Lyapunov type inequality for the equation including 1-dim \(p\)-Laplacian (Q2911482)

The authors obtain a Lyapunov-type inequality for equations of the form NEWLINE\[NEWLINE(-1)^m(|u^{(m)}(x)|^{p-2}u^{(m)}(x))^{(m)}=r(x)|u(x)|^{p-2}u(x),\quad a\leq x\leq b,\tag{\(*\)} NEWLINE\]NEWLINE under clamped boundary condition \(u^{(i)}(a)=u^{(i)}(b)=0\), \(i = 0,1,2,\dots,m-1\). The main result in the paper is stated as follows:NEWLINENEWLINETheorem. Let \(m = 1,2, 3\), \(p > 1\), \(q\) be a conjugate exponent of \(p\), that is, \(\frac1p+\frac1q=1\). If \(u\in W_0^{m,p}(a,b)\) is a non-trivial solution of \((*)\), then NEWLINE\[NEWLINE\frac{1}{C(m,p)^p}<\int_a^br(x)dx,NEWLINE\]NEWLINE where \(C(m,p)\) is the best constant of the \(L^p\)-Sobolev inequality NEWLINE\[NEWLINE(\sup_{a\leq x\leq b}|v(x)|)\leq C \biggl(\int_a^b|v^{(m)}(x)|^pdx\biggr)^{\frac1p}, \quad v\in W_0^{m,p}(a,b).NEWLINE\]NEWLINE \(W^{m,p}(a,b)\) is the Sobolev space defined on the interval \([a, b]\), that is, \(W^{m,p}(a,b)=\{u\mid u^{(i)}\in L^p(a,b), i=0,1,\dots,m\}\) and \(W_0^{m,p}(a,b)\) is a sub-space of \(W^{m,p}(a,b)\) whose derivatives up to \(m-1\) vanish at \(x = a, b\), where \(u^{(i)}\) denotes \(i\)-th derivative of \(u\) in the sense of distributions.