Critical Kirchhoff-type equation with singular potential
In this paper, we deal with the following Kirchhoff-type equation: equation* -(1 +_R^3 | abla u|^2dx) Δ u +A|x|^α u =f(u),xR^3, equation* where A> 0 is a real parameter and α(0,1)(4/3,2). Remark that f(u)=|u|^2_α^*-2u +λ|u|^q-2u +|u|^4u, where λ> 0, q(2_α^*,6), 2_α^*=2+4α/(4-α) is the embedding bottom index, and 6 is the embedding top index and Sobolev critical exponent. We point out that the nonlinearity f is the almost ``optimal’’ choice. First, for α(4/3,2), applying the generalized version of Lions-type theorem and the Nehari manifold, we show the existence of nonnegative Nehari-type ground sate solution for above equation. Second, for α(0,1), using the generalized version of Lions-type theorem and the Pohožaev manifold, we establish the existence of nonnegative Pohožaev-type ground state solution for above equation. Based on our new generalized version of Lions-type theorem, our works extend the results in Li-Su [Z. Angew. Math. Phys. 66 (2015)]., number=2, journal=Topological Methods in Nonlinear Analysis, author=Su, Yu and Liu, Senli, year=2023, month=Jun., pages=611–636