Avoiding multiple If statements in Java

Using a HashMap perhaps?

This way you could do myMap.get(mystr);

You can use a Map to hold your solutions:

Map<String,String> extensionToMimeType = new HashMap<String,String>();
extensionToMimeType.put("pdf", "application/pdf");
extensionToMimeType.put("doc", "application/msword");
// and the rest

int lastDot = fileName.lastIndexOf(".");
String mimeType;
if (lastDot == -1) {
    mimeType = NO_EXTENSION_MIME_TYPE;
} else {
    String extension = fileName.substring(lastDot+1);
    mimeType = extensionToMimeType.getOrDefault(extension, 
                                                UNKNOWN_EXTENSION_MIME_TYPE);
}

For this code to work you'll need to have defined NO_EXTENSION_MIME_TYPE and UNKNOWN_EXTENSION_MIME_TYPE as in your class, somewhat like this:

private static final String NO_EXTENSION_MIME_TYPE = "application/octet-stream";
private static final String UNKNOWN_EXTENSION_MIME_TYPE = "text/plain";