On a two-dimensional elliptic problem with large exponent in nonlinearity
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- by Xiaofeng Ren and Juncheng Wei PDF
- Trans. Amer. Math. Soc. 343 (1994), 749-763 Request permission
Abstract:
A semilinear elliptic equation on a bounded domain in ${R^2}$ with large exponent in the nonlinear term is studied in this paper. We investigate positive solutions obtained by the variational method. It turns put that the constrained minimizing problem possesses nice asymptotic behavior as the nonlinear exponent, serving as a parameter, gets large. We shall prove that ${c_p}$, the minimum of energy functional with the nonlinear exponent equal to p, is like ${(8\pi e)^{1/2}}{p^{ - 1/2}}$ as p tends to infinity. Using this result, we shall prove that the variational solutions remain bounded uniformly in p. As p tends to infinity, the solutions develop one or two peaks. Precisely the solutions approach zero except at one or two points where they stay away from zero and bounded from above. Then we consider the problem on a special class of domains. It turns out that the solutions then develop only one peak. For these domains, the solutions enlarged by a suitable quantity behave like a Green’s function of $- \Delta$. In this case we shall also prove that the peaks must appear at a critical point of the Robin function of the domain.References
- Haïm Brezis and Frank Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u=V(x)e^u$ in two dimensions, Comm. Partial Differential Equations 16 (1991), no. 8-9, 1223–1253. MR 1132783, DOI 10.1080/03605309108820797
- Haïm Brézis and Walter A. Strauss, Semi-linear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan 25 (1973), 565–590. MR 336050, DOI 10.2969/jmsj/02540565
- D. G. de Figueiredo, P.-L. Lions, and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9) 61 (1982), no. 1, 41–63. MR 664341
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Zheng-Chao Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré C Anal. Non Linéaire 8 (1991), no. 2, 159–174 (English, with French summary). MR 1096602, DOI 10.1016/S0294-1449(16)30270-0
- J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. MR 301504, DOI 10.1512/iumj.1971.20.20101
- Wei-Ming Ni and Izumi Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819–851. MR 1115095, DOI 10.1002/cpa.3160440705 S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Soviet Math. Dokl. 6 (1965), 1408-1411. X. Ren and J. Wei, On a semilinear elliptic problem in ${R^2}$ when the exponent approaches infinity, preprint.
- Olivier Rey, The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), no. 1, 1–52. MR 1040954, DOI 10.1016/0022-1236(90)90002-3
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 343 (1994), 749-763
- MSC: Primary 35J65; Secondary 35B30, 35J20
- DOI: https://doi.org/10.1090/S0002-9947-1994-1232190-7
- MathSciNet review: 1232190