一类退缩椭圆方程正解的多重性

Multiplicity of Positive Solutions of a Class of Degenerate Elliptic Equations

讨论了ＲＮ（Ｎ≥３）中有界区域Ω上一类带临界增长的拟线性退缩的椭圆方程－Ｄｉ［ｇ（｜ｕ｜ｐ）｜ｕ｜ｐ－２Ｄｉｕ］＝λｕα＋ｕｑ－１＋ｆ（ｘ，ｕ）的Ｄｉｒｉｃｈｌｅｔ问题正解的存在性．其中１＜ｐ＜Ｎ＜２ｐ，ｑ＝Ｎｐ／（Ｎ－ｐ），由于ｑ是Ｗ１，ｐ（Ω）嵌入到Ｌｑ（Ω）的极限指数．此时嵌入非紧，方程对应的变分泛函在Ｗ１，ｐ（Ω）中不满足（ｐ，ｓ）条件，这给寻求方程的正解造成了困难，文中用没有（ｐ，ｓ）条件的山路引理和Ｌｉｏｎｓ的集中紧性原理证明了方程的能量泛函至少有两个临界点。

The existence of positive solutions on a class of degenerate quasilinear elliptic equations involving critical Sobolev exponents - D i[g(|FDA1u| p)|FDA1u| p-2 D iu]=λu α-1 +u q-1 +f(x,u) under Dirichlet boundary condition was discussed, where ΩR N(N≥3) is a bounded domain, 1 <p<N<2p, q=NpN-p . Because q is the limiting Sobolev exponent for the embedding W 1,p (Ω)L q(Ω) , the embedding is not compact, which brings about many difficulties to seek the positive solutions of the equation. By a mountain pass lemma without (p,s) condition and Lions' concentration compactness principle, this paper proved that the corresponding functional of the equation has at least two critical points, so the two positive solutions of the problem are obtained.