Asymptotic Estimates and Qualitatives Properties of an Elliptic Problem in Dimension Two

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Mehdi, Khalil El; Grossi, Massimo;
  • Subject: Mathematics - Analysis of PDEs | 35J60

In this paper we study a semilinear elliptic problem on a bounded domain in $\R^2$ with large exponent in the nonlinear term. We consider positive solutions obtained by minimizing suitable functionals. We prove some asymtotic estimates which enable us to associate a "li... View more
  • References (12)
    12 references, page 1 of 2

    [6] E. N. Dancer, On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mount. J. Math.25, 1995, 957-975.

    [7] 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. 61 (1982), 41-63.

    [8] H. Federer, Geometric measure theory , Spinger-Verlag, 1969.

    [9] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243.

    [10] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Par. Diff. Eqns. 6 (1981), 883-901.

    [11] D. Gilbarg and N. Trudinger, Elliptic partial differential equation of second order, second edition, Springer Verlag 1983.

    [12] M. Grossi and R. Molle, On the shape of the solutions of some semilinear elliptic problems, Preprint 2001.

    [13] C. S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in R2, Manuscripta Math. 84 (1994), 13-19.

    [14] L. E. Payne, On two conjectures in the fixed membrane eigenvalue problem, Z. Angew. Math. Phys. 24 (1973), 721-729.

    [15] 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.

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