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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 $$-(pu')'+ru'+qu=\lambda g(u), \;\; u'(0)=u'(1)=0,\tag1$$ where $p$ is a $C^1 $ positive function in $[0,1]$, $r$ and $q$ are continuous in $[0,1]$, $r$ is positive; $\lambda $ is a positive parameter. The solutions to (1) are the critical points of a functional of the form $\Phi-\lambda\Psi$, $\Phi(u)=\|u\|^2/2$, $\Psi(u)=\int_0^1F(x,u(x))\,dx$. Here, $\|\cdot\|$ is a norm of $W^{1,2}(0,1)$ 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 $F(x,u)/u^2$, 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 {\it G. Bonanno} and {\it P. Candito} [J. Differ. Equations 244, No. 12, 3031--3059 (2008; Zbl 1149.49007)].

34B15Nonlinear boundary value problems for ODE
58E05Abstract critical point theory
34B24Sturm-Liouville theory
Full Text: DOI
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