臺灣博碩士論文加值系統

在這篇論文中，我們考慮一個半線性橢圓方程式在一些無界定義域非零解的存在性問題。為了找出半線性橢圓方程式的非零解，許多人用了各種不同的方法，而我們則是利用維塔利收斂定理和巴萊斯麥爾理論，也就是有巴萊斯麥爾數列、巴萊斯麥爾條件，藉由函數收斂的緊緻性質來找尋此方程式的非零解。所以在這篇論文中，我們呈現了一些巴萊斯麥爾數列的新結果，而且我們還證明了巴萊斯麥爾條件、維塔利收斂定理、定義域的指數、半線性橢圓方程在一些其它無界定義域正解的存在性問題和之前的人所做的方法是等價的，這樣將有助於我們尋找此半線性橢圓方程式非零解的存在性。

In this article, we present several new results for Palais─Smale sequences. Consequently, we unify the Vitali convergence theorem and many main concepts in the variational methods by Lions, Lien-Tzeng-Wang, del Pino-Felmer and Chabrowski.

1 Introduction …………………………………………… 2
2 Compactness Theorems ………………………………… 3
3 Palais─Smale Values ………………………………… 6
4 Palais─Smale Conditions …………………………… 11
References …………………………………………………… 23

[1] A. Bahri and P. L. Lions, On the existence of positive solutions of semilinear elliptic equations in unbounded domains, Ann. I. H. P. Analyse Nonlineaire 14 (1997), 365-413.
[2] A. K. Ben-Naoum, C. Troestler and M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains, Nonlinear Anal. 26 (1996), 823-833.
[3] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of ground state, Arch. Rat. Mech. Anal., 82(1983), 313-345.
[4] H. Brézis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44(1991), 939-963.
[5] J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. PDE 3 (1995), 493-512.
[6] J. Chabrowski, Variational Methods for Potential Operator Equations, Walter de Gruyter, Berlin, New York, 1999.
[7] J. Chabrowski, Weak Convergence Methods for Semilinear Elliptic Equations, World Scientific, 1999.
[8] K. J. Chen and H. C. Wang, A necessary and sufficient condition for Palais─Smale conditions, SIAM Journal on Math. Anal. 31 (1999), 154-165.
[9] M. A. Del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. PDE 4 (1996), 121-137.
[10] M. A. Del Pino and P. L. Felmer, Least energy solutions for elliptic equations in unbounded domains, Proc. Royal Society Edinburgh, 126A (1996), 195-208.
[11] W. C. Lien, S. Y. Tzeng, and H. C. Wang, Existence of solutions of semilinear elliptic problems on unbounded domains, Diff. and Integ. Eqns. 6 (1993), 1281-1298.
[12] 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.
[13] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conference Series in Mathematics, American Mathematical Society, 1986.
[14] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-162.
[15] C. A. Stuart, Bifurcation in Lp (RN)for a semilinear elliptic equation}, Proc. London Math. Soc. 45 (1982), 169-192.
[16] H. C. Wang, A Palais─Smale approach to problems in Esteban-Lions domains with holes, Trans. Amer. Math. Soc. 352 (2000), 4237-4256.
[17] M. Willem, Minimax Theorems, Birkhauser Verlag, Basel, 1996.
[18] E. Zeidler, Nonlinear Functional Analysis and its Applications II/A, Springer-Verlag, New York, 1989.