Hi Govand,
In order to keep the Bootstrap Modal Popup open after submt you need to be re-opened it.
Check this example. Now please take its reference and correct your code.
Controller
public class HomeController : Controller
{
    public IActionResult Index()
    {
        return View();
    }
    [HttpPost]
    public IActionResult Index(string name)
    {
        ViewBag.Inserted = !string.IsNullOrEmpty(name) ? true : false;
        return View();
    }
}
View
@addTagHelper*, Microsoft.AspNetCore.Mvc.TagHelpers
@{
    Layout = null;
}
<!DOCTYPE html>
<html>
<head>
    <meta name="viewport" content="width=device-width" />
    <title>Index</title>
</head>
<body>
    <script src="https://code.jquery.com/jquery-3.2.1.slim.min.js" integrity="sha384-KJ3o2DKtIkvYIK3UENzmM7KCkRr/rE9/Qpg6aAZGJwFDMVNA/GpGFF93hXpG5KkN" crossorigin="anonymous"></script>
    <link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/css/bootstrap.min.css" integrity="sha384-Gn5384xqQ1aoWXA+058RXPxPg6fy4IWvTNh0E263XmFcJlSAwiGgFAW/dAiS6JXm" crossorigin="anonymous" />
    <script src="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/js/bootstrap.min.js" integrity="sha384-JZR6Spejh4U02d8jOt6vLEHfe/JQGiRRSQQxSfFWpi1MquVdAyjUar5+76PVCmYl" crossorigin="anonymous"></script>
    <button type="button" class="btn btn-primary" data-toggle="modal" data-target="#exampleModal">Insert</button>
    <form asp-action="Index" asp-controller="Home" method="post">
        <div id="exampleModal" class="modal" tabindex="-1" role="dialog">
            <div class="modal-dialog" role="document">
                <div class="modal-content">
                    <div class="modal-header">
                        <h5 class="modal-title">Details</h5>
                        <button type="button" class="close" data-dismiss="modal" aria-label="Close">
                            <span aria-hidden="true">×</span>
                        </button>
                    </div>
                    <div class="modal-body">
                        <div class="form-group">
                            <label>Name:</label>
                            <input type="text" name="name" class="form-control" />
                        </div>
                    </div>
                    <div class="modal-footer">
                        <button type="submit" class="btn btn-primary">Save</button>
                        <button type="button" class="btn btn-secondary" data-dismiss="modal">Close</button>
                    </div>
                </div>
            </div>
        </div>
    </form>
    @if (ViewBag.Inserted != null && !ViewBag.Inserted)
    {
        <script type="text/javascript">
            $(function () {
                $('#exampleModal').modal('show');
            });
        </script>
    }
</body>
</html>
Screenshot
