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A class of second order BVPs on infinite intervals. (English) Zbl 1134.34018
This paper concerns the study of the solvability of the following boundary value problem on the whole real line
\[ -u'' + cu'+\lambda u = h(x,u),\qquad u(\pm \infty)=0, \] where \(h\) is a continuous function such that \(h(\pm\infty,0)=0\), and \(c,\lambda\) are positive parameters. Such a problem is associated to a generalized Fisher-type equation. The parameter \(c\) can be viewed as a wave speed, whereas \(\lambda\) is an eigenvalue of the problem. Sufficient conditions for the existence of solutions and of positive solutions, classical or weak, are established. Such conditions mainly depend on the growth of the source term \(h\) with respect to the second argument.

MSC:
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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