Multiplicity results for asymmetric boundary value problems with indefinite weights (Q1773556)

The nonlinear Dirichlet problem \[ u''+f(t,u)=0,\quad u(0)=0=u(T),\tag{1} \] is considered. The nonlinear part of the equation is asymmetrically linear both at \(0^\pm\) and at \(\pm\infty\). The approach is via the positively homogeneous eigenvalue problem (where \(\nu\) denotes \(<\) or \(>\)) \[ u''+\lambda(\varphi(t)u^+-\psi(t)u^-)=0,\quad u(0)=0=u(T), \quad u'(0)\nu0.\tag \(P_{\varphi,\psi,\nu}\) \] In a first step, the author proves the existence of a sequence of eigenvalues \(\lambda^\nu_i(\varphi,\psi)\) and nodal properties of the corresponding eigenfunctions. This is done by using a Prüfer change of variables. A rotation number, defined for nontrivial solutions of the homogeneous differential equation, and counting half turns, is then used to estimate the eigenvalues. The role of \(\varphi,\,\psi\) will then be played by the functions that bound the ratio \(\frac{f(t,u)}{u}\) as \(u\to0^\pm\) or \(u\to\pm\infty\). Call them, with a (hopefully) obvious notation, \(a_j^\pm\leq b_j^\pm\) where \(j=0\) or \(j=\infty\). A careful comparison of several rotation numbers involved allows one to prove the main result, where the number of solutions of (1) for which \(u'(0)\) has a given sign is given by (at least) \(n-m+1\), where \(\lambda^\nu_n(a_0^+,a_0^-)<1<\lambda^\nu_m(b_\infty^+,b_\infty^-)\) or \(\lambda^\nu_n(a_\infty^+,a_\infty^-)<1<\lambda^\nu_m(b_0^+,b_0^-)\). A continuation principle due to Mawhin, Rebelo and Zanolin is involved in the argument. This statement improves several earlier results; relations of these with the main result of the paper are discussed in the final remarks.