A Sturm type comparison theorem for nonlinear problems
A Sturmian comparison theorem is proved for a pair of nonlinear initial value problems of type \[ \begin{cases} -\bigl[p_i(t) f(u_i) |u_i' |^{m-2} u_i' \bigr]' = g_i(t,u_i), \quad t>0 \\ u_i(0) = a_i, \quad u_i'(0) = b_i, \quad i=1,2, \end{cases} \leqno (1)_i \] where \(a_i,b_i,m\) are constants with \(a_1\geq a_2>0\), \(b_1\geq b_2\leq 0\), \(m>1\) and \(f\in C (\mathbb{R}_+, \mathbb{R}_+)\), \(g_i \in C(\mathbb{R}^2_+, \mathbb{R}_+)\), \(p_i\in C (\mathbb{R}_+, \mathbb{R}_+)\), \(p_1^{-1} (r) \int^r_0 p_1(t) dt \to 0\) as \(r\to 0_+\), \(i=1,2\). Additional technical hypotheses (not duplicated here) generalize the classical notion that \((1)_1\) is a ``Sturmian majorant'' of \((1)_2\); for the prototype \(f(s)= s^\alpha\), \(g_i(t,s) = q_i(t) s^\beta\), these hypotheses are \(0<\beta \leq \alpha+ m-1\), \(p_1(t) \geq p_2(t) >0\), \(q_2(t) \geq q_1(t) \geq 0\) for all \(t>0\). If \(b_1\geq 0\), the main theorem states that \(u_1(t) \geq u_2(t)\) in an interval \([0,t_0)\) of joint positivity of \(u_1\) and \(u_2\). If instead \(b_1< 0\), an extra hypothesis is required for the same conclusion. Furthermore, in both cases, \(u_1^\lambda (t)- u_2^\lambda (t) \geq a_1^\lambda - a_2^\lambda\) in \([0,t_0)\), where \(\lambda = (\alpha - \beta+ m-1)/(m-1)\) (stated for the prototype above). Applications of this theorem include existence and uniqueness results for radial solutions to Dirichlet problems involving the \(m\)-Laplacian. The novel methods contain a Riccati-type transformation for nonlinear equations. Recent related results were obtained by \textit{Ch. Gui} [J. Differ. Equations 99, 245-280 (1992; Zbl 0754.35017)], \textit{G. Makay} [Am. Math. Mon. 99, 218-219 (1992; Zbl 0756.34015)], \textit{R. Filippucci} et al. [Arch. Ration. Mech. Anal. 127, 255-280 (1994; Zbl 0820.34012)], and \textit{R. Filippucci} and \textit{R. Ghiselli Ricci} [Arch. Ration. Mech. Anal. 127, 289-295 (1994; Zbl 0820.34013)].
scientific article; zbMATH DE number 933273