B737 Captain reported the alternate gear extension door failed to remain close.

Narrative

After takeoff out of ZZZ; the FO PM raised the gear handle. When running the after takeoff flow; he noticed that all of the red and green gear lights were illuminated with the handle in the UP position. I decided to continue the climb to retract the flaps and continue to a safer altitude. We leveled at 10;000ft (4500 ft above filed elevation) and I called for the GEAR DISAGREE checklist. The FO performed the checklist starting at the handle UP condition. This quickly led to the step to check the Alternate Gear Extension Door. He made several attempts to open/close the door before it remained closed. It seemed to want to keep popping up about 1/8 inch. Once the door was fully closed; he continued with the QRH checklist; and the gear came up. We continued to ZZZ1 and made an ELB (Electronic Log Book) entry.Upon block in at ZZZ1; we noticed the door was slightly open again. I spoke to Maintenance Control and he told me that he would have MX (Maintenance) clean the door. The aircraft departed the next day with no issues.