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On a two-dimensional elliptic problem with large exponent in nonlinearity


Authors: Xiaofeng Ren and Juncheng Wei
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
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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.


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  • [1] H. Brezis and F. Merle, Uniform estimate and blow-up behavior for solutions of $ - \Delta u = V(x){e^u}$ in two dimensions, Comm. Partial Differential Equations 16 (1991), 1223-1253. MR 1132783 (92m:35084)
  • [2] H. Brezis and W. Strauss, Semilinear second-order elliptic equations in $ {L^1}$, J. Math. Soc. Japan 25 (1973), 565-590. MR 0336050 (49:826)
  • [3] D. G. DeFigueiredo, P. L. Lions, and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures. Appl. 61 (1982), 41-63. MR 664341 (83h:35039)
  • [4] H. Federer, Geometric measure theory, Springer-Verlag, Berlin-Heidelberg-New York, 1969. MR 0257325 (41:1976)
  • [5] B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243. MR 544879 (80h:35043)
  • [6] D. Gilbarg and S. N. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York, 1983. MR 737190 (86c:35035)
  • [7] Z. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Analyse Nonlinéaire 8-2 (1991). MR 1096602 (92c:35047)
  • [8] J. Moser, A sharp form of an inequality by Trudinger, Indiana Univ. Math. J. 20 (1971), 1077-1092. MR 0301504 (46:662)
  • [9] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure. Appl. Math. 44 (1991). MR 1115095 (92i:35052)
  • [10] S. Pohozaev, Eigenfunctions of the equation $ \Delta u + \lambda f(u) = 0$, Soviet Math. Dokl. 6 (1965), 1408-1411.
  • [11] X. Ren and J. Wei, On a semilinear elliptic problem in $ {R^2}$ when the exponent approaches infinity, preprint.
  • [12] O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), 1-52. MR 1040954 (91b:35012)

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DOI: https://doi.org/10.1090/S0002-9947-1994-1232190-7
Article copyright: © Copyright 1994 American Mathematical Society

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