Studies on Some Degenerate Quasilinear Elliptic Equations in Unbounded Domains

研究了分别在原点和无穷远处渐近等于p-拉普拉斯算子的拟线性退化椭圆方程非平凡解的一些性质。得到的结果如下：首先，我们考虑了在原点和无穷远处，主分量和外力项的渐近阶分别相等的情况。通过给出方程解的梯度的一致估计，推广了Leray-Schauder的阶理论和Rabinowitz的全局分岔理论，得到了从平凡解和无穷分岔的正解分支的结构。也就是说，正解分别从零解和无穷远处的极限p-拉普拉斯算子的第一特征值处分叉。给出了从高特征值出发的进一步分岔现象。与线性方程不同的是，对于p-拉普拉斯方程，获得更高的特征值并非易事。利用Ljusternik-Schnirelman理论构造了一系列特征值。虽然不知道与这些特征值相对应的特征函数系是否完备，但我们已经证明了非平凡解至少在这些特征值处从平凡解和无穷远处分叉。其次，研究了主体阶数与外力项不同时的情况。利用上述第一分岔结果，给出了各参数正解的多重性和不存在性，以及最小解的存在性。在论证中，首先讨论的方程正解分支的结构起着至关重要的作用。利用比较定理，推广Brezis-Nirenberg给出的“H^1与C^1局部极小值”原理，对该类拟线性椭圆方程进行退化。应用这些结果，我们得到了上述正解的一些性质。少

We have investigated some properties of nontrivial solutions of quasilinear degenerate elliptic equations which are equal to p-Laplacians asymptotically at the origin and infinity respectively. The results obtained in this project are as follows:First we have considered the case when the asymptotic orders of the principal part and the term of the exterior force are equal at the origin and the infinity respectively. By giving the the uniform estimate of the gradients of solutions for the equations and by generalizing the degree theory of Leray-Schauder and global bifurcation theory by Rabinowitz, we have obtained the structure of the branches of positive solutions bifurcating from the trivial solution and infinity. Namely, positive solutions bifurcate from the zero solution and the infinity at the first eigenvalues of the limitting p-Laplacians at zero and infinity respectively. Further bifuracation phenomina from the higher eigenvalues are also given. Differed from the linear equations … More , it is not trivial to obtain higher eigenvalues for the p-Laplacian. The Ljusternik-Schnirelman theory is used to construct a series of eigenvalues. Although it is not known that the system of eigenfunctions corresponding to those eigenvalues are complete or not, we have shown that nontrivial solutions bifurcate at least at those eigenvalues from the trivial solution and infinity.Secondary the case when the orders of the principal part and the term of the exterior force are different has been investigated. By using the first bifurcation results stated above, we have given the the multiplicity and nonexistence of positive solutions for each parameters and existence of the minimal solutions. In the argument the structure of the branch of positive solutions for the equations discussed at first plays a crucial role. We showed the comparison theorem and extend the principle of "H^1 versus C^1 local minimizers" given by Brezis-Nirenberg to degenerate qusilinear elliptic equations of this type. Applying these results, we obtain some properties of positive solutions stated above. Less