| Summary: | In previous work by Castro, Cossio, and Neuberger cite{ccn}, it was shown that a superlinear Dirichlet problem has at least three nontrivial solutions when the derivative of the nonlinearity at zero is less than the first eigenvalue of $-Delta$ with zero Dirichlet boundry condition. One of these solutions changes sign exactly-once and the other two are of one sign. In this paper we show that when this derivative is between the $k$-th and $k+1$-st eigenvalues there still exists a solution which changes sign at most $k$ times. In particular, when $k=1$ the sign-changing {it exactly-once} solution persists although one-sign solutions no longer exist.
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