## Multiplicity results for asymmetric boundary value problems with indefinite weights.(English)Zbl 1079.34007

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.

### MSC:

 34B09 Boundary eigenvalue problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations

### Keywords:

multiplicity; weighted eigenvalues; rotation number
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