Sharp critical exponents for nonlinear equations with the fractional Laplacian

Abstract

In this paper we consider two classes of nonlinear partial differential equations with the fractional Laplacian, namely (−Δ)α2(um)=u|u|q−1+w(x),x∈RN,1≤mα and ∂ku∂tk+(−Δ)α2u=uq,(x,t)∈RN×(0,+∞),k≥1, 0<α≤2, N>α. Solutions defined for all x∈RN of the first equation are referred to as entire solutions, while solutions defined for all (x,t)∈RN×[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.KEYWORDS: Fractional Laplaciancritical exponentexistencenonexistenceAMS SUBJECT CLASSIFICATIONS: 35B0835B3335R11 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was partially supported by the Special Project for Local Science and Technology Development Guided by the Central Government of Sichuan Province [grant number 2021ZYD0014].