B757 Captain reported receiving a leading edge slat asymmetry message during climb out. The flight crew discussed options with company and continued to destination where an uneventful landing was made.

Narrative

On departure; after the flap indicator showed UP; at approximately 4000 feet MSL; an amber LE SLAT ASYM message posted. We complied with the QRH. We stopped our climb and coordinated with maintenance and dispatch via ACARS for input regarding best maintenance location and fuel requirements. The QRH was not restrictive concerning when to land. We discussed options and solicited additional input from dispatch. Both the dispatcher and maintenance control seemed to support continuing to destination. We briefly considered diverting to ZZZ1 but there were isolated thunderstorms in close vicinity to the airport. The weather at ZZZ was currently nice and forecast to remain nice. We decided to continue to destination. We coordinated the limit speed (240 KIAS) and a cruise altitude of FL190 with ATC. Because of light chop; we set 235 KIAS as the target speed. Since there was no need for priority handling in cruise; we waited until initial contact with approach control to [request priority handling]. We asked for equipment to be standing by. Made an uneventful F20 [Flaps 20] landing on Rwy XXR using ALTN flaps (Per the QRH). CFR (Crash Fire Rescue) was standing by but no action was taken. We taxied off and to the gate on our own.