Multiplicity of solutions for a fractional Kirchhoff type equation with a critical nonlocal term

This paper is concerned with the multiplicity of solutions for the fractional Kirchhoff type equations with a critical nonlocal term {M (parallel to u parallel to 2/Z) L-K u = lambda f (x, u) + (integral(Omega) |u(y)|(2)*(alpha,s)/|x - y|(alpha) dy) |u|2*(-2)(alpha,s)u, x is an element of Omega, u(x) = 0, x is an element of R-N\Omega, where LK is an integrodifferential operator with kernel K, Omega is an open bounded subset of R-N, M and f are continuous functions. parallel to center dot parallel to Z is a functional norm and 2*(alpha,s) = 2N-alpha/N-2s is the Hardy-Littlewood-Sobolev critical exponent with N > 2s, s is an element of(0, 1) and alpha is an element of (0, N). We first deal with the case that, function f satisfies some superlinear and quasi-critical growth conditions, in which case we prove that, for any given k is an element of N, there exists at least k pairs of nontrivial solutions for lambda large enough; while in the case that f is sublinear growth, we obtain the existence of infinitely many solutions which tend to zero under a suitable small value of lambda. To establish these multiplicity results, we use a new concentration-compactness principle dealing with nonlocal critical problems from the paper (X. He, X. Zhao, W. Zou, The Benci-Cerami problem for the fractional Choquard equationwith critical exponent, ManuscriptMath. 2022), together with Kajikiya's new version of the symmetric mountain pass lemma and the truncation skill.