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In this paper, we study the existence of positive ground state solutions and infinitely many geometrically distinct solutions for the following fractional Schrödinger–Poisson system (- Δ) s u + V (x) u + ϕ u = f (x , u) , in R 3 , (- Δ) s ϕ = u 2 , in R 3 , where s ∈ (3 4 , 1) is a fixed constant, f is continuous, superlinear at infinity with subcritical or critical growth and V and f are asymptotically periodic in x. Applying the method of Nehari manifold and Lusternik–Schnirelmann category theory, three existence results are given. [ABSTRACT FROM AUTHOR]

We study the following Kirchhoff equation involving fractional Laplacian in R N where N ≥ 2 , a ≥ 0 , b , μ > 0 , 0 < s < 1 , and (- Δ) s is the fractional Laplacian with order s. By reducing (K) to an equivalent system, we obtain the existence of a positive solution of (K) with general nonlinearities. The positive solution is unique if g (u) = | u | p - 1 u , 1 < p < N + 2 s N - 2 s . Moreover, if the function g is odd, the existence of infinitely many (sign-changing) solutions is concluded. As we shall see, for the case where 0 < s ≤ N 4 , a necessary condition of existence of nontrivial solutions of (K) is that b is small. Our method works well for the so-called degenerate case a = 0 . [ABSTRACT FROM AUTHOR]

In this paper, we consider the following general nonlocal problem -LKu=f(x,u)inΩ,u=0inRN\Ω,where Ω⊂RN is a bounded domain with Lipschitz boundary ∂Ω, s∈(0,1) with 2s and LK is a nonlocal integrodifferential operator of fractional Laplacian type. We obtain the existence of infinitely many sign-changing solutions by combining critical point theory and invariant sets of descending flow. [ABSTRACT FROM AUTHOR]

In this paper, we establish a sharp inequality for some multilinear singular integral operators with non-smooth kernels. As an application, we obtain the weighted [InlineEquation not available: see fulltext.]-norm inequality and [InlineEquation not available: see fulltext.]-type inequality for the multilinear operators. MSC: 42B20, 42B25. [ABSTRACT FROM AUTHOR]