Recent results in the theory of semilinear elliptic degenerate differential equations. (English) Zbl 1179.35144

Nguyen, Minh Tri

Vietnam J. Math. 37, No. 2-3, 387-397 (2009).

Summary: We give a survey on recent study of semilinear elliptic degenerate differential equations. Here we discuss the critical exponents phenomenon for boundary value problems and interior regularities of solutions of various classes of such equations. Similar problems for nonlinear elliptic equations were studied in [S. Bernstein, Math. Ann. 59, 20–76 (1904; JFM 35.0354.01); M. Z. 28, 330–348 (1928; JFM 54.0506.02); A. Friedman, J. Math. Mech. 7, 43–59 (1958; Zbl 0078.27702); M. Gevrey, Ann. Éc. Norm. (3) 35, 129–190 (1918; JFM 46.0721.01); G. Giraud, Ann. Éc. Norm. (3) 43, 1-128 (1926; JFM 52.0483.01); E. Hopf, Math. Z. 34, 194–233 (1931; Zbl 0002.34003); H. Lewy, Nachrichten Göttingen 1927, 178–186 (1927; JFM 53.0474.01); Math. Ann. 101, 609–619 (1929; JFM 55.0882.03), C. B. Morrey jun., Am. J. Math. 80, 198–218, 219–237 (1958; Zbl 0081.09402), I. G. Petrowsky, Rec. Math. Moscou, n. Ser. 5, 3–68 (1939; Zbl 0022.22601); s. I. Pokhozaev, Eigenfunctions for the equation \(\Delta u+\lambda f(u)=0\), Russ. Dokl. Akad. Nauk SSSR 165, 33–36 (1965); The solvability of nonlinear equations with odd operators, Funk. Anal. Prilozh. 1, 66–73 (1967); Ngyuen Minh Tri, Russ. J. Math. Phys. 10, No. 3, 353-358 (2003; Zbl 1039.35002)].

MSC:

35J70	Degenerate elliptic equations
35J60	Nonlinear elliptic equations
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35B33	Critical exponents in context of PDEs
35A08	Fundamental solutions to PDEs
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

elliptic degenerate equations; semilinear equations; critical exponents; regularity of solutions

Citations:

Zbl 0078.27702; Zbl 0002.34003; Zbl 0081.09402; Zbl 0022.22601; Zbl 1039.35002; JFM 35.0354.01; JFM 54.0506.02; JFM 46.0721.01; JFM 52.0483.01; JFM 53.0474.01; JFM 55.0882.03