Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $

Abstract

This paper considers the following fractional $ (p, q) $-Laplacian equation: $ (-\Delta)_{p}^{s} u+(-\Delta)_{q}^{s} u+V(x)\left(|u|^{p-2} u+|u|^{q-2} u\right) = \lambda f(u)+|u|^{q^*_s-2}u \quad \text { in } \mathbb{R}^{N}, $ where $ s \in(0, 1), \lambda > 0, 2 < p < q < \frac{N}{s} $, $ (-\Delta)_{t}^{s} $ with $ t \in\{p, q\} $ is the fractional $ t $-Laplacian operator, and potential $ V $ is a continuous function. Using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove that the above problem has a least energy sign-changing solution $ u_{\lambda} $ under suitable conditions on $ f $, $ V $ and $ \lambda $. Moreover, we show that the energy of $ u_{\lambda} $ is strictly larger than two times the ground state energy.

Keywords

• fractional $ (p， q) $-laplacian
• sign-changing solutions
• critical problem