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Second order, Sturm-Liouville problems with asymmetric, superlinear nonlinearities. II. (English) Zbl 1065.34021
This paper considers the nonlinear Sturm-Liouville problem \[ -[p(x)u'(x)]' + q(x) u(x) = f(x, u(x)) + h(x), x \in (0, \pi), \] under the boundary conditions \[ c_{00} u(0) + c_{01} u'(0) = 0, c_{10} u(\pi) + c_{11}u'(\pi) = 0, \] where \(p \in C^1[0, \pi], q \in C^0 [0, \pi]\) with \(p > 0\) for all \(x \in [0, \pi]\); \(c_{i0}^2 + c_{i1}^2 > 0, i = 0, 1; h \in L^2 (0, \pi)\). It is assumed that \(f\) is continuous and there exist increasing functions \(\zeta_l, \zeta_u:[0, \infty) \rightarrow \mathbb R\), and positive constants \(A, B\) such that \(\lim_{t \rightarrow \infty} \zeta_l (t) = \infty\) and \(-A + \zeta_l (\xi) \xi \leq f(x, \xi) \leq A + \zeta_u (\xi) \xi, \xi \geq 0\) as well as \(| f(x, \xi) | \leq A + B | \xi |, \xi \leq 0\). Thus, the nonlinearity of the equation is superlinear as \(u(x) \rightarrow \infty\) and linearly bounded as \(u(x) \rightarrow - \infty\), so it is said to be asymmetric.
On the other hand, let \(E\) denote the set of \(u \in C^1 [0, \pi]\) satisfying the same boundary conditions described above. \(H = H^2 (0, \pi) \cap E\). The operator \(L\) is defined by \(L u = -(pu')' + q u\) for \(u \in H\). Suppose \((a, b) \in L^\infty (0, \pi)^2\) and let \(\sum_H (a, b)\) denote the set of \(\lambda \in \mathbb R\) for which the equation \(Lu = au^+ - bu^- + \lambda u\) has a nontrivial solution \(u \in H\). Such a \(\lambda\) is called a half-eigenvalue while \(u\) is called a half-eigenfunction. When \(f\) is linearly bounded as \(| \xi |\) tends to \(\infty\), the relation between the sign of certain half-eigenvalues and the existence of the solution has been investigated by the author in [J. Differ. Equations 161, 87–109 (2000; Zbl 0976.34024); ibid. 170, 215–227 (2001; Zbl 0986.34021)].
In this paper, conditions for the existence and nonexistence of the solutions are obtained and are expressed in terms of the signs of the asymptotes of the half-eigenvalues, where these asymptotes are obtained by letting the linear bound on \(f\) tend to \(\infty\) as \(\xi\) tends to \(\infty\).

MSC:
34B24 Sturm-Liouville theory
34B15 Nonlinear boundary value problems for ordinary differential equations
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