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Half-linear Sturm-Liouville problem with weights. (English) Zbl 1252.34034

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 \[ \lim_{t\to b-} \Big( \int_a^t \sigma(\tau) d\tau\Big)\Big( \int_t^b \rho^{1-q}(\tau) d\tau\Big)^{p-1}=0. \]

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
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Full Text: Euclid