Laotian citizens Khampheng and Sisong Phokhasombath, husband and wife, petition for review of an order of the Board of Immigration Appeals (BIA) affirming an Immigration Judge’s (IJ’s) denial of their applications for asylum, withholding of removal, and relief under the Convention Against Torture (CAT), as well as the IJ’s denial of voluntary departure. Having carefully reviewed the record, we deny the petition. See Menendez-Donis v. Ashcroft, 360 F.3d 915, 917-19 (8th Cir. 2004) (standard of review).

We defer to the IJ’s credibility findings, because the IJ gave specific, cogent reasons for disbelief, see Nyama v. Ashcroft, 357 F.3d 812, 817 (8th Cir. 2004) (per curiam); and we find that the noted discrepancies between Khampheng’s application and asylum-interview statements, in contrast to the testimony he gave at the merits hearing, were material to the asylum claim, see Prawira v. Gonzales, 405 F.3d 661, 662-63 (8th Cir. 2005) (inconsistencies or omissions that relate to basis of persecution are not minor but are at heart of asylum claim). Because the Phokhasombaths did not establish past persecution, they were not entitled to a presumption that their fear of future persecution is well founded, and they failed to offer specific credible evidence that reasonable persons in their position would fear persecution if returned. See Kondakova v. Ashcroft, 383 F.3d 792, 798 (8th Cir. 2004), cert. denied, — U.S. —, 125 S.Ct. 894, 160 L.Ed.2d 775 (2005). In this regard, the IJ reasonably concluded that the Phokhasombaths’ problems were limited to the region where they had lived in Laos, given that Khampheng testified his *564 problems were with local authorities. See 8 C.F.R. § 208.13(b)(3)(i) (2005).

The remaining arguments the Phokhasombaths raise provide no basis for granting their petition; they failed to raise to the BIA their challenges to the denial of withholding of removal and CAT relief, see Ateka v. Ashcroft, 384 F.3d 954, 957 (8th Cir. 2004); and we lack jurisdiction to consider the denial of voluntary departure, see Lorenzo-Gonzales v. Gonzales, 419 F.3d 754, 756 (8th Cir. 2005) (per curiam).

Accordingly, we deny the petition.