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

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
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