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Characteristic numbers of non-autonomous Emden-Fowler type equations. (English) Zbl 1107.34024
Summary: We consider the Emden-Fowler equation $$x''=-q(t)|x|^{2 \varepsilon}x$$, $$\varepsilon>0$$, in the interval $$[a,b]$$. The coefficient $$q(t)$$ is a positive-valued continuous function. The Nehari characteristic number $$\lambda_n$$ associated with the Emden-Fowler equation coincides with a minimal value of the functional $$\frac{\varepsilon}{1+ \varepsilon}\int^b_ax^{\prime 2}(t)dt$$ over all solutions of the boundary value problem $x''=-q(t)|x|^{2\varepsilon}x,\quad x(a)=x(b) =0,\quad x(t)\text{ has exactly }(n-1)\text{ zeros in }(a,b).$ The respective solution is called the Nehari solution. We construct an example which shows that the Nehari extremal problem may have more than one solution.

##### MSC:
 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
Nehari solution