Existence of a Least Energy Nodal Solution for a Class of p&q-Quasilinear Elliptic Equations

Abstract

Abstract In this paper we prove an existence result for a least energy nodal (or sign-changing) solution for the class of p&q problems given by where Ω is a smooth bounded domain in ℝN, N ≥ 3 and 2 ≤ p < N. The function a : ℝ+ → ℝ+ grows like as t → +∞ for some p ≤ q < N, the case q = p meaning that a is bounded away from zero and infinity. The nonlinearity f : ℝ → ℝ grows like |t|m-1 at infinity with Moreover, we find that u has exactly two nodal domains or changes sign exactly once in Ω. The functions f and a satisfy suitable additional growth and monotonicity conditions which allow this result to extend previous ones to a larger class of p&q type problems. The proof is based on a minimization argument and a variant of quantitative deformation lemma.