PHP

where post_type like

function title_filter( $where, &$wp_query )
{
    global $wpdb;
    // 2. pull the custom query in here:
    if ( $search_term = $wp_query->get( 'search_prod_title' ) ) {
        $where .= ' AND ' . $wpdb->posts . '.post_title LIKE '%' . esc_sql( like_escape( $search_term ) ) . '%'';
    }
    return $where;
}

$args = array(
    'post_type' => 'product',
    'posts_per_page' => $page_size,
    'paged' => $page,
    // 1. define a custom query var here to pass your term through:
    'search_prod_title' => $search_term,
    'post_status' => 'publish',
    'orderby'     => 'title', 
    'order'       => 'ASC'
);

add_filter( 'posts_where', 'title_filter', 10, 2 );
$wp_query = new WP_Query($args);
remove_filter( 'posts_where', 'title_filter', 10 );
return $wp_query;