此論文中，我們分析在一些情況之下一非線性橢圓方程的基態解為古典解。

In this thesis, we present the concept of indexes of domains. Then we characterize achieved domains. In an achieve domain, there is a ground state solution. We develop various analysis to assert that a ground state solution is a classical solution.

Adams, R. A., Sobolev space, Academic Press, New York 1975.
A. Ambrosetti and P. H. Rabinowitz, Dual variational method in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
H. Brézis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44(1991), 939-963.
K. -J. Chen and H. -C. Wang, A necessary and sufficient condition for Palais-Smale conditions, SIAM J. Math. Anal., 31 (1999), 154-165.
M. J. Esteban and P. -L. Lions, Existence and non-existence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh, Sect. A, 93 (1982), 1-12.
B. Gidas, W. -M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in R^{N}, Adv. in Math. Suppl. Stud., 7A (1981), 369-402.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second Edition, Springer Verlag, New York, 1983.
M. -K. Kwong, Uniqueness of positive solutions of Δu-u+u^{p}=0 in Rⁿ, Arch. Ration. Mech. Anal., 105 (1989), 243-266.
W. -C. Lien, S. -Y. Tzeng, and H. -C. Wang, Existence of solutions of semilinear elliptic problems in unbounded domains, Differential Integral Equations, 6 (1993), 1281-1298.
P. -L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145; 223-283.
P. -L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, II, Rev. Mat. Iberoamericana, 1, No. 1 (l 985), 145-20 1; No. 2 (1985), 45-121.
Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123.
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conference Series in Mathematics, American Mathematical Society, 1986.
C. A. Stuart, Bifurcation in L^{p}(R^{N}) for a semilinear elliptic equation, Proc. London Math. Soc., 45 (1982), 169-192.
H. -C. Wang, A Palais-Smale approach to problems in Esteban-Lions domains with holes, Trans. Amer. Math. Soc., 352 (2000), 4237-4256.
H. -C. Wang, Palais-Smale approachs to semilinear elliptic equations in unbounded domains, Preprint.
M. Willem, Minimax theorems, Birkhauser Verlag, Basel, 1996.