla Chiara Polvara et al.· 0 citations
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