
677 F.3d 346 (2012)
Larry NEELY, Appellant,
v.
Dustin McDANIEL, Arkansas Attorney General, Appellee.
No. 10-3227.
United States Court of Appeals, Eighth Circuit.
Submitted: January 12, 2012.
Filed: May 2, 2012.
*348 J. Thomas Sullivan, argued and on the brief, Little Rock, AR, for appellant.
Christian Harris, AAG, argued and on the brief, Little Rock, AR, for appellee.
Before BYE, SMITH, and COLLOTON, Circuit Judges.
COLLOTON, Circuit Judge.
Larry Neely pleaded guilty in Arkansas state court to sexual indecency with a child. He later filed a petition for writ of habeas corpus, pursuant to 28 U.S.C. § 2254, challenging the constitutionality of the statute under which he was convicted. The district court[1] dismissed the action, and we affirm.

I.
During the autumn of 2003, Neely made a series of telephone calls from his home in Albuquerque, New Mexico, to minors living in Lonoke, Arkansas. In some of the calls, Neely offered to perform oral sex on the minors, including one boy who was *349 fourteen years old at the time. After police traced the calls to Neely, he pleaded guilty to five misdemeanor counts of harassing communications and two felony counts of sexual indecency with a child, in violation of Ark.Code Ann. §§ 5-71-209 and 5-14-110, respectively.
The Arkansas felony statute states, in relevant part:
(a) A person commits sexual indecency with a child if:
(1) Being eighteen (18) years of age or older, the person solicits another person who is less than fifteen (15) years of age or who is represented to be less than fifteen (15) years of age to engage in:
(A) Sexual intercourse;
(B) Deviate sexual activity; or
(C) Sexual contact
Ark.Code Ann. § 5-14-110(a)(1). An Arkansas state court sentenced Neely to serve five years' probation, and ordered him to register as a sex offender. Pursuant to an agreement with the prosecuting attorney, supervision of Neely's probation was transferred to New Mexico. See Neely v. McCastlain, 2009 Ark. 189, 306 S.W.3d 424, 425 (2009).
Less than a year later, an Arkansas prosecuting attorney sought to revoke Neely's probation because he violated conditions of probation. See id. Neely resisted revocation on the ground that New Mexico authorities had imposed conditions that were not authorized by the judgment. The revocation petition was eventually dismissed, id. at 425 n. 1, but Neely filed a federal habeas petition arguing that § 5-14-110 is unconstitutional, and that his felony convictions should be vacated. The district court stayed the proceedings to allow Neely to exhaust his remedies in state court.
An Arkansas trial court rejected Neely's challenge, and the Supreme Court of Arkansas affirmed, holding that neither declaratory nor habeas relief was available because Neely was not in state custody. Neely, 306 S.W.3d 424. The district court then concluded that § 5-14-110 is not unconstitutional on its face and dismissed Neely's habeas petition. This court granted Neely a certificate of appealability with respect to his constitutional challenges to Ark.Code Ann. § 5-14-110, and on the issue whether we should certify a question to the Supreme Court of Arkansas.

II.
A defendant may not collaterally attack a guilty plea by raising "independent claims relating to the deprivation of constitutional rights that occurred prior to the entry of the guilty plea." Tollett v. Henderson, 411 U.S. 258, 267, 93 S.Ct. 1602, 36 L.Ed.2d 235 (1973). Because Neely entered a voluntary and intelligent guilty plea, he cannot challenge the constitutionality of Ark.Code Ann. § 5-14-110 as applied in his case. See Weisberg v. State of Minn., 29 F.3d 1271, 1279-80 (8th Cir. 1994). Neely's guilty plea, however, does not foreclose certain types of challenges: "`[A] plea of guilty to a charge does not waive a claim thatjudged on its face the charge is one which the State may not constitutionally prosecute.'" United States v. Broce, 488 U.S. 563, 575, 109 S.Ct. 757, 102 L.Ed.2d 927 (1989) (quoting Menna v. New York, 423 U.S. 61, 62 n. 2, 96 S.Ct. 241, 46 L.Ed.2d 195 (1975) (per curiam)).

III.
Neely challenges Ark.Code Ann. § 5-14-110 as unconstitutionally vague, in violation of the Due Process Clause, and overbroad, in violation of the First Amendment. He also argues the statute violates *350 the Due Process Clause because it includes no requirement that the defendant know the victim is underage and instead places the burden on the defendant to prove reasonable mistake of age as an affirmative defense. To succeed on his facial challenge to the statute, Neely must establish that "no set of circumstances exists under which the Act would be valid." United States v. Salerno, 481 U.S. 739, 745, 107 S.Ct. 2095, 95 L.Ed.2d 697 (1987). We review the district court's legal conclusions de novo and its factual findings for clear error. Hanegan v. Miller, 663 F.3d 349, 354 (8th Cir.2011).

A.
Neely's first argument is that the term "solicits" in Ark.Code Ann. § 5-14-110 provides insufficient notice of the conduct that the statute prohibits. Under the vagueness doctrine, a penal statute must define the prohibited conduct "with sufficient definiteness that ordinary people can understand what conduct is prohibited and in a manner that does not encourage arbitrary and discriminatory enforcement." Kolender v. Lawson, 461 U.S. 352, 357, 103 S.Ct. 1855, 75 L.Ed.2d 903 (1983). The inherent uncertainty of language often will impart some degree of vagueness to a statute, but that uncertainty alone does not mean that a statute is unconstitutional. Recourse to additional sources like dictionaries or judicial opinions may provide sufficient warning. See Rose v. Locke, 423 U.S. 48, 49-50, 96 S.Ct. 243, 46 L.Ed.2d 185 (1975).
Arkansas courts have resorted to dictionary definitions to elucidate the meaning of the term "solicits" in § 5-14-110. Applicable definitions included:
"3: to make petition to: ENTREAT, IMPORTUNE ...; esp.: to approach with a request or plea (as in selling or begging) . . .
4: to move to action . . .
7: to endeavor to obtain by asking or pleading: plead for ...; also to seek eagerly or actively
10: to serve as a temptation . . . [.]"
Heape v. State, 87 Ark. App. 370, 192 S.W.3d 281, 285 (2004) (quoting Webster's Third New International Dictionary 2169 (1993)); see also Renderos v. State, 92 Ark. App. 293, 213 S.W.3d 37, 38-39 (2005) (relying on the definition of "solicit" used in Heape). These decisions demonstrate that "solicits" has an ordinary and usually accepted meaning in common language that can be drawn from dictionaries, and that "men may conduct themselves so as to avoid that which is forbidden." Rose, 423 U.S. at 50, 96 S.Ct. 243. The statute is not "impermissibly vague in all of its applications." Vill. of Hoffman Estates v. Flipside, 455 U.S. 489, 497, 102 S.Ct. 1186, 71 L.Ed.2d 362 (1982), and Neely's challenge fails.

B.
Neely next asserts that § 5-14-110 is overbroad. The overbreadth doctrine permits a court to invalidate a statute that restricts expression if "a substantial number of its applications are unconstitutional" in relation to its "plainly legitimate sweep." United States v. Stevens, ___ U.S. ___, 130 S.Ct. 1577, 1587, 176 L.Ed.2d 435 (2010) (internal quotation omitted). A plaintiff may invoke the overbreadth doctrine on the ground that "the statute's very existence may cause others not before the court to refrain from constitutionally protected speech or expression." Broadrick v. Oklahoma, 413 U.S. 601, 612, 93 S.Ct. 2908, 37 L.Ed.2d 830 (1973). But application of the doctrine is "strong medicine" that should be employed "sparingly and only as a last resort." Id. at 613, 93 S.Ct. 2908.
*351 By pleading guilty to a violation of § 5-14-110, Neely admitted to "solicit[ing] another person who is less than fifteen (15) years of age . . . to engage in: (A) Sexual intercourse; (B) Deviate sexual activity; or (C) Sexual contact." Ark.Code Ann. § 5-14-110(a)(1). The statute prohibits the solicitation of conduct that is already criminal under Arkansas law. See Ark. Code Ann. § 5-14-127(a)(1). Offers to engage in illegal transactions enjoy no First Amendment protection, United States v. Williams, 553 U.S. 285, 297, 128 S.Ct. 1830, 170 L.Ed.2d 650 (2008), and the conduct to which Neely pleaded guilty falls squarely within this category.
Nor does § 5-14-110 prohibit a substantial amount of protected speech. In his brief, Neely conjures up examples of § 5-14-110's potential overreach, suggesting, for instance, that "expression that could be construed as leading fourteen year olds to engage in sexual activitysuch as that included in television advertising, films or popular music with sexual themes," would fall within the statute's scope. But by criminalizing the solicitation of minors to engage in sexual activity, § 5-14-110 targets primarily, if not exclusively, illicit activity within the State's power to regulate. If the statute actually encompasses any of Neely's hypothetical scenarios, then such cases would constitute a "tiny fraction" of affected speech. See New York v. Ferber, 458 U.S. 747, 773, 102 S.Ct. 3348, 73 L.Ed.2d 1113 (1982). We therefore reject Neely's overbreadth challenge and conclude that "whatever overbreadth may exist should be cured through case-by-case analysis of the fact situations to which its sanctions, assertedly, may not be applied." Broadrick, 413 U.S. at 615-16, 93 S.Ct. 2908.

C.
The last issues on appeal relate to the mens rea requirements of § 5-14-110. Neely claims that the statute abridges an accused's right to due process under the Fourteenth Amendment by imposing criminal liability without proof that a defendant knew the solicited victim was underage. He further asserts that although Arkansas law offers an affirmative defense for individuals who "reasonably believed the child to be of the critical age or over" when "criminality of conduct depends on a child's being below a critical age older than fourteen," Ark.Code Ann. § 5-14-102(d)(1), the defense impermissibly places the burden on the defendant to disprove an essential element of the crime. See Mullaney v. Wilbur, 421 U.S. 684, 701, 95 S.Ct. 1881, 44 L.Ed.2d 508 (1975). The district court rejected Neely's argument, concluding a default statute in the Arkansas Code supplies § 5-14-110 with a mens rea element. The default statute provides that "if the statute defining an offense does not prescribe a culpable mental state a culpable mental state is nonetheless required and is established only if a person acts purposely, knowingly, or recklessly." Ark.Code Ann. § 5-2-203(b).
A federal court considering an application for writ of habeas corpus is bound by a state supreme court's interpretations of state law, see Barrett v. Acevedo, 169 F.3d 1155, 1163 (8th Cir.1999) (en banc), but the Supreme Court of Arkansas has not spoken directly to whether § 5-14-110 requires mens rea with respect to the victim's age. Compare Short v. State, 349 Ark. 492, 79 S.W.3d 313, 316-17 (2002) (characterizing both first-degree sexual abuse and statutory rape as strict-liability crimes with respect to knowledge of the victim's age), and Wright v. State, 98 Ark. App. 271, 254 S.W.3d 755, 757-58 (2007) (reaching the same conclusion regarding Arkansas's fourth-degree sexual abuse statute), with Heape, 192 S.W.3d at 286 *352 (applying Arkansas's mens rea default statute to § 5-14-110 in the context of whether the defendant intended his solicitations to actually result in sexual contact with the minor victim). We need not predict how the state court would resolve the issue, because we conclude that Neely's arguments fail under either interpretation of § 5-14-110.[2]

1.
Assuming, as Neely argues, that § 5-14-110 imposes strict criminal liability on an accused who is ignorant of the age of the victim, the lack of a mens rea requirement does not violate the accused's due process rights. Courts generally presume that a criminal statute should be construed to require proof of mens rea, see Staples v. United States, 511 U.S. 600, 605-06, 114 S.Ct. 1793, 128 L.Ed.2d 608 (1994), but this presumption has not been applied to "sex offenses, such as rape, in which the victim's actual age was determinative despite defendant's reasonable belief that the girl had reached age of consent." Morissette v. United States, 342 U.S. 246, 251 n. 8, 72 S.Ct. 240, 96 L.Ed. 288 (1952). As a result, criminal statutes prohibiting production of child pornography or statutory rape need not require that the defendant know the victim is underage. See United States v. Wilcox, 487 F.3d 1163, 1174 (8th Cir.2007).
Seeking to distinguish this line of cases, Neely cites United States v. X-Citement Video, Inc., 513 U.S. 64, 115 S.Ct. 464, 130 L.Ed.2d 372 (1994), for the proposition that strict criminal liability may apply only to sex offenses in which "the perpetrator confronts the underage victim personally and may reasonably be required to ascertain the victim's age." See id. at 72 n. 2, 115 S.Ct. 464. The issue in X-Citement Video was whether 18 U.S.C. § 2252, which prohibits, inter alia, the knowing distribution of any visual depiction where production of the depiction involved the use of a minor engaging in sexually explicit conduct, applies only where the distributor knows that the person depicted was a minor. The Court concluded that the common-law treatment of sex offenses did not militate against construing § 2252 to include a scienter requirement, because "[t]he opportunity for reasonable mistake as to age increases significantly once the victim is reduced to a visual depiction." Id. Neely reasons that because § 5-14-110 prohibits conduct that need not involve face-to-face contact between the perpetrator and the victim, the statute's lack of a mens rea requirement concerning the victim's age violates due process.
To prevail on a facial challenge, Neely must show that § 5-14-110 would be unconstitutional in all of its applications. Solicitation under § 5-14-110 often will involve precisely the face-to-face confrontation contemplated by X-Citement Video. In many instances, perhaps most, the prohibited conduct is analogous to statutory rape, which requires no proof that the accused knew the victim's age. Section 5-14-110, moreover, provides those charged with violating the statute an affirmative defense for reasonable mistake of age. See Ark.Code Ann. § 5-14-102(d)(1). A defendant charged with child pornography production or statutory rape has no constitutional right to this defense, see United States v. McCloud, 590 F.3d 560, 567-68 (8th Cir.2009); Wilcox, 487 F.3d at 1174, so its availability to Arkansas defendants offers protection beyond that required by the Due Process Clause. *353 Because it is clear that there are many applications of § 5-14-110 that pass constitutional muster under the precedents governing face-to-face sex offenses, we need not decide whether the statute would be constitutional as applied to a perpetrator who never personally confronts the minor victim.
We reject Neely's contention that the reasonable mistake of age defense in § 5-14-102(d) violates the Due Process Clause by shifting the burden of proof on an essential element to the defendant, as in Mullaney v. Wilbur. See 421 U.S. at 686-87, 95 S.Ct. 1881. If the Arkansas statute employs a strict-liability standard concerning the victim's age, then the State retains the burden of proving all elements of the offense, and "[n]o further facts are either presumed or inferred in order to constitute the crime." Patterson v. New York, 432 U.S. 197, 205-06, 97 S.Ct. 2319, 53 L.Ed.2d 281 (1977). The defendant's reasonable ignorance of the victim's age would therefore mitigate the offense, not rebut a presumed element, and Mullaney is inapposite.

2.
If, as the district court thought, Arkansas's default statute provides a mens rea requirement in § 5-14-110 with respect to the victim's age, then Neely's claims still fail. As a facial matter, the statute adequately respects due process even if it imposes strict liability, so the inclusion of a more demanding mens rea element cannot render § 5-14-110 unconstitutional. Nor would the affirmative defense for reasonable mistake of age deprive defendants of due process under this view. Interpreting § 5-14-110 to include a mens rea requirement might render the affirmative defense for reasonable mistake of age superfluous, but its availability would not relieve the State of its obligation to prove all elements of the crime beyond a reasonable doubt. Accordingly, if § 5-14-110 requires the State to prove that the defendant knew the victim was underage, then Arkansas's treatment of reasonable mistake of age as an affirmative defense does not violate the Due Process Clause.

* * *
The judgment of the district court is affirmed.
NOTES
[1]  The Honorable Jerome T. Kearney, United States Magistrate Judge for the Eastern District of Arkansas, sitting by consent of the parties pursuant to 28 U.S.C. § 636(c).
[2]  Because Neely's claim fails under either interpretation of § 5-14-110, we decline to certify any question to the state supreme court.
