Sharp critical exponents for nonlinear equations with the fractional Laplacian.

In this paper we consider two classes of nonlinear partial differential equations with the fractional Laplacian, namely $$\begin{align*} \left(-\Delta\right)^{\frac{\alpha}{2}}\left(u^{m} \right) &=u\left|u\right|^{q-1}+w\left(x\right), \quad x\in \mathbb{R}^{N},\\ &\qquad 1\leq m (− Δ) α 2 (u m) = u | u | q − 1 + w (x) , x ∈ R N , 1 ≤ m < q , 0 < α ≤ 2 , N > α and $$\begin{align*} &\frac{\partial^{k}u}{\partial t^{k}}+\left(-\Delta\right)^{\frac{\alpha}{2}}u=u^{q}, \quad \left(x,t\right)\in \mathbb{R}^{N}\times\left(0,+\infty\right),\\ &\qquad k\geq 1, \ 0 ∂ k u ∂ t k + (− Δ) α 2 u = u q , (x , t) ∈ R N × (0 , + ∞) , k ≥ 1 , 0 < α ≤ 2 , N > α. Solutions defined for all $ x\in \mathbb {R}^{N} $ x ∈ R N of the first equation are referred to as entire solutions, while solutions defined for all $ \left (x,t\right)\in \mathbb {R}^{N}\times \left [0,+\infty \right) $ (x , t) ∈ R N × [ 0 , + ∞) of the second equation are referred to as global solutions. Several existence and nonexistence theorems are established over different ranges of q, and thus the respective relations between the existence, nonexistence of solutions for these equations and the index q in the nonlinear terms are obtained. It is illustrated that our results are sharp in cases of m = 1 and k = 1 respectively. In addition, we prove the positivity, symmetry and odevity of solutions we constructed for the first equation with m = 1 associated with the inhomogeneous term w. [ABSTRACT FROM AUTHOR]