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

We consider the nonlinear Sturm–Liouville problem View the MathML source where p∈C1[0,π],q∈C0[0,π], with p(x)>0 for all x∈[0,π]; ci02+ci12>0, i=0,1; h∈L2(0,π). We suppose that View the MathML source is continuous and there exist increasing functions View the MathML source, and positive constants A, B, such that View the MathML source and View the MathML source 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.