Heteroclinic solutions for a class of boundary value problems associated with singular equations

On the solvability of singular boundary value problems on the real line in the critical growth case

“…Finally, we explicitly note that Theorem 2.3 comprehends also the case when a(t, x) = a(t), that is a is independent of x. As a consequence, the result in this paper complete the study started in [6,7]. To the best of our knowledge it remains open the problem when f (t, ·, ·) ≈ |t| γ as t → ∞, for suitable γ < −1.…”

“…The literature concerning BVP for singular ODEs of the form (4) (both on compact and non-compact intervals) seems to be rather incomplete. To the best of our knowledge, we mention the paper [15] in which the authors obtained the solvability of (4) assuming that a(t, x) = k(t) and I = [0, T ] for some T > 0; more recently, this paper has been generalized in [7] to the case where a depends on (t, x) (and again I = [0, T ]).…”

“…As regards the case I = (0, ∞) or I = R, we highlight the paper [6] in which the authors establish the existence of at least one weak solution of…”

On the existence of weak solutions for singular strongly nonlinear boundary value problems on the half-line

“…In view of their several applications, ODEs of the form (3) have been deeply studied in the literature since the early 1990s, and it is beyond our scopes to present here an exhaustive list of references; in the context of non-singular ODEs we highlight the papers [7,8,13,[17][18][19][20][28][29][30], where the techniques exploited are somehow similar to ours (which shall be described later on).…”

“…It is also worth observing that, in addition to ODEs associated with Φ-Laplace operators (which are assumed to be homeomorphisms from R onto itself), several generalizations of (3) have been proposed and studied, for example, ODEs involving singular Φ-Laplace operators (i.e., with bounded domain) or non-surjective Φ-Laplace operators; see, e.g., [1,9,16].…”