Solution surfaces for semilinear elliptic equations on rotated domains

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.