Single-point condensation and least-energy solutions

Ren, Xiaofeng; Wei, Juncheng

doi:10.1090/S0002-9939-96-03156-5

Single-point condensation and least-energy solutions
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by Xiaofeng Ren and Juncheng Wei PDF

Proc. Amer. Math. Soc. 124 (1996), 111-120 Request permission

Abstract:

We prove a conjecture raised in our earlier paper which says that the least-energy solutions to a two-dimensional semilinear problem exhibit single-point condensation phenomena as the nonlinear exponent gets large. Our method is based on a sharp form of a well-known borderline case of the Sobolev embedding theory. With the help of this embedding, we can use the Moser iteration scheme to carefully estimate the upper bound of the solutions. We can also determine the location of the condensation points.

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
Luis A. Caffarelli and Avner Friedman, Convexity of solutions of semilinear elliptic equations, Duke Math. J. 52 (1985), no. 2, 431–456. MR 792181, DOI 10.1215/S0012-7094-85-05221-4
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
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, DOI 10.1007/BF01221125
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
C.-S. Lin, Uniqueness of solutions minimizing the functional $\int _{\Omega }|\nabla u|^{2}$ $/(\int _{\Omega }u^{p+1})^{2/p+1}$ in $R^{2}$, preprint.
Xiaofeng Ren and Juncheng Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc. 343 (1994), no. 2, 749–763. MR 1232190, DOI 10.1090/S0002-9947-1994-1232190-7
—, Counting peaks of solutions to some quasilinear elliptic equations with large exponents, J. Differential Equations 117 (1995), 28–55.
—, Asymptotic behavior of energy solutions to a two dimensional semilinear problem with mixed boundary condition, Nonlinear Anal.: TMA 24 (1995), 587–604.