Nonlinear 𝑝-Laplacian problems on unbounded domains

Yu, Lao Sen

doi:10.1090/S0002-9939-1992-1162957-9

Nonlinear -Laplacian problems on unbounded domains

Author: Lao Sen Yu
Journal: Proc. Amer. Math. Soc. 115 (1992), 1037-1045
MSC: Primary 35J40; Secondary 35B45, 35J65, 58E05
DOI: https://doi.org/10.1090/S0002-9939-1992-1162957-9
MathSciNet review: 1162957
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Abstract: We consider the -Laplacian problem

$\displaystyle - div(a(x)\vert\nabla u{\vert^{p - 2}}\nabla u) + b(x)\vert u{\ve... ...al \Omega }} = 0,\quad \mathop {\lim }\limits_{\vert x\vert \to \infty } u = 0,$

where $1 < p < n,\Omega ( \subset {R^n})$ is an exterior domain. Under certain conditions, we show the existence of solutions for this problem via critical point theory.

[1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
[2] J. P. García Azorero and I. Peral Alonso, Existence and nonuniqueness for the 𝑝-Laplacian: nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987), no. 12, 1389–1430. MR 912211, https://doi.org/10.1080/03605308708820534
[3] Melvyn S. Berger and Martin Schechter, Embedding theorems and quasi-linear elliptic boundary value problems for unbounded domains, Trans. Amer. Math. Soc. 172 (1972), 261–278. MR 312241, https://doi.org/10.1090/S0002-9947-1972-0312241-X
[4] Marie-Françoise Bidaut-Véron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal. 107 (1989), no. 4, 293–324. MR 1004713, https://doi.org/10.1007/BF00251552
[5] Henrik Egnell, Existence and nonexistence results for 𝑚-Laplace equations involving critical Sobolev exponents, Arch. Rational Mech. Anal. 104 (1988), no. 1, 57–77. MR 956567, https://doi.org/10.1007/BF00256932
[6] Mohammed Guedda and Laurent Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), no. 8, 879–902. MR 1009077, https://doi.org/10.1016/0362-546X(89)90020-5
[7] Satyanad Kichenassamy and Laurent Véron, Singular solutions of the 𝑝-Laplace equation, Math. Ann. 275 (1986), no. 4, 599–615. MR 859333, https://doi.org/10.1007/BF01459140
[8] Gong Bao Li and Shu Sen Yan, Eigenvalue problems for quasilinear elliptic equations on 𝑅^{𝑁}, Comm. Partial Differential Equations 14 (1989), no. 8-9, 1291–1314. MR 1017074, https://doi.org/10.1080/03605308908820654
[9] Gary M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203–1219. MR 969499, https://doi.org/10.1016/0362-546X(88)90053-3
[10] V. G. Maz'ja, Sobolev spaces, Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1985.
[11] W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case, Accad. Naz. Lincei 77 (1986), 231-257.
[12] Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 845785
[13] James Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. MR 170096, https://doi.org/10.1007/BF02391014
[14] François de Thélin, Local regularity properties for the solutions of a nonlinear partial differential equation, Nonlinear Anal. 6 (1982), no. 8, 839–844. MR 671725, https://doi.org/10.1016/0362-546X(82)90068-2
[15] Neil S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747. MR 226198, https://doi.org/10.1002/cpa.3160200406