一类椭圆方程正解的多重性

Multiplicity of Positive Solutions of a Class of Elliptic Equations

椭圆问题因其广泛的物理背景而受到普遍的关注．近十几年来，关于具临界增长的椭圆问题正解的研究是该领域中的热点之一．当非线性项是次临界增长时，相应的能量泛函可以满足一定的紧性条件，变分方法、上下解方法、拓扑度理论及畴数理论等标准方法已被广泛地应用于研究解的存在及多重性问题．如果非线性项的增长阶是关于Ｓｏｂｏｌｅｖ嵌入的临界指数，这时的嵌入连续而不紧，故能量泛函不满足紧性条件，以上的标准方法失效．本文通过没有（Ｐ．Ｓ）条件的山路引理和对最佳Ｓｏｂｏｌｅｖ常数及能量泛函的细致分析，得到了一类具有次线性及临界增长组合非线性项的齐次椭圆问题的能量泛函至少有一个具正能量的鞍点和一个具负能量的局部极值点，从而得到该问题的两个非平凡正解．

Elliptic problems have been attracting more attention by its general physcical background. The problem of positive solution to elliptic equations involving critical exponents has been extensively studied over the past two decades. When the nonlinearity grows subcritical, the corresponding functional satisfies some compact condition, existence and multiplicity of solutions have been researched by standard methods such as variational argument, sub super solution, degree and category theory. If the nonlinearity grows at the rate of the critical Sobolev imbedding exponent, the imbedding mapping is then continuous but not compact so the above standard methods fail. In this papaer, by a mountain pass lemma without(P.S) condition and by analysis of the best Sobolev constant and energy functionally, carefully, we obtain that the functional which corresponding to a class of elliptic problems with the nonlinearity combined of sublinear and critical growth has at least one saddle point with positive energy and one local minimal point with negative energy, so two nontrivial positive solutions of the problem are obtained.