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EnglishWe consider the following perturbed critical Dirichlet problem involving the Hardy-Schr\"odinger operator on a smooth bounded domain $\Omega \subset \mathbbR^N$, $N\geq 3$, with $0 \in \Omega$: $$ \left\ \beginarrayll-\Delta u-\gamma \fracu|x|^2-\epsilon u=|u|^\frac4N-2u &\hboxin \Omega u=0 & \hboxon \partial \Omega, \endarray\right. $$ when $\epsilon>0$ is small and $\gamma< (N-2)^2\over4$. Setting $ \gamma_j= \frac(N-2)^24\left(1-\fracj(N-2+j)N-1\right)\in(-\infty,0]$ for $j \in \mathbbN,$ we show that if $\gamma\leq \frac(N-2)^24-1$ and $\gamma \neq \gamma_j$ for any $j$, then for small $\epsilon$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $\gamma<\frac(N-2)^24-4,$ then for any integer $k \geq 2$, the equation has for small enough $\epsilon$, a sign-changing solution that develops into a superposition of $k$ bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition that $\gamma\neq \gamma_j$ is not necessary. Indeed, it is known that, if $\gamma > \frac(N-2)^24-1$ and $\Omega$ is a ball $B$, then there is no radial positive solution for $\epsilon>0$ small. We complete the picture here by showing that, if $\gamma\geq \frac(N-2)^24-4$, then the above problem has no radial sign-changing solutions for $\epsilon>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $\gamma=0$.
| Titolo: | Sign-Changing Solutions for Critical Equations with Hardy Potential |
| Autori: | |
| Data di pubblicazione: | 2017 |
| Citazione: | Esposito, P., Ghoussoub, N., Pistoia, A., & Vaira, G. (2017). Sign-Changing Solutions for Critical Equations with Hardy Potential. |
| Abstract: | We consider the following perturbed critical Dirichlet problem involving the Hardy-Schr\"odinger operator on a smooth bounded domain $\Omega \subset \mathbbR^N$, $N\geq 3$, with $0 \in \Omega$: $$ \left\ \beginarrayll-\Delta u-\gamma \fracu|x|^2-\epsilon u=|u|^\frac4N-2u &\hboxin \Omega u=0 & \hboxon \partial \Omega, \endarray\right. $$ when $\epsilon>0$ is small and $\gamma< (N-2)^2\over4$. Setting $ \gamma_j= \frac(N-2)^24\left(1-\fracj(N-2+j)N-1\right)\in(-\infty,0]$ for $j \in \mathbbN,$ we show that if $\gamma\leq \frac(N-2)^24-1$ and $\gamma \neq \gamma_j$ for any $j$, then for small $\epsilon$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $\gamma<\frac(N-2)^24-4,$ then for any integer $k \geq 2$, the equation has for small enough $\epsilon$, a sign-changing solution that develops into a superposition of $k$ bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition that $\gamma\neq \gamma_j$ is not necessary. Indeed, it is known that, if $\gamma > \frac(N-2)^24-1$ and $\Omega$ is a ball $B$, then there is no radial positive solution for $\epsilon>0$ small. We complete the picture here by showing that, if $\gamma\geq \frac(N-2)^24-4$, then the above problem has no radial sign-changing solutions for $\epsilon>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $\gamma=0$. |
| Handle: | http://hdl.handle.net/11590/340925 |
| Appare nelle tipologie: | 5.12 Altro |
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