In this paper, the existence of a nontrivial least energy solution is considered for the nonlinear fractional Schrödinger-Poisson systems (−Δ)su + V(x)u + ϕu = |u|p−1u and (−Δ)tϕ = u2 in R3, where (−Δ)α is the fractional Laplacian for α = s, t ∈ (0, 1) with s < t and 2s + 2t > 3. Under some appropriate assumptions on the non-constant potential V(x), we prove the existence of a nontrivial least energy solution when 2<p<2s*−1=(3+2s)/(3−2s) by using some new analytical skills and the Nehari-Pohožaev type manifold.

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In this paper, we study a class of critical elliptic problems of Kirchhoff type: where , , , and are constants and is the Hardy–Sobolev exponent in . For a suitable function , we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard as a parameter to obtain the convergence property of solutions for the given problem as by the mountain pass theorem and Ekeland’s variational principle.

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