Osaka Journal of Mathematics

Non-local elliptic problem in higher dimension

Tosiya Miyasita

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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

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

First available in Project Euclid: 19 March 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory


Miyasita, Tosiya. Non-local elliptic problem in higher dimension. Osaka J. Math. 44 (2007), no. 1, 159--172.

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