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

在这项研究中，我们考虑了半线性抛物方程的时间整体解的渐近行为，其Lyapnov泛函具有紧性，类似于具有缺乏紧性的变分泛函的椭圆型问题。对于抛物型问题，我们处理了含临界Sobolev指数的半线性抛物型问题，给出了时间整体解的能量量子化现象。我们还研究了定义在外部区域上的具有有界补且权函数在空间无穷远处衰减的椭圆问题。证明了当权函数的衰减非常慢时，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.