Gemara

Rav Papa and Rav Huna brei d'Rav Yehoshua would eat Chodosh on the night after the 16th They hold that Chodosh is only mid'Rabanan in Chutz la'Aretz, so we are lenient about a Safek (which day is really the 16th).

Sukah 29b (Mishnah): A stolen or dry Lulav is Pasul.

Inference: The Mishnah disqualifies Stam, i.e. even on Yom Tov Sheni.

Question: Granted, dry is Pasul because it is not Hadar (beautiful). Gazul (stolen) is Pasul on Yom Rishon, for it is not "Lachem" (yours). However, why is it Pasul on Yom Tov Sheni?

Answer (R. Yochanan, citing R. Shimon): It is Pasul due to Mitzvah ha'Ba'ah b'Aveirah.

30a (R. Ami): Dry is Pasul because it is not Hadar. Gazul is Pasul due to Mitzvah ha'Ba'ah b'Aveirah.

They argue with R. Yitzchak;

(R. Yitzchak citing Shmuel): The Mishnah disqualifies (Gazul) on Yom Rishon. On Yom Tov Sheni, just like one is Yotzei with borrowed Minim, he is Yotzei with Gazul.

Question (Rav Nachman bar Yitzchak - Mishnah): A stolen or dry Lulav is Pasul.

Inference: A borrowed Lulav is Kosher.

This cannot be on Yom Rishon, for it is not "Lachem"! Rather, it is on Yom Sheni, and Gazul is Pasul!

Answer (Rava): Really, it is on Yom Tov Rishon. Not only borrowed is Pasul, rather, even Gazul. One might have thought that in a Stam case of open robbery, the owner despairs, and it is as if the Gazlan owns it (and he is Yotzei with it)! The Mishnah teaches that this is not so.

36b - Version #1 (Rav): If mice nibbled on an Esrog, it is not Hadar. (It is Pasul.)

Question: R. Chanina would eat part of his Esrog, and be Yotzei with the remains!

Question (against R. Chanina - Mishnah): A Chaser (deficient) Esrog is Pasul.

Answer: The Mishnah discusses Yom Rishon. He was Yotzei with it on Yom Sheni.

Answer: If mice nibbled on it, (this is worse than not Hadar;) it is Ma'os (repulsive).

Version #2 (Rav): If mice nibbled on an Esrog, it is Hadar.

Rishonim

Rif (Sukah 17b): R. Chanina was Yotzei with the remains of his Esrog on Yom Sheni, but not on Yom Rishon. This is the Halachah.

Ran (13b DH u'Mihu): Our Mishnah equates Yom Rishon and Sheni. The Gemara accepted this regarding Hadar. On 36b, we require Hadar only on Yom Rishon! Really, it is needed on Yom Sheni. The remainder of the Esrog was Hadar. Dry is not Hadar at all. It is worse than what mice nibbled on. The Yerushalmi disqualifies it due "Lo ha'Mesim Yehalelu Kah." Version #2 proves this. Had the Torah explicitly disqualified any hole or Chisaron (lack), we would disqualify it even on Yom Sheni mid'Rabanan. Chachamim enact like Torah law. Rather, the Torah Stam required Hadar, and left it to Chachamim (to define what is Hadar). They enacted (Hadar for Yom Sheni,) like Torah law, but it would not be proper to be as stringent (as for Yom Rishon). However, the Gemara says that the Mishnah applies to Yom Rishon and Sheni. How can we be Machshir Chaser on Yom Sheni? Perhaps they relied on Rav Nachman bar Yitzchak's Diyuk, that borrowed is Kosher (to prove that our Mishnah discusses Yom Sheni), but there is no source to say so for the Mishnah about Chaser. Also, surely is Pasul on Yom Sheni. Since Gazul was taught with it, also Gazul is Pasul on Yom Sheni. This does not apply to Chaser, since the remainder is Hadar.

Ran (ibid.): The Ramban says that the entire Perek applies also to Yom Tov Sheni (due to Safek) mid'Oraisa. The first Mishnah to discuss the Mitzvah mid'Rabanan to take the Lulav all seven days is on 41a. The Sugya of an Esrog that mice nibbled on discusses Yom Tov mid'Rabanan. We disqualify an Esrog of Orlah and Tamei Terumah because one may not eat it, or it is not yours. This applies only to Yom Rishon. However, since it is Asur b'Hana'ah, and it must be burned, so Kesusei Michtas Shi'urei (it is considered not to have the required size), so it is Pasul all seven days. All Pesulim due to Hadar are Kosher on Yom Tov Sheni mid'Rabanan. If Chazazis (like warts) covers the majority, the Yerushalmi disqualifies only on Yom Rishon. The Ge'onim rely on this, since the Bavli is Machshir Chaser and what mice nibbled on, which is repulsive. How can we disqualify Chazazis and a missing Pitom?! We are stringent only when Kesusei Michtas Shi'urei applies, or if it is the wrong species or less than a Shi'ur, or Mitzvah ha'Ba'ah b'Aveirah according to the Ramban. How can one bless on what Hash-m despises?! The Ramban is Machshir dry on Yom Sheni. I disqualify. The Ra'avad proved that it is Pasul even for Yom Tov Sheni mid'Rabanan. R. Yehudah tried to prove that dry is Kosher from a place where people bequeathed Lulavim to their children. Chachamim said 'we do not bring a proof from pressed circumstances.' They did not say that they were used for Yom Tov Sheni mid'Rabanan! This shows that it is Pasul all seven days.

Ran (ibid.): The Ba'al ha'Ma'or is Machshir Gazul on Yom Sheni, for Rav Ashi explained Shmuel. Some say that nowadays the Pesulim of Yom Tov Sheni do not apply, for we have a fixed calendar and there is no Safek. We keep Yom Tov Sheni merely to continue our ancestors' custom. We are stringent about Kedushas ha'Yom, but not about Lulav, lest this lead to a leniency, i.e. one who lacks a Hadar Lulav will be Batel from the Mitzvah. We have no proof to exempt ourselves. I disagree. We, like our ancestors who had a Safek about the day, take on Yom Sheni what is Pasul on Yom Rishon, but without a Berachah.

Rambam (Hilchos Lulav 8:9): Whatever is Pasul due to blemishes that we explained, or due to Gazul, is Pasul only on Yom Rishon. On Yom Tov Sheni and all other days it is Kosher.

Ra'avad: Not all agree about Gazul and dry. They are not Pasul due to Hadar. Rather, Gazul is Mitzvah ha'Ba'ah b'Aveirah, and dry is like dead. The Yerushalmi says so.

Hagahos Maimoniyos (100): The Rambam rules like Shmuel. The Re'em disagrees. In the Yerushalmi, Shmuel bar Nachmani says that all Pesulim are only on Yom Rishon. R. Chanina ate part of his Esrog each day (and was Yotzei with what remained). He holds like the other opinion that is Machshir on Yom Sheni even if mice nibbled on it, unlike the version that disqualifies due to Mius. If one did not have an Esrog, and he was not lazy and negligent about the Mitzvah, logic dictates that he may bless even on an Esrog with a hole or another Pesul, even though it is not Hadar. R. Yehudah and Rabanan argue about whether we require Hadar for the Lulav and Esrog. We may rely on R. Yehudah in pressed circumstances, even though Rabim disagree. Another proof is from a dry Lulav. It is Pasul on Yom Rishon. Surely, it is Kosher on other days. He also brought a proof from brothers who bought an Esrog from the estate. We exclude only a quince or pomegranate, but not Chaser. The same applies to a slightly broken Lulav, if he does not have another, like we say about dry. We say that one cannot bring a proof from pressed circumstances (then it is Kosher). However, if mice nibbled on it, it is Pasul even in pressed circumstances, due to Mi'us, like the Gemara says.

Magid Mishneh: The Ge'onim learned from R. Chanina that all blemishes are Kosher on Yom Sheni. This (it was partially eaten) is the biggest blemish. However, one is not Yotzei with Pesulim due to Shi'ur, i.e. Kesusei Michtas Shi'urei. All agree to this. Some disqualify also Gazul and dry on Yom Sheni, like the Ra'avad says. The Ge'onim say like the Rambam. They are Machshir on Yom Sheni everything except for what one may not eat. In the Gemara, they argue about Yom Tov Sheni in the Mikdash, even though it is mid'Oraisa (to take the Lulav all seven days in the Mikdash). We keep two days of Yom Tov. Some say that the second day is like the first. If not, people would disgrace it. Others say that it is like other days, for we know when the month began, and that (mid'Oraisa) the first day is Yom Tov. Therefore, we are not stringent about these blemishes (on Yom Sheni). Also, according to the first opinion, if he cannot find Minim without a blemish, should he not take anything?! The stringency leads to a leniency! The Rambam's words connotes that Yom Sheni is like other days. There is no proof for this from the Gemara. What the Rambam disqualified in Perek 7, because it is the wrong species, or it is less than the Shi'ur, is Pasul on the second day like on the first.

Rosh (Sukah 3:3): For us, Yom Tov Sheni is like Yom Rishon, for it is a Safek. I saw other opinions, but I did not write them, for they are not proper in my eyes. What I wrote is primary.

Rashba (Teshuvah 1:23): Since we keep Yom Tov Sheni because it is a Safek if it is really Yom Tov, it is reasonable that one must use a Lulav that is Kosher for Yom Rishon.

Poskim

Shulchan Aruch (OC 649:5): Those who observe two days of Yom Tov, they may take on Yom Tov Sheni Minim Pasul on Yom Tov Rishon, but they do not bless.

Taz (10): We have no Safek about the day (so Pesulim of Yom Rishon are Kosher). Or, perhaps Chachamim enacted like Torah law. Therefore, we take it without a Berachah.

Gra (DH veha'Osim): The Rashba explains that we keep our ancestors' custom, i.e. and it is as if it is a Safek which day is Yom Tov. This is why we do not take it when Yom Rishon on Shabbos (even though they used to where they had no Safek - Sukah 41b). The Rif (Megilah 2b) says that this is one reason why we do not fulfill Shofar on Shabbos. We do not refrain from Bris Milah on Shabbos, for there was never Safek about the day.

Teshuvas Pnei Yehoshua (3, cited in R. Akiva Eiger): The Beis Yosef and Poskim say that one takes on Yom Sheni Pesulim of Yom Rishon, but he does not bless. If Kosher Minim are available, he should bless on them! If Kosher Minim are not available, this is pressed circumstances, and even on Yom Rishon one takes them without a Berachah! We can say that with difficulty, one could get Kosher Minim. (On Yom Sheni) one is not obligated to do so merely for the Berachah, since he is Yotzei with Pesulim of Yom Rishon.

Kaf ha'Chayim (86): Some say that on Yom Rishon, only the Shali'ach Tzibur takes a Pasul Lulav, but on Yom Sheni everyone takes it.

Eshel Avraham (8): In Chutz la'Aretz, one is not Yotzei with a borrowed Lulav on the second day. Therefore, one cannot give his Lulav even after he was Yotzei on the first day. To fulfill Chinuch one must buy a Lulav for the child himself

Birkei Yosef (657:2): On Yom Tov Sheni, one may ne train his son with borrowed Minim. We are lenient about Safek Chodosh mid'Rabanan (Menachos 68b). Likewise, we are lenient about Chinuch mid'Rabanan.

Meshiv Davar (1:41 DH Asher): He permits to let the child bless on it, unlike an adult, who may not bless on it. I disagree, for one must train a child to do what he must do when he is an adult.

Ha'aros on Birkei Yosef (2): One need not tell the child that the Lulav is borrowed, just like the child need not know all the laws when he takes it for Chinuch. Batei Kehunah says that one may not be Mechanech with borrowed Minim on Yom Rishon, for the source of the Mitzvah is mid'Oraisa. Yom Tov Sheni is not considered to have a source in the Torah.

Bi'ur Halachah (DH Pesulei): One may not bless on a borrowed Lulav on Yom Tov Sheni. The Rema (below) connotes like this. People are not careful about this.

Rema: If someone else has a Kosher Lulav and Esrog, he blesses on it with the owner' consent.

Taz (11): He should acquire it, like for the first day. If the owner does not know, he consents that others to do a Mitzvah with his property, but the borrower does not own it.

Mishnah Berurah (51): First one borrows a Kosher Lulav and blesses on it, then he takes his own for Hallel and the Hoshanos.

Kaf ha'Chayim (85): If one takes the Pasul Lulav first, afterwards he cannot bless, for perhaps he was already Yotzei.