## Osaka Journal of Mathematics

### Non-local elliptic problem in higher dimension

Tosiya Miyasita

#### Abstract

Non-local elliptic problem, $-\Delta v = \lambda \bigl(e^{v}\big/\bigl(\int_{\Omega}e^{v} dx \bigr)^{p}\bigr)$ with Dirichlet boundary condition is considered on $n$-dimensional bounded domain $\Omega$ with $n \geq 3$ for $p>0$. If $\Omega$ is the unit ball, $3 \leq n \leq 9$ and $2/n \leq p \leq 1$, we have infinitely many bendings in $\lambda$ of the solution set in $\lambda-v$ plane. Finally if $\Omega$ is an annulus domain and $p \geq 1$, we show that a solution exists for all $\lambda>0$.

#### Article information

Source
Osaka J. Math., Volume 44, Number 1 (2007), 159-172.

Dates
First available in Project Euclid: 19 March 2007

https://projecteuclid.org/euclid.ojm/1174324329

Mathematical Reviews number (MathSciNet)
MR2313033

Zentralblatt MATH identifier
1213.35229

#### Citation

Miyasita, Tosiya. Non-local elliptic problem in higher dimension. Osaka J. Math. 44 (2007), no. 1, 159--172. https://projecteuclid.org/euclid.ojm/1174324329

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