How to determine if asp.net core has been installed on a windows server

You can search Microsoft .NET Core 1.1.1 - Windows Server Hosting registry key under HKEY_LOCAL_MACHINE\SOFTWARE\WOW6432Node\Microsoft\Updates\.NET Core path like screenshot below.

Also you can use PowerShell to determine the whether the key existed or not.

$DotNETCoreUpdatesPath = "Registry::HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Updates\.NET Core"
$DotNetCoreItems = Get-Item -ErrorAction Stop -Path $DotNETCoreUpdatesPath
$NotInstalled = $True
$DotNetCoreItems.GetSubKeyNames() | Where { $_ -Match "Microsoft .NET Core.*Windows Server Hosting" } | ForEach-Object {
    $NotInstalled = $False
    Write-Host "The host has installed $_"
}
If ($NotInstalled) {
    Write-Host "Can not find ASP.NET Core installed on the host"
}

And you can download sample from How to determine ASP.NET Core installation on a Windows Server by PowerShell.