On the existence of solutions to higher-order regular nonlinear Emden–Fowler type equations with given number of zeros on the prescribed interval. (English) Zbl 1394.34059
From the introduction: Consider the equation
\[
y^{(n)}+ p(t,y,y',\dots, y^{(n-1)})|y|^k\,\text{sgn\,}y= 0,\tag{1}
\]
where \(n\geq 2\), \(k\in (1,+\infty)\), the function \(p(t,y_1,y_2,y_3,\dots, y_n)\in C(\mathbb{R}^{n-1})\) is Lipschitz cntinuous in \((y_1,y_2,y_3,\dots, y_n)\) and for some \(m\), \(M>0\) satisfies the inequalities
\[
0<m\leq p(t,y_1,y_2,\dots, y_n)\leq M<+\infty.
\]
The problem of the existence of solutions to (1) with the given number of zeros on a prescribe domain is investigated.
MSC:
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
