Advances in Differential Equations

Solution surfaces for semilinear elliptic equations on rotated domains

Seth Armstrong and Renate Schaaf

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Abstract

For the problem $$ \begin{align} \Delta u + \lambda f(u) &= 0 \quad\text{ in } \ \Omega, \;\ \lambda \in {\mathbf {R}} ^{+}, \\ u & = 0 \quad\text{ on } \ \partial \Omega, \end{align} $$ if $\Omega$ and $f$ satisfy certain hypotheses, a parameterized curve of positive solutions $\alpha\mapsto (u(\alpha),\lambda(\alpha))$ has been shown to exist, where $\alpha=max_{\Omega}u$. If $\Omega\subset{\mathbf {R}}^{n}$ is translated by $1/\epsilon$ and then rotated about a coordinate axis to obtain a new domain $\Omega_{\epsilon}\subset{\mathbf {R}}^{n+1}$, it can be shown that a surface of positive rotationally invariant solutions $(\alpha,\epsilon)\mapsto (\hat{u}(\alpha,\epsilon),\hat{\lambda}(\alpha,\epsilon))$ exists for the resulting problem $$ \begin{align} \Delta_{\epsilon}\hat{u} + \hat{\lambda} f(\hat{u}) & = 0 \quad\text{ in } \ \Omega_{\epsilon}, \ \hat{\lambda} \in {\mathbf {R}} ^{+} \\ \hat{u} & = 0 \quad\text{ on } \ \partial \Omega_{\epsilon}, \end{align} $$ where $\Delta_{\epsilon}$ is the Laplacian in the new variables and $(\hat{u}(\alpha,0),\hat{\lambda}(\alpha,0))=(u(\alpha),\lambda(\alpha))$. From this, we can give various examples of problems on domains with a large hole for which the structure of solutions can be well described.

Article information

Source
Adv. Differential Equations Volume 4, Number 2 (1999), 251-274.

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366291415

Mathematical Reviews number (MathSciNet)
MR1674343

Zentralblatt MATH identifier
0953.35045

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 34B15: Nonlinear boundary value problems 47H15

Citation

Armstrong, Seth; Schaaf, Renate. Solution surfaces for semilinear elliptic equations on rotated domains. Adv. Differential Equations 4 (1999), no. 2, 251--274. https://projecteuclid.org/euclid.ade/1366291415.


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