High-order Bahri–Lions Liouville-type theorems View Full Text


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Article Info

DATE

2019-03-28

AUTHORS

Abdellaziz Harrabi

ABSTRACT

Consider the following polyharmonic equations: (-Δ)ru=f(u)inO,(0.1)where O=Rn or O=R+n with the Dirichlet boundary conditions, r∈N∗ and n≥2r+1. We prove some Liouville-type theorems classifying stable (or stable at infinity) solutions, possibly unbounded and sign-changing. Regarding the class of stable solutions, we focus on the case of superlinear nonlinearities f with subcritical or critical growth near zero, like f(s)=|s|p-1s(1+c0|s|q)orf(s)=|s|p-1sexp(s2),where10andc0≥0.Our approach to get the main integral estimates makes use of delicate analysis with appropriate test functions and weighted seminorms. We also establish a variant of Pohozaev identity (Pohozaev in Sov Math Dokl 5:1408–1411, 1965). This permits us to get classification result for stable at infinity solutions under the global subcritical condition: 2nn-2rF(s)-f(s)s>0,∀s≠0,whereF(s)=∫0sf(t)dt.Our assumptions can be verified by many nonlinearities very close to the critical growth, like f(s)=c|s|p-1s+|s|4rn-2rslnq(s2+a),where 11, with 4rn-2r-2qaln(a)>0;orc>0ifq=0. More... »

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1-18

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http://scigraph.springernature.com/pub.10.1007/s10231-019-00839-8

DOI

http://dx.doi.org/10.1007/s10231-019-00839-8

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