A Singular Sturm–liouville Equation Involving Measure Data

Abstract

Let α > 0 and let µ be a bounded Radon measure on the interval (−1, 1). We are interested in the equation −(|x| 2α u ) + u = µ on (−1, 1) with boundary condition u(−1) = u(1) = 0. We identify an appropriate concept of solution for this equation, and we establish some existence and uniqueness results. The cases 0 < α < 1 and α ≥ 1 must be considered separately. We also study the limiting behavior of two different approximation schemes: one is the elliptic regularization and the other is to approximate a measure µ by a sequence of L ∞ -functions.

A semilinear singular Sturm–Liouville equation involving measure data

“…We start with a few results from [23] about the linear operator. The investigation of the linear operator can also be found in [11,12].…”

“…In the previous work [23], we studied the corresponding linear equation (i.e., p = 1 in (1.1)). For the linear case, we defined a notion of solution for all α > 0 and a notion of good solution for 0 < α < 1.…”

“…We also presented a necessary and sufficient condition on μ for the existence of the solution when α ≥ 1. For the semilinear equation (1.1), we can adapt from [23] the notion of solution and the notion of good solution. Rewrite (1.1) as −(|x| 2α u ) + u = u − |u| p−1 u + μ.…”

“…Rewrite (1.1) as −(|x| 2α u ) + u = u − |u| p−1 u + μ. Then according to [23], a function u is a solution of (1.1) if u ∈ L p (−1, 1) ∩ W 1 In this work, we are interested in the question of existence and uniqueness, the limiting behavior of three different approximation schemes, and the classification of the isolated singularity at 0.…”

“…Rewrite (1.11) as −(|x| 2α u n ) + u n = u n − g p,n (u n ) + μ. Then according to [23], a function u n is a solution of (1.11) if u n ∈ L 1 (−1, 1) ∩ W 1,1 loc ([−1, 1]\ {0}), |x| 2α u n ∈ BV (−1, 1), and u satisfies (1.11) in the usual sense. When 0 < α < 1, a solution u n of (1.11) is called a good solution if it satisfies in addition (1.3).…”