Generalization of the Gekhtman theorem for a nonlinear Sturm-Liouville equation (Q804785)

The author considers the nonlinear Sturm-Liouville problem \((\dot x(t))^._{(p)}+\lambda^{-1}a(t)(x(t))_{(p)}=0,\quad x(0)=x(1)=0,\) where a is a continuous strictly positive function on \(I=[0,1]\) and \((z)_{(p)}=| z|^{p-1}sgn z\) (z\(\in {\mathbb{R}})\). Using previous results that the spectrum of this problem consists of the eigenvalues \(\lambda_ 1>\lambda_ 2>...>0\), the following results are proved where \(x_ n\) is an eigenfunction to \(\lambda_ n\) with \(\| x_ n\|_{L_ 2(I,a)}=0:\) (a) \(\lim_{n\to \infty}\| x_ n\|_{C(I)}\lambda_ n^{1/p^ 2}=0\). (b) For any \(\epsilon >0\), \(0<\epsilon <1/p^ 2\), there exists a positive continuous function a on I such that \(\limsup_{n\to \infty}| x_ n(1/2)| \lambda_ n^{1/p^ 2-\epsilon}=\infty\).