Drábek, Pavel; Kuliev, Komil Half-linear Sturm-Liouville problem with weights. (English) Zbl 1252.34034 Bull. Belg. Math. Soc. - Simon Stevin 19, No. 1, 107-119 (2012). The authors consider the following half-linear equation of \(p\)-Laplacian type \[ (\rho(t) |u'(t)|^{p-2}u'(t))' + \lambda \sigma(t) |u(t)|^{p-2}u(t)=0, \quad t\in (a,b),\tag{1} \] subject to the Neumann-Dirichlet boundary conditions \[ \lim_{t\to a+} \rho(t)|u'(t)|^{p-2}u'(t)=\lim_{t\to b-} u(t)=0.\tag{2} \] Here, \(p>1\) is a real number, \(-\infty \leq a<b\leq +\infty\), and \(\rho,\sigma: (a,b)\to (0,+\infty)\) are continuous functions. It is also assumed that \(\sigma \in L^1(a,x)\) and \(\rho^{1-q}\in L^1(x,b)\) for every \(x\in (a,b)\), where \(\frac{1}{p}+\frac{1}{q}=1\).Definition. Problem (1)–(2) has the (S.L.) property if the following conditions are satisfied. (i) The set of eigenvalues of (1)–(2) forms an increasing sequence \(\{\lambda_n\}_1^\infty\) such that \(\lambda_1>0\) and \(\lim_{n\to\infty}\lambda_n=+\infty\).(ii)Eeach eigenvalue is simple in the sense that all eigenfunctions associated with an eigenvalue \(\lambda_n\) are mutually proportional.(iii) The eigenfunction \(u_{\lambda_n}\) associated with \(\lambda_n\) has precisely \(n-1\) zeros in \((a,b)\) and, moreover, the zero points of \(u_{\lambda_{n-1}}\) separate the zero points of \(u_{\lambda_{n}}\) whenever \(n\geq 3\). The main result of the paper under review is the following theorem.Theorem. Problem (1)–(2) has the (S.L.) property if and only if \[ \lim_{t\to a+} \Big( \int_a^t \sigma(\tau) d\tau\Big)\Big( \int_t^b \rho^{1-q}(\tau) d\tau\Big)^{p-1}=0, \] and \[ \lim_{t\to b-} \Big( \int_a^t \sigma(\tau) d\tau\Big)\Big( \int_t^b \rho^{1-q}(\tau) d\tau\Big)^{p-1}=0. \] Reviewer: Aleksey Kostenko (Donetsk) Cited in 5 Documents MSC: 34B24 Sturm-Liouville theory 34B40 Boundary value problems on infinite intervals for ordinary differential equations 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators Keywords:\(p\)-Laplacian; Sturm-Liouville problem; Hardy inequality; weighted spaces; variational eigenvalues; oscillation theory PDFBibTeX XMLCite \textit{P. Drábek} and \textit{K. Kuliev}, Bull. Belg. Math. Soc. - Simon Stevin 19, No. 1, 107--119 (2012; Zbl 1252.34034) Full Text: Euclid