高 黎
(中國海洋大學數(shù)學科學學院,山東 青島266100)
本文考慮一類半線性橢圓型耦合方程組
其中:N≥3;p(x)和q(x)是RN上的非負連續(xù)函數(shù);fi和gi(i=1,2)是[0,∞)上單調不減的連續(xù)函數(shù),且滿足如下條件:
方程組(1)描述許多物理現(xiàn)象,比如在非線性光學的應用中雙折射光纖、光折變介質中脈沖的傳播等等[1-2]。由于非線性橢圓型方程(組)中局部解的存在性不一定能夠保證全局解的存在性,從而使全局解的存在性問題成為國內外許多學者關注的熱點。
對于單個方程的情形,Keller[3]和 Osserman[4]于1957年首次提出方程Δu=f(u)在正則有界區(qū)域Ω下,當f滿足(H1)時,存在大解的充要條件為
對于方程組而言,當方程組(1)不含加權系數(shù)p和q,且f2=g1≡1,f1(v)=vα,g2(u)=uβ,α>0,β>0時,文獻[8]證明了,當αβ>1時,方程組(1)在有界區(qū)域Ω下存在大解。當f1(v)=vα1,f2(u)=uα2,g1(v)=vβ1,g2(u)=uβ2,其中α1>0,β2>0,α2>1,β1>1時,文獻[9]證明了,當α1<β1-1,β2<α2-1時,方程組(1)在有界區(qū)域Ω下存在大解。當方程組(1)帶加權系數(shù)p(x)=p(|x|)和q(x)=q(|x|),且f2=g1≡1,f1(v)=vα,g2(u)=uβ,0<α≤β<1時,文獻[10]研究了在整個RN空間上徑向全局解的存在性問題,證明了當函數(shù)函數(shù)p,q∈C(RN),滿足∫∞0tp(t)dt時,方程組(1)存在徑向有界全局解;而 當 p,q ∈ C(RN),滿 足)時,方程組(1存在徑向整大解。對于一般非線性項形式的耦合方程組,文獻[11]同樣給出了方程組(1)存在徑向有界全局解和徑向整大解的充分條件,其研究結果后來被推廣到完全非線性項為變量分離形式的耦合方程組[12]。據(jù)查閱文獻發(fā)現(xiàn),大多數(shù)學者側重于對方程組徑向解的研究,僅有少數(shù)文獻涉及到方程組的非徑向解。受此啟發(fā),本文將討論帶加權系數(shù)且完全非線性項為變量分離形式的半線性橢圓型耦合方程組,首先利用上下解方法證明方程組存在非徑向有界全局解,對于徑向情形得到方程組徑向整大解的不存在性結果。
致謝:在該論文的寫作過程中,始終得到了樸大雄教授的悉心指導,在此,謹向樸老師致以崇高的敬意和真摯的感謝。
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