Sign-Changing Solutions for Critical Equations with Hardy Potential
| Autor: | Angela Pistoia, Nassif Ghoussoub, Giusi Vaira, Pierpaolo Esposito |
|---|---|
| Přispěvatelé: | Esposito, Pierpaolo, Ghoussoub, Nassif, Pistoia, Angela, Vaira, Giusi |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Dirichlet problem
Numerical Analysis Applied Mathematics 010102 general mathematics Sign changing 01 natural sciences Omega Combinatorics Mathematics - Analysis of PDEs Bounded function 0103 physical sciences Domain (ring theory) FOS: Mathematics 010307 mathematical physics Ball (mathematics) 0101 mathematics Analysis Mathematics Analysis of PDEs (math.AP) |
| Popis: | We consider the following perturbed critical Dirichlet problem involving the Hardy-Schr\"odinger operator on a smooth bounded domain $\Omega \subset \mathbb{R}^N$, $N\geq 3$, with $0 \in \Omega$: $$ \left\{ \begin{array}{ll}-\Delta u-\gamma \frac{u}{|x|^2}-\epsilon u=|u|^{\frac{4}{N-2}}u &\hbox{in }\Omega u=0 & \hbox{on }\partial \Omega, \end{array}\right. $$ when $\epsilon>0$ is small and $\gamma< {(N-2)^2\over4}$. Setting $ \gamma_j= \frac{(N-2)^2}{4}\left(1-\frac{j(N-2+j)}{N-1}\right)\in(-\infty,0]$ for $j \in \mathbb{N},$ we show that if $\gamma\leq \frac{(N-2)^2}{4}-1$ 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 $\gamma \frac{(N-2)^2}{4}-1$ 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 $\gamma\geq \frac{(N-2)^2}{4}-4$, 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 $\gamma=0$. Comment: 41 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/ |
| Databáze: | OpenAIRE |
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