Research on quasilinear elliptic equations with rapidly growing principal parts

1.研究了在全空间上具有非幂型增主部的拟线性椭圆型方程，证明了正解的存在性。在通常的Sobolev空间中，能量泛函很难处理。因此，我们精确地研究了Orlicz-Sobolev空间和这些空间上的泛函的性质。以这些性质为基础，应用P. L. Lions的集中-紧性论证，我们证明了正解的存在性. 2.虽然上述方程的主部相同，但当强迫项的增长阶比主部的增长阶小时，我们证明了多个正解的存在性.这里所述的区域是有界的。利用这些方程解的正则性和比较定理，以及Orlicz-Sobolev空间及其上泛函的性质，利用变分方法和上下解方法，我们证明了多个正解的存在性. 3.在主部增长很慢的情况下，如主部增长很快的情况，Orlicz-Sobolev不是自反的，泛函不是Frechet可微的.这些事实使分析变得困难。K. Le利用变分不等式分析了这类具有次临界非线性项的方程。本文利用K. Le，我们证明了具有临界非线性项的方程正解的存在性。关于原点附近外力行为的假设可以被取消。

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.