A study on the elliptic and the parabolic equations associated with noncompact variational structures

本文研究了一类半线性抛物型方程的Lyapnov泛函缺乏紧性时的时间整体解的渐近性态，以及一类变分泛函缺乏紧性时的椭圆型方程的时间整体解的渐近性态.在变分问题（极小化问题）方面，我们研究了与Sobolev，哈代和Trudinger-Moser型等临界泛函不等式有关的变分问题（极小化问题），在抛物问题方面，我们研究了具有临界Sobolev指数的半线性抛物问题，并证明了时间整体解的能量量子化现象。另外，我们还研究了定义在外部区域上的椭圆问题，其权函数在空间无穷远衰减，其补有界。证明了当权函数的衰减足够慢时，方程有多个正解。

In this research, we are concerned with the asymptotic behavior of time-global solutions for semilinear parabplic equations whose Lyapnov functional are suffered from the lack of compactness, togehter with the elliptic problem with variational functional with lack of compactness. Also we studeid the variational problem (minimizing problem) associated with the critical functional inequalities such as Sobolev, Hardy and the Trudinger-Moser type.As for the parabolic problems, we treated the semilinear parabolic problem involving critical Sobolev exponent and showed the energy quantization phenomena for the time-global solutions. Also we studied the elliptic problem defined in the exterior domain with bounded complement with weight function decaying at the spatial infinity. We proved that if the decay of the weight function is sufiiciently slow, then the equations have multiple positive solutions.