Sukkah 3:12 teaches:
בָּרִאשׁוֹנָה הָיָה לוּלָב נִטָּל בַּמִּקְדָּשׁ שִׁבְעָה, וּבַמְּדִינָה יוֹם אֶחָד. מִשֶּׁחָרַב בֵּית הַמִּקְדָּשׁ, הִתְקִין רַבָּן יוֹחָנָן בֶּן זַכַּאי שֶׁיְּהֵא לוּלָב נִטָּל בַּמְּדִינָה שִׁבְעָה, זֵכֶר לַמִּקְדָשׁ. וְשֶׁיְּהֵא יוֹם הָנֵף כֻּלּוֹ אָסוּר:
In earlier times the lulav was taken for seven days in the Temple, and in the provinces for one day only. When the temple was destroyed, Rabbi Yohanan ben Zakkai decreed that the lulav should be taken in the provinces for seven days in memory of the Temple, [He also decreed] that on the whole of the day of waving it be forbidden [to eat the new produce].
So we see taking the lulav in the provinces for seven days is Rabbinic.
Rashi explains that in Toras Kohanim it is derived from the verse "You shall rejoice before Hashem seven days" (Leviticus 23:40) is referring to the Biblical obligation to take the four species in the temple for seven days, and the inference is that outside the Temple it isn't for seven days.
Some say (I forget who) that according to the Rambam there is a Biblical mitzvah to take the lulav all seven days in (the old city of) Jerusalem. This is because in his commentary to the Mishnah he says the mitzvah is only one day everywhere but Jerusalem. This implies he understood "in the Temple" to not be literal but referring to all of Jerusalem.