Estimate, existence and nonexistence of positive solutions of Hardy–Hénon equations

We consider the boundary Hardy–Hénon equation \[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \] where $B_1(0)\subset \mathbb {R}^{N}$ $(N\geq 3)$ is a ball of radial $1$ centred at $0$ , $p>0$ and $\alpha \in \mathbb {R}$ . We are concerned with the estimate, existence and nonexistence of positive solutions of the equation, in particular, the equation with Dirichlet boundary condition. For the case $0< p<({N+2})/({N-2})$ , we establish the estimate of positive solutions. When $\alpha \leq -2$ and $p>1$ , we give some conclusions with respect to nonexistence. When $\alpha >-2$ and $1< p<({N+2})/({N-2})$ , we obtain the existence of positive solution for the corresponding Dirichlet problem. When $0< p\leq 1$ and $\alpha \leq -2$ , we show the nonexistence of positive solutions. When $0< p<1$ , $\alpha >-2$ , we give some results with respect to existence and uniqueness of positive solutions.

Singular solutions of a Hénon equation involving a nonlinear gradient term

<p style='text-indent:20px;'>Here, we consider positive singular solutions of</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{lcc} -\Delta u = |x|^\alpha |\nabla u|^p &amp; \text{in}&amp; \Omega \backslash\{0\},\\ u = 0&amp;\text{on}&amp; \partial \Omega, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a small smooth perturbation of the unit ball in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> are parameters in a certain range. Using an explicit solution on <inline-formula><tex-math id="M5">\begin{document}$ B_1 $\end{document}</tex-math></inline-formula> and a linearization argument, we obtain positive singular solutions on perturbations of the unit ball.</p>