Positive Normalized Solutions to a Kind of Fractional Kirchhoff Equation with Critical Growth

Abstract

In this paper, we have investigated the existence of normalized solutions for a class of fractional Kirchhoff equations involving nonlinearity and critical nonlinearity. The nonlinearity satisfies L2-supercritical conditions. We transform the problem into an extremal problem within the framework of Lagrange multipliers by utilizing the energy functional of the equation in the fractional Sobolev space and applying the mass constraint condition (i.e., for given m>0,∫RN|u|2dx=m2). We introduced a new set and proved that it is a natural constraint. The proof is based on a constrained minimization method and some characterizations of the mountain pass levels are given in order to prove the existence of ground state normalized solutions.

Keywords

• normalized solutions
• nonlinearity
• Kirchhoff equation
• critical growth
• L2-supercritical