The traditional interpretation of these verses is that they describe Christ. This is supported by passages in the book of Revelation:
15 And out of his mouth goeth a sharp sword, that with it he should smite the nations: and he shall rule them with a rod of iron: and he treadeth the winepress of the fierceness and wrath of Almighty God.
16 And he hath on his vesture and on his thigh a name written, KING OF KINGS, AND LORD OF LORDS. [Revelation 19:15–16; see also 2:16]
However, Isaiah 11:2–4 must be understood within the context established in verse 1, including the “rod” (servant) mentioned there, and verse 2 through 4 especially may refer to Joseph Smith. “The Spirit of the Lord shall rest upon him [the servant]” (v. 2). The Lord revealed: “this generation shall have my word through [Joseph Smith]” (D&C 5:10). After summarizing his message to the Gentiles, the Lamanites, and the Jews (2 Nephi 30:1–7), all to be fulfilled in the latter days, Nephi says: “ the Lord God shall commence his work among all nations … to bring about the restoration of his people upon the earth” (2 Nephi 30:8). Joseph Smith headed the latter-day work among all nations to restore God’s people. He was given the keys of the kingdom, “which kingdom is coming forth for the last time,” and promised that “the keys of this kingdom shall never be taken from you, while thou art in the world, neither in the world to come” (D&C 90:2–3).
Nephi quoted Isaiah 11:4–9 after his prophesy of the restoration of God’s people and added his comments (2 Nephi 30:9–15). In 2 Nephi 30:9 (Isaiah 11:4), Nephi uses “the Lord God” instead of the pronoun “he will judge the people.” After all, the servant was to be an instrument in the hands of Christ to “commence his work” (v. 8). Since the Book of Mormon was brought forth by Joseph Smith to begin the Lord’s work, and Jesus Christ will cleanse the earth and slay the wicked (v. 2) to conclude his work and bring about the millennium, Nephi’s change seems appropriate. As the Spirit of the Lord came upon Joseph Smith, he did attain wisdom, understanding, and knowledge as he began the work (2 Nephi 21:2). Furthermore, Joseph did also judge in righteousness through the Spirit of the Lord (2 Nephi 21:3).