Lyapunov inequalities for nonlinear p-Laplacian problems with weight functions (Q2901491)

The paper contains different generalizations of the classical \(L^1\)-Lyapunov inequality for the boundary value problem NEWLINE\[NEWLINE u''(x) + r(x)u(x) = 0, \;x \in (a,b), \;\;u(a) = u(b) = 0, \tag{1}NEWLINE\]NEWLINE which states that, if \(r \in L^1 (a,b)\) and \(u\) is a nontrivial solution of (1), then NEWLINE\[NEWLINE\int_a^b \;r^+(x) \, dx > \frac{4}{b-a} NEWLINE\]NEWLINE (here, \(r^+\) means the positive part of the function \(r\)). Moreover, the constant \( \frac{4}{b-a}\) is optimal. There have been many generalizations of this result. In particular, it can be proved that, if \(r \in L^1 (a,b)\) and \(u\) is a nontrivial solution of (1), then NEWLINE\[NEWLINE\int_a^b \;(x-a)(b-x)r^+(x) \, dx >b-aNEWLINE\]NEWLINE (see, for instance Theorem 5.1 in [\textit{P. Hartman}, Ordinary differential equations. New York-London-Sydney: John Wiley and Sons (1964; Zbl 0125.32102)]).NEWLINENEWLINEIn this paper, the author considers the boundary value problem NEWLINE\[NEWLINE (s(x)\varphi_p (u'(x)))' + r(x)\varphi_p (u(x)) = 0, \;x \in (a,b), \;\;u(a) = u(b) = 0, \tag{2}NEWLINE\]NEWLINE where \(p > 1,\) \(\varphi_p (y) = | y|^{p-2} y\), \(s \in C^1([a,b],[k,\infty))\), \( k > 0\) and \(r \in C([a,b],(0,\infty))\). Assuming some additional restrictions on the function \(r\) and by using Hölder's inequality, it is proved that, if (2) has some positive solution in the interval \((a,b),\) then NEWLINE\[NEWLINE \int_a^b (x-a)^{p-1}(b-x)^{p-1} r(x) \, dx \geq \frac{k(b-a)^{p-1}}{2^{p-2}}. NEWLINE\]NEWLINE In addition, some extensions of this result for the case of systems of equations are shown.