Abstract
This paper is mainly concerned with the natural order relationship between positive solutions of the elliptic eigenvalue Dirichlet problem: \(-\Delta u=\lambda f(u)\) in \(\Omega\) and u=0 on \(\partial\Omega\). Under suitable conditions, we prove that there are 2m-1 positive solutions satisfying \(\hat u_1 < u^*_2 < \hat u_2 < \cdots < u^*_m < \hat u_m\). It seems that standard arguments do not provide such a result. Several authors, including P. Hess, proved the existence of equal number of positive solutions without such a relationship between them. We also prove that in Hess's result as well as in ours some sufficient condition is also necessary if the domain possesses a particular shape. At last, as an illustrative example, we study the diagram of positive solutions when \(\lambda f(u)=\lambda (d+ cos u)\) with \(\lambda\) and d being both parameters.
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