A Variational ODE and its Application to an Elliptic Problem

Abstract

In this paper, we consider the following ODE problem

$$ \left\{ {\begin{array}{*{20}c} {{ - {u}\ifmmode{''}\else$''$\fi{\left( r \right)} + \frac{{{\left( {N - 1} \right)}{\left( {N - 3} \right)}}} {{4r^{2} }}u{\left( r \right)} + \lambda u{\left( r \right)} = f{\left( {r,r^{{\frac{{1 - N}} {2}}} u} \right)}u{\left( r \right)},}} & {{r > 0,}} & {{u \in H,}} & {{N \geqslant 3.}} \\ \end{array} } \right. $$

((P))

where f ∈ C((0,+∞) × ℝ,ℝ), f(r, s) goes to p(r) and q(r) uniformly in r > 0 as s → 0 and s → +∞, respectively, 0 ≤ p(r) ≤ q(r) ∈ L ^∞(0,∞). Moreover, for r > 0, f(r, s) is nondecreasing in s ≥ 0. Some existence and non-existence of positive solutions to problem (P) are proved without assuming that p(r) ≡ 0 and q(r) has a limit at infinity. Based on these results, we get the existence of positive solutions for an elliptic problem.