Liouville type theorem for critical order Hénon-Lane-Emden type equations on a half space and its applications

Abstract

In this paper, we are concerned with the critical (i.e., n-th) order Hénon-Lane-Emden type equations with Navier boundary conditions on a half space R+n:(0.1){(−Δ)n2u(x)=f(x,u(x)),u(x)≥0,x∈R+n,u(x)=−Δu(x)=⋯=(−Δ)n2−1u(x)=0,x∈∂R+n, where u∈Cn(R+n)∩Cn−2(R+n‾) and n≥2 is even. We first consider the typical case f(x,u)=|x|aup with 0≤a<∞ and 1<p<∞. We prove the super poly-harmonic properties and establish the equivalence between (0.1) and the corresponding integral equations(0.2)u(x)=∫R+nG+(x,y)f(y,u(y))dy, where G+(x,y) denotes the Green function for (−Δ)n2 on R+n with Navier boundary conditions. Then, we establish Liouville theorem for (0.2) and hence obtain the Liouville theorem for (0.1) on R+n. As an application of the Liouville theorem on R+n (Theorem 1.6) and Liouville theorems in Rn, we derive a priori estimates via blowing-up methods for solutions (possibly change signs) to Navier problems involving critical order uniformly elliptic operators L. Consequently, by using the Leray-Schauder fixed point theorem, we derive existence of positive solutions to critical order Lane-Emden equations in bounded domains for all n≥2 and 1<p<∞. In contrast to the subcritical order cases, our results seem to be the first work on Navier problems for critical order equations on R+n, which is the critical-order counterpart to those results on subcritical order cases in [6,20,21]. Extensions to IEs and PDEs with general nonlinearities f(x,u) are also included. Surprisingly, there are no growth conditions on u and hence f(x,u) can grow exponentially (or even faster) on u.