Research on quasilinear elliptic equations with rapidly growing principal parts

1.在整个空间上研究了一类主部非幂次递增的拟线性椭圆方程，并证明了其正解的存在性。附加在方程上的能量泛函很难在通常的Sobolev空间中处理。因此，我们精确地研究了Orlicz-Sobolev空间及其上泛函的性质。以这些性质为基础，应用pll lions的浓度-紧性论证，我们证明了正解的存在性。虽然上述方程的主成分是相同的，但当强迫项的增长阶比主成分的增长阶小时，我们已经证明了多个正解的存在性。这里所述的区域是有界的。利用这些方程的解的正则性和比较定理，结合Orlicz-Sobolev空间及其泛函的性质，应用变分方法和上、次解方法，证明了这些方程的多个正解的存在性。进一步证明了极大解的存在性。在主函数增长缓慢的情况下，在主函数增长迅速的情况下，Orlicz-Sobolev函数不是自反的，泛函也不是Frechet可微的。这些事实使分析变得困难。K. Le用变分不等式分析了这类具有亚临界非线性的方程。本文利用K. Le给出的解，证明了临界非线性方程正解的存在性。关于在原点附近的外力行为的假设可以去掉。

1.We have investigated quasilinear elliptic equations with non-power like increasing principal parts on the whole space and showed the existence of positive solutions. The energy functionals attached to the equations are difficult to be treated in the usual Sobolev spaces. Therefore we investigated precisely the properties of Orlicz-Sobolev spaces and functionals on those spaces. Using these properties as an basis and applying concentration-compactness arguments by P.L.Lions, we have shown the existence of positive solutions.2.Although the principal parts are the same in the equations stated above, when the growing orders of the forcing terms are small compared to those of principal parts, we have shown the existence of multiple positive solutions. The regions stated here are bounded. Using the regularity of solutions and comparison theorem for these equations, in addition to the properties of Orlicz-Sobolev spaces and the functionals on those spaces, and applying both variational methods and super-, sub-solution methods, we have shown the existence of multiple positive solutions. Further the existence of maximal solution is proved.3.In the case when the principal part grows very slowly, as in the case when the principal part grows rapidly, the Orlicz-Sobolev is not reflexive, and the functional is not Frechet differentiable. These facts make analysis difficult. K. Le has analyzed the equations of this type with subcritical nonlinearity by using variational inequalities. Here, making use of the solutions given by K. Le, we showed the existence of positive solutions of the equations with critical nonlinearities. The hypothesis on the behavior of exterior forces near the origin can be removed.