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In this paper, we consider the existence and multiplicity of solutions for the following perturbed fractional p-Laplacian equation { ϵ s p (− Δ) p s u + V (x) | u | p − 2 u = A (x) | u | p s ∗ − 2 u + h (x , u) , x ∈ R N , u (x) → 0 , as | x | → ∞. Under some mild conditions on V, A and h, we show that the problem has at least one positive weak solution provided ϵ ≤ ε σ , and for any m ∗ ∈ N , it has m ∗ pairs of solutions if ϵ ≤ ε m ∗ σ , where ε σ and ε m ∗ σ are sufficiently small positive numbers. Moreover, these solutions u ϵ → 0 in W s , p (R N) as ϵ → 0. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and multiplicity of solutions for the following perturbed fractional p-Laplacian equation { ϵ s p (− Δ) p s u + V (x) | u | p − 2 u = A (x) | u | p s ∗ − 2 u + h (x , u) , x ∈ R N , u (x) → 0 , as | x | → ∞. Under some mild conditions on V, A and h, we show that the problem has at least one positive weak solution provided ϵ ≤ ε σ , and for any m ∗ ∈ N , it has m ∗ pairs of solutions if ϵ ≤ ε m ∗ σ , where ε σ and ε m ∗ σ are sufficiently small positive numbers. Moreover, these solutions u ϵ → 0 in W s , p (R N) as ϵ → 0. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Choquard-type equation with the fractional Laplacian (− Δ) s u (x) + a u (x) = 1 | x | n − α ∗ u p u p − 1 , x ∈ R n , u (x) > 0 , x ∈ R n , where 0

In this paper, we study a nonlinear elliptic system involving a fractional Laplacion: where 0 < α < 2, p, q > 0 and max{p, q} ≥ 1, τ ≥ 0, n ≥ 2. There are two cases to be considered. The first one is where the domain is bounded, and the second one is where the domain is the whole space. First of all, we consider the above system in the star-shaped and bounded domain Ω, by using the Pohozaev identity. We prove the nonexistence of a positive solution in the critical and supercritical case p + q + 1 ≥. In addition, we show that the positive solutions of the above system are radially symmetric and decreasing about the origin by using the method of moving planes in ℝn. Moreover, while in the subcritical case , we prove the nonexistence of a positive solution for the above system in ℝn. Then, through the doubling lemma we obtain the singularity estimates of the positive solutions on a bounded domain. [ABSTRACT FROM AUTHOR]