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A Neumann boundary value problem for the Sturm-Liouville equation. (English) Zbl 1176.34020

The authors deal with the Neumann boundary value problem

$-{\left(p{u}^{\text{'}}\right)}^{\text{'}}+r{u}^{\text{'}}+qu=\lambda g\left(u\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{u}^{\text{'}}\left(0\right)={u}^{\text{'}}\left(1\right)=0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $p$ is a ${C}^{1}$ positive function in $\left[0,1\right]$, $r$ and $q$ are continuous in $\left[0,1\right]$, $r$ is positive; $\lambda$ is a positive parameter.

The solutions to (1) are the critical points of a functional of the form ${\Phi }-\lambda {\Psi }$, ${\Phi }\left(u\right)={\parallel u\parallel }^{2}/2$, ${\Psi }\left(u\right)={\int }_{0}^{1}F\left(x,u\left(x\right)\right)\phantom{\rule{0.166667em}{0ex}}dx$. Here, $\parallel ·\parallel$ is a norm of ${W}^{1,2}\left(0,1\right)$ and $F$ is a primitive of $f$ with respect to $u$. It is shown that, under assumptions that very roughly speaking express an oscillating character of $F\left(x,u\right)/{u}^{2}$, problem (1) may have multiple solutions (infinitely many, or at least three, depending on the particular assumptions). The main tool in the proof are variants of the Ricceri variational principle as given in the paper of G. Bonanno and P. Candito [J. Differ. Equations 244, No. 12, 3031–3059 (2008; Zbl 1149.49007)].

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 58E05 Abstract critical point theory 34B24 Sturm-Liouville theory