On sign-changing solutions of certain semilinear elliptic problems. (English) Zbl 0835.35051

Dancer, E. N.; Du, Yihong

Appl. Anal. 56, No. 3-4, 193-206 (1995).

We continue our work on when the equation \(-\Delta u= f(u)\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\) has a changing sign solution. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^n\). In the first part, we use ideas from H. Hofer [Math. Ann. 261, 493-514 (1982; Zbl 0488.47034)] to prove the existence of changing sign solutions when \(f\) is sublinear at infinity. This improves an earlier work of the authors [J. Math. Anal. Appl. 189, No. 3, 848-871 (1995)]. In the second part, we obtain results for some superlinear subcritical problems. This depends on a variant of a result of K.-C. Chang [Infinite-dimensional Morse theory and its applications (Universite de Montreal, Canada, 1985; Zbl 0609.58001)] for the degree of maps which are of variational type.

Reviewer: E.N.Dancer (Sydney)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
47J25	Iterative procedures involving nonlinear operators

Keywords:

fixed point index; existence of changing sign solutions

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