Действие при нажатии или навидении на button появление дива div

.main-menu {
  display: block; /* display: none; */
}

.btn-menu:hover+.main-menu {
  display: block;
}

.main-menu div .list-menu {
  position: relative;
  padding: 0;
  margin: 0;
  list-style-type: none;
}

.main-menu div .list-menu::after {
  content: '';
  position: absolute;
  left: 48%;
  top: -1.2rem;
  border: 10px solid transparent;
  border-top-color: transparent;
  border-top-style: solid;
  border-top-width: 10px;
  border-top: 10px solid rgb(248 249 250);
  -moz-transform: rotate(180deg);
  -ms-transform: rotate(180deg);
  -webkit-transform: rotate(180deg);
  -o-transform: rotate(180deg);
  transform: rotate(180deg);
}

.main-menu div .list-menu li {
  border-left: 1px solid rgb(222 226 230);
  border-top: 1px solid rgb(222 226 230);
  border-right: 1px solid rgb(222 226 230);
}

.main-menu div .list-menu li:last-child {
  border-bottom: 1px solid rgb(222 226 230);
  -webkit-border-bottom-right-radius: .25rem;
  -webkit-border-bottom-left-radius: .25rem;
  -moz-border-radius-bottomright: .25rem;
  -moz-border-radius-bottomleft: .25rem;
  border-bottom-right-radius: .25rem;
  border-bottom-left-radius: .25rem;
}

.main-menu div .list-menu li a {
  width: 100%;
  display: block;
  padding: .2rem .5rem;
  color: rgb(0 0 0);
  text-transform: uppercase;
  text-decoration: none;
  transition: 0.5s;
  -webkit-transition: 0.5s;
  -moz-transition: 0.5s;
  -o-transition: 0.5s;
}

.main-menu div .list-menu li:hover {
  border-right: 0;
  -webkit-border-bottom-right-radius: 0;
  -moz-border-radius-bottomright: 0;
  border-bottom-right-radius: 0;
}

.main-menu div .list-menu li a:hover {
  padding-left: 1rem;
}

.main-menu div .content-menu {
  display: block; /* display: none; */
}

.main-menu div .list-menu li:hover .content-menu {
  display: block;
}

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<link href="https://cdn.jsdelivr.net/npm/[email protected]/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-1BmE4kWBq78iYhFldvKuhfTAU6auU8tT94WrHftjDbrCEXSU1oBoqyl2QvZ6jIW3" crossorigin="anonymous">
<script src="https://cdn.jsdelivr.net/npm/[email protected]/dist/js/bootstrap.bundle.min.js" integrity="sha384-ka7Sk0Gln4gmtz2MlQnikT1wXgYsOg+OMhuP+IlRH9sENBO0LRn5q+8nbTov4+1p" crossorigin="anonymous"></script>
<script src="https://cdn.jsdelivr.net/npm/@popperjs/[email protected]/dist/umd/popper.min.js" integrity="sha384-7+zCNj/IqJ95wo16oMtfsKbZ9ccEh31eOz1HGyDuCQ6wgnyJNSYdrPa03rtR1zdB" crossorigin="anonymous"></script>

<div class="container">
  <div class="row">
    <div class="col-3 btn btn-primary btn-menu">Nanigation</div>
    <form class="d-flex col-6">
      <input class="form-control me-2" type="search" placeholder="Search" aria-label="Search">
      <button class="btn btn-outline-success" type="submit">Search</button>
    </form>
    <button type="button" class="col-3 btn btn-danger">Button</button>
  </div>
</div>
<div class="container position-relative main-menu px-0">
  <div class="row position-absolute w-100 g-0">
    <nav class="col-3 list-menu bg-light">
      <li><a href="">Link 1</a></li>
      <li><a href="">Link 2</a></li>
      <li><a href="">Link 3</a></li>
      <li><a href="">Link 4</a></li>
    </nav>
    <div class="col-9 border border-start-0 rounded content-menu bg-light">
      <h3 class="h-100 text-center">title category 1</h3>
    </div>
    <div class="col-9 border border-start-0 rounded content-menu bg-light">
      <h3 class="h-100 text-center">title category 2</h3>
    </div>
  </div>
</div>
<div class="container">
</div>