某些擬線性橢圓型問題其強解的存在性

Abstract

令.OMEGA.為一個在■上的有界集合,而且其邊界.prtl..OMEGA. .in.■. 我們研究底下這個擬線性橢圓問題:■其中■.in.■,.lgvert.g(x,r,. xi.).lgvert..ltorsim.h.(.lgvert. r.lgvert.)(1+■),0.ltorsim.. theta..ltorsim.2,並且h是個局部有界函數.假設對所有(x,r,.xi.).in.. OMEGA.*R*■,g滿足■此處■和■是非負函數.當0.ltorsim..theta.. ltorsim.,我們可証得在■.intersection.■中u的強解存在性,並且若p> N/2時■.intersection.■中u的強解皆■有界.接下來我們考慮下列擬線 性橢圓特徵值問題:■其中■.in.■,且g.in.■.那麼對所有p,p. ltorsim.<.inf.,存在■使得當 0.ltorsim..lambda.<■此特徵值有解u. in.■.intersection.■. Let .OMEGA. be a bounded set in ■ with its boundary .prtl.. OMEGA..in.■. We study the following qualinear ellptic problem: ■ where ■.in.■,.lgvert.g(x,r,.xi.).lgvert..ltorsim.h(. lgvert. r.lgvert.)(1+■),0.ltorsim..theta..ltorsim.2 and h is a locally bounded function. Suppose that g satisfies the following condition: g(x,r,.xi.) sign r .ltorsim.■+■.lgvert.. xi..lgvert. for all (x,r,.xi.).in..OMEGA.*R*■where■and■are nonnegative constants. Whenever 0<.theta.<2,the existence of strong solutions u.in.■(.OMEGA.).intersection.■(.OMEGA.) is proved and all the solutions u.in.■(.OMEGA.).intersection.■(. OMEGA.) are ■ bounded for p>N/2. Consider the following quasilinear elliptic eigenvalue problem:■ where ■and■. Thus for each p,1.ltorsim.p<.inf.,there exists a ■>0 such that the eigenvalue problem has a solution u.in.■.intersection.■ whenever 0.ltorsim..lambda.<■.