A bifurcation phenomenon for the critical Laplace and $p$-Laplace equation in the ball
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| Format: | Preprint |
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2024
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| _version_ | 1866916465912840192 |
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| author | Dalbono, Francesca Franca, Matteo Sfecci, Andrea |
| author_facet | Dalbono, Francesca Franca, Matteo Sfecci, Andrea |
| contents | In this paper we show that the number of radial positive solutions of the following critical problem $$ Δ_p u(x) + λK(|x|) \,u(x) \, |u(x)|^{q-2} =0\,,$$ $$ u(x)>0 \quad |x|<1,$$ $$ u(x)=0 \quad |x|=1,$$ where $q= \frac{np}{n-p}$, $\frac{2n}{n+2} \le p \le 2$ and $x \in \mathbb{R}^n$, undergoes a bifurcation phenomenon. Namely, the problem admits one solution for any $λ>0$ if $K$ is steep enough at $0$, while it admits no solutions for $λ$ small and two solutions for $λ$ large if $K$ is too flat at $0$.
The existence of the second solution is new, even in the classical Laplace case. The proofs use Fowler transformation and dynamical systems tools. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_01186 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A bifurcation phenomenon for the critical Laplace and $p$-Laplace equation in the ball Dalbono, Francesca Franca, Matteo Sfecci, Andrea Analysis of PDEs 35J92, 35J62, 35B33, 35B09, 34C45 In this paper we show that the number of radial positive solutions of the following critical problem $$ Δ_p u(x) + λK(|x|) \,u(x) \, |u(x)|^{q-2} =0\,,$$ $$ u(x)>0 \quad |x|<1,$$ $$ u(x)=0 \quad |x|=1,$$ where $q= \frac{np}{n-p}$, $\frac{2n}{n+2} \le p \le 2$ and $x \in \mathbb{R}^n$, undergoes a bifurcation phenomenon. Namely, the problem admits one solution for any $λ>0$ if $K$ is steep enough at $0$, while it admits no solutions for $λ$ small and two solutions for $λ$ large if $K$ is too flat at $0$. The existence of the second solution is new, even in the classical Laplace case. The proofs use Fowler transformation and dynamical systems tools. |
| title | A bifurcation phenomenon for the critical Laplace and $p$-Laplace equation in the ball |
| topic | Analysis of PDEs 35J92, 35J62, 35B33, 35B09, 34C45 |
| url | https://arxiv.org/abs/2411.01186 |