Stop sending email everytime I updates my post

You need to use {old_status}to{new_status} hook. And use draft and publish statuses. This hook will only be executed when your post will change status from draft to publish.

function wpse_288250_send_email( $post ) {

    $post_id = $post->ID;

    if( wp_is_post_revision( $post_id ) ) {
        return;
    }

    $post_url = get_permalink( $post_id );
    $subject="Your post is published!";

    $message = "Testing!";
    $message .= "<a href="". $post_url. "">Click here to view</a>";

    $email = get_post_meta($post_id, 'author_email', true ); 

    wp_mail($email, $subject, $message );
}

add_action(  'draft_to_publish',  'wpse_288250_send_email' );

Keep in mind if you change your post status back to draft and again to publish e-mail will be send again. To prevent this you can update post meta which will tell you if e-mail was already sent.

function wpse_288250_send_email_once( $post ) {

    $post_id = $post->ID;

    if( wp_is_post_revision( $post_id ) ) {
        return;
    }

    $email_sent = get_post_meta( $post_id, 'email_sent' );

    if( $email_sent ) {
        return;
    }

    $post_url = get_permalink( $post_id );
    $subject="Your post is published!";

    $message = "Testing!";
    $message .= "<a href="". $post_url. "">Click here to view</a>";

    $email = get_post_meta($post_id, 'author_email', true ); 

    wp_mail($email, $subject, $message );
    update_post_meta( $post_id, 'email_sent', true );
}

add_action(  'draft_to_publish',  'wpse_288250_send_email_once' );