| Abstract: |
| The boundary value problem $u`` + (|x|^l +\lambda)u^p=0$ for $x\in(-1,1)$, $u(-1)=u(1)=0$ is considered, where $l\ge0$, $\lambda\ge0$ and $p>1$.
This problem always has a positive even solution.
At first, the uniqueness of positive even solutions is proved on the majority part of $(l,\lambda)\in [0,\infty) \times [0,\infty)$ for fixed $p>1$.
Then a very narrow set remains as the possible region for which the problem has at least two positive even solutions.
Therefore it is natural to expect that the uniqueness of positive even solutions holds for each $(l,\lambda)\in [0,\infty) \times [0,\infty)$.
Contrary to this expectation, it is shown that there exists at least three positive even solutions with the aid of numerical verification method.
This is a joint work with Naoki Shioji (Yokohama National University) and Kohtaro Watanabe (National Defense Academy). |
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