Second order, Sturm–Liouville problems with asymmetric, superlinear nonlinearities II

We consider the nonlinear Sturm–Liouville problem −(p(x)u′(x))′+q(x)u(x)=f(x,u(x))+h(x), in (0,π),c 00u(0)+c 01u′(0)=0, c 10u(π)+c 11u′(π)=0, where p∈C 1[0,π], q∈C 0[0,π], with p( x)>0 for all x∈[0, π]; c i0 2+ c i1 2>0, i=0, 1 ; h∈ L 2(0, π). We suppose that f:[0,π]× R→ R is continuous and there exist increasing functions ζ l, ζ u:[0,∞)→ R , and positive constants A, B, such that lim t→∞ ζ l(t)=∞ and −A+ζ l(ξ)ξ⩽f(x,ξ)⩽A+ζ u(ξ)ξ, ξ⩾0,|f(x,ξ)|⩽A+B|ξ|, ξ⩽0, for all x∈[0, π] (thus the nonlinearity is superlinear as u( x)→∞, but linearly bounded as u( x)→−∞). Existence and non-existence results are obtained for the above problem. Similar results have been obtained before for problems in which f is linearly bounded as | ξ|→∞, and these results have been expressed in terms of ‘half-eigenvalues’ of the problem. The results obtained here for the superlinear case are expressed in terms of certain asymptotes of these half-eigenvalues.