if get_post_meta function returns empty – Do Not Display HTML

If the custom field will be always empty when there is no pdf file, you can check if it is not empty before displaying the link:

    <?php
    $pdf_name = get_post_meta($post->ID, 'pdf_name', true);
    if($pdf_name) {
    ?>
    <a class="pdfDownload" href="<?php bloginfo('url'); ?>/wp-content/uploads/<?php echo $pdf_name; ?>">PDF Download</a>
    <?php
    }
    ?>

But, if there is a possibility that the custom field can hold a value even if there is no actual pdf file attached, then you can check if the pdf file actually exists by its path before displaying the download URL:

    <?php
    $pdf_name = get_post_meta($post->ID, 'pdf_name', true);
    $uploads_dir = wp_upload_dir();
    $pdf_path = $uploads_dir['basedir'] . "https://wordpress.stackexchange.com/" . $pdf_name;
    if(is_file($pdf_path)) { ?>
        <a class="pdfDownload" href="<?php bloginfo('url'); ?>/wp-content/uploads/<?php echo $pdf_name; ?>">PDF Download</a>
   <?php
    }
    ?>

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