AN ASYMPTOTIC MAJORANT FOR SOLUTIONS OF STURM–LIOUVILLE EQUATIONS IN $L_P(\mathbb{R})$

Abstract

Under certain assumptions on $g(x)$, we obtain an asymptotic formula for computing integrals of the form

$$ F(x,\alpha)=\int_{-\infty}^\infty g(t)^\alpha\exp\biggl(-\bigg|\int_x^tg(\xi)\,\mathrm{d}\xi\bigg|\biggr)\,\mathrm{d}t,\quad\alpha\in\mathbb{R}, $$

as $|x|\to\infty$. We use this formula to study the properties (as $|x|\to\infty$) of the solutions of the correctly solvable equations in $L_p(\mathbb{R})$, $p\in[1,\infty]$,

\begin{equation} -y''(x)+q(x)y(x)=f(x),\quad x\in\mathbb{R}, \tag{1} \end{equation}

where $0\le q\in L_1^{\mathrm{loc}}(\mathbb{R})$, and $f\in L_p(\mathbb{R})$. (Equation (1) is called correctly solvable in a given space $L_p(\mathbb{R})$ if for any function $f\in L_p(\mathbb{R})$ it has a unique solution $y\in L_p(\mathbb{R})$ and if the following inequality holds with an absolute constraint $c_p\in(0,\infty)$: $\|y\|_{L_p(\mathbb{R})}\leq c(p)\|f\|_{L_p(\mathbb{R})}$, $\forall f\in L_p(\mathbb{R})$.)