The nodal solution for a problem involving the logarithmic and exponential nonlinearities

Abstract

ABSTRACT In this work, we study the existence of the least energy sign-changing solution for the following degenerate Kirchhoff-type problem involving the fractional N/s-Laplacian with logarithmic and both subcritical and critical exponential nonlinearities: { M ( ∫ R 2 N | u ( x ) − u ( y ) | N s | x − y | 2 N d x d y ) ( − Δ ) N / s s u = | u | q − 2 uln ⁡ | u | 2 + μf ( u ) , in Ω , u = 0 , in R N ∖ Ω , where Ω ⊂ R N is a bounded domain. The proof is based on constrained variational method, fractional Trudinger–Moser inequality, quantitative deformation lemma and Brouwer's degree theory in Nehari sets. To be more precise, the least energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold.