A sign-changing solution for a superlinear Dirichlet problem, II

• Main Authors: Alfonso Castro, Pavel Drabek, John M. Neuberger
• Format: Article
• Language: English
• Published: Texas State University 2003-02-01
• Series: Electronic Journal of Differential Equations
• Subjects: Dirichlet problem; superlinear; subcritical; sign-changing solution; deformation lemma.
• Online Access: http://ejde.math.txstate.edu/conf-proc/10/c3/abstr.html

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.