Nodal Solutions for a Quasilinear Elliptic Equation Involving the p-Laplacian and Critical Exponents

Abstract

Abstract This paper is concerned with the following type of quasilinear elliptic equations in ℝ N {\mathbb{R}^{N}} involving the p-Laplacian and critical growth: - Δ p ⁢ u + V ⁢ ( | x | ) ⁢ | u | p - 2 ⁢ u - Δ p ⁢ ( | u | 2 ) ⁢ u = λ ⁢ | u | q - 2 ⁢ u + | u | 2 ⁢ p * - 2 ⁢ u , -\Delta_{p}u+V(|x|)|u|^{p-2}u-\Delta_{p}(|u|^{2})u=\lambda|u|^{q-2}u+|u|^{2p^{% *}-2}u, which arises as a model in mathematical physics, where 2 < p < N {2<p<N} , p * = N ⁢ p N - p {p^{*}=\frac{Np}{N-p}} . For any given integer k ≥ 0 {k\geq 0} , by using change of variables and minimization arguments, we obtain, under some additional assumptions on p and q, a radial sign-changing nodal solution with k + 1 {k+1} nodal domains. Since the critical exponent appears and the lower order term (obtained by a transformation) may change sign, we shall use delicate arguments.