Research on quasilinear elliptic equations with rapidly growing principal parts

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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.