On nondecreasing singular solutions of the Emden-Fowler equation (Q2712572)

The author studies the solutions to the Emden-Fowler equation. For \(a< b\), let \(p:(a, b)\to (0,+\infty)\) be a locally Lebesgue integrable function on \((a,b)\) that differing from zero on a set of positive measure in any left neighborhood of \(b\). Let \(\lambda> 0\) and \(n\) be an integer \(\geq 2\). The author considers the problem NEWLINE\[NEWLINEu^{(n)}= p(t)|u|^\lambda \text{sgn }u,\quad t\in (a,b),\tag{1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEu^{(i)}(t)> 0,\quad 1\leq i\leq n-1,\quad t\in (a,b),\quad \lim_{t\to b} u(t)=+\infty,\tag{2}NEWLINE\]NEWLINE where \(u: [a,b)\to (0,+\infty)\) is an unknown function. Equation (1) is called the Emden-Fowler equation, and a solution \(u\) to equation (1) satisfying (2) is called a nonoscillatory singular solution of the second kind. Denote NEWLINE\[NEWLINEJ(s,t)= \int^t_s p(\tau)(b- \tau)^{n- 1}d\tau,\quad (s,t)\in (a,b)\times (a,b),NEWLINE\]NEWLINE and NEWLINE\[NEWLINEv_i(t)= \sum^i_{\ell= 0} u^{(\ell)}(t) {(b- t)^\ell\over \ell!},\quad 1\leq i\leq n-1,\quad t\in (a,b).NEWLINE\]NEWLINE The author proves that, if \(J(a,b)< \infty\), then every solution \(u\) to (1), (2) admits the lower estimate NEWLINE\[NEWLINE{(n-1)!\over\lambda- 1} J(t, b)< (v_{n- 1}(t))^{1- \lambda},\quad t\in (a,b).NEWLINE\]NEWLINE Denote \(n_1= 1+ (n-1)\lambda\), \(\varphi:(a, b)\to (0,+\infty)\) a nondecreasing function on \((a,b)\), NEWLINE\[NEWLINE\varphi_M(t)= \min\Biggl\{{1\over b-t}, {\dot\varphi(t)\over M\varphi(t)}\Biggr\},\quad t\in (a,b),NEWLINE\]NEWLINE and NEWLINE\[NEWLINEF_{\nu,\mu,\sigma, M}(\varphi)(t)= \varphi^\sigma(t) \int^b_t {(p(\tau)(b- \tau)^{n- 1})^\mu\over \varphi^\sigma(\tau) \varphi^{\mu+ \nu-1}_M(\tau)} d\tauNEWLINE\]NEWLINE with \(M> 0\), \(\nu\in [0, 1)\), \(\mu\in ({1-\nu\over n_1}, {1-\nu\over n})\), \(\sigma> 0\).NEWLINENEWLINENEWLINEThe main result of this paper proves that, if \(u\) is a solution to (1), (2) and \(\varphi\) is any nondecreasing function on \((a,b)\), then for any numbers \(n_1\), \(\nu\), \(\mu\), \(M\), \(\sigma\) as given above the equality \(\lim_{t\to b} F_{\nu,\mu, \sigma,M}(\varphi)(t)= 0\) is true and the upper estimate \(u(t)= \gamma(F_{\nu, \mu, \sigma,M}(\varphi)(t))^{{1\over (1-\lambda)\mu}}\) is fulfilled, where \(\gamma\) depends only on \(n\), \(\lambda\), \(\mu\), \(\nu\).NEWLINENEWLINENEWLINEAs an application of the above results, the author establishes that, if \(p(t)(b- t)^n< cJ(a,t)\), \(t\in [t_c, b)\subset (a,b)\), \(c> 0\), and equation (1) has a solution satisfying (2), then the condition \(J(a,b)< +\infty\) holds and in some neighborhood of \(b\) we have \(u(t)< \gamma J^{{1\over (1-\lambda)\mu}}(t, b)\), where \(\gamma\) depends on \(n\), \(\lambda\), \(\mu\). Moreover, if \(J(a,b)< +\infty\) and \(p(t)(b- t)^n< cJ(t, b)\), \(t\in [t_c, b)\subset (a,b)\), \(c>0\), hold, then the two-sided estimate \(0< \gamma_1< u(t) J^{{1\over 1-\lambda}}(t, b)< \gamma_2\) holds, where \(\gamma_1\) and \(\gamma_2\) depend on \(n\) and \(\lambda\).NEWLINENEWLINENEWLINEFor the general case, the author introduces into consideration the nonnegative functions NEWLINE\[NEWLINE(p_f)_*(t)= \min\Biggl\{p(t), {f(t)\over (b- t)^n}\Biggr\},\quad (p_f)^*(t)= p(t)- (p_f)_*(t),\quad t\in (a,b),NEWLINE\]NEWLINE and the integrals NEWLINE\[NEWLINE(J_f)_*(s, t)= \int^t_s {(p_f)_*(x)(b- x)^{n- 1}\over f(x)} dx,\quad (J_f)^*(s, t)= \int^t_s {(p_f)^*(x)(b- x)^{n-1}\over f(x)} dx,NEWLINE\]NEWLINE where \(f:(a, b)\to (0,+\infty)\) is an arbitrary nondecreasing function. Then if equation (1) has a solution \(u\) of type (2) for an arbitrary nondecreasing positive function \(f\) and all \(\mu\in (0,{1\over n})\) we have NEWLINE\[NEWLINE(J_f)_*(a, b)<+\infty,\quad \lim_{t\to b} f^\mu(t)(J_f)_*(t, b)= 0,NEWLINE\]NEWLINE and in some neighborhood of \(b\) the estimate NEWLINE\[NEWLINEu(t)< \gamma(f(t)(J_f)^{{1\over \mu}}_*(t, b))^{{1\over 1-\lambda}}NEWLINE\]NEWLINE is true. The author gives also an estimate on \(u\) using the integral \((J_f)^*\).