Normalized solutions for nonlinear Kirchhoff type equations with low-order fractional Laplacian and critical exponent

Abstract

In this paper, we study the following fractional Kirchhoff equation with critical growth a+b∫R3|(−Δ)s2u|2dx(−Δ)su+λu=μ|u|q−2u+|u|2s∗−2uinR3,under the constraint ∫R3|u|2dx=c2, where s∈(0,1), a,b,c>0, 2<q<2s∗, μ>0, and λ∈R appears as a Lagrange multiplier. The equation has been extensively studied in the case s∈(34,1). In contrast, no existence result of normalized solutions is available for the case s∈(0,34]. Because the complicated competition between the Kirchhoff nonlocal term and the Sobolev critical term, such problem cannot be studied by applying standard variational methods, even by restricting its underlying energy functional on the Pohožaev manifold. In this paper, by using appropriate transform, we first get the equivalent system of the above problem. With the equivalence result, we obtain the nonexistence, existence and multiplicity of normalized solutions in the case s∈(0,34].