Characteristic numbers of non-autonomous Emden-Fowler type equations. (English) Zbl 1107.34024

Gritsans, A.; Sadyrbaev, F.

Math. Model. Anal. 11, No. 3, 243-252 (2006).

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.)

Keywords:

Nehari solution