The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems. (English) Zbl 1160.35417
The aim of this paper is to study the precise asymptotic behaviour of the solutions near the boundary to the model problems
\[
\Delta u= k(x)g(u),\quad x\in\Omega,
\]
\[ u|_{\partial\Omega}= +\infty, \] where \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb{R}^N\) \((N\geq 1)\), \(g\in C^1([0,\infty))\) and satisfies additional natural assumptions.
\[ u|_{\partial\Omega}= +\infty, \] where \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb{R}^N\) \((N\geq 1)\), \(g\in C^1([0,\infty))\) and satisfies additional natural assumptions.
Reviewer: Messoud A. Efendiev (Berlin)
References:
| [1] | Bandle, C.; Marcus, M., Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. Anal. Math., 58, 9-24 (1992) · Zbl 0802.35038 |
| [2] | Bandle, C.; Giarrusso, E., Boundary blowup for semilinear elliptic equations with nonlinear gradient terms, Adv. Differential Equations, 1, 133-150 (1996) · Zbl 0840.35034 |
| [3] | Cirstea, F.; Radulescu, V. D., Uniqueness of the blow-up boundary solution of logistic equations with absorption, C. R. Acad. Sci. Paris Sér. I Math., 335, 447-452 (2002) · Zbl 1183.35124 |
| [4] | Cirstea, F.; Radulescu, V. D., Asymptotics for the blow-up boundary solution of the logistic equation with absorption, C. R. Acad. Sci. Paris Sér. I Math., 336, 231-236 (2003) · Zbl 1068.35035 |
| [5] | Cirstea, F.; Radulescu, V. D., Solutions with boundary blow-up for a class of nonlinear elliptic problems, Houston J. Math., 29, 821-829 (2003) · Zbl 1119.35018 |
| [6] | Diaz, G.; Letelier, R., Explosive solutions of quasilinear elliptic equation: existence and uniqueness, Nonlinear Anal., 20, 97-125 (1993) · Zbl 0793.35028 |
| [7] | Giarrusso, E., Asymptotic behavior of large solutions of an elliptic quasilinear equation with a borderline case, C. R. Acad. Sci. Paris Sér. I Math., 331, 777-782 (2000) · Zbl 0966.35041 |
| [8] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0691.35001 |
| [9] | Greco, A.; Porru, G., Asymptotic estimates and convexity of large solutions to semilinear elliptic equations, Differential Integral Equations, 10, 219-229 (1997) · Zbl 0889.35028 |
| [10] | Keller, J. B., On solution of \(\Delta u = f(u)\), Comm. Pure Appl. Math., 10, 503-510 (1957) · Zbl 0090.31801 |
| [11] | Lair, A. V., A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations, J. Math. Anal. Appl., 240, 205-218 (1999) · Zbl 1058.35514 |
| [12] | Lasry, J. M.; Lions, P. L., Nonlinear elliptic equation with singular boundary conditions and stochastic control with state constraints, Math. Z., 283, 583-630 (1989) · Zbl 0688.49026 |
| [13] | Lazer, A. C.; McKenna, P. J., On a problem of Bieberbach and Rademacher, Nonlinear Anal., 21, 327-335 (1993) · Zbl 0833.35052 |
| [14] | Lazer, A. C.; McKenna, P. J., Asymptotic behavior of solutions of boundary blowup problems, Differential Integral Equations, 7, 1001-1019 (1994) · Zbl 0811.35010 |
| [15] | Loewner, C.; Nirenberg, L., Partial differential equations invariant under conformal or projective transformations, (Contributions to Analysis (a collection of papers dedicated to Lipman Bers) (1974), Academic Press: Academic Press New York), 245-272 · Zbl 0298.35018 |
| [16] | López-Gómez, J., The boundary blow-up rate of large solutions, J. Differential Equations, 195, 25-45 (2003) · Zbl 1130.35329 |
| [17] | Marcus, M.; Véron, L., Uniqueness of solutions with blowup on the boundary for a class of nonlinear elliptic equations, C. R. Acad. Sci. Paris Sér. I Math., 317, 557-563 (1993) · Zbl 0803.35041 |
| [18] | Maric, V., Regular Variation and Differential Equations, Lecture Notes in Math., vol. 1726 (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 0946.34001 |
| [19] | Osserman, R., On the inequality \(\Delta u \geqslant f(u)\), Pacific J. Math., 7, 1641-1647 (1957) · Zbl 0083.09402 |
| [20] | Seneta, R., Regular Varying Functions, Lecture Notes in Math., vol. 508 (1976), Springer-Verlag: Springer-Verlag Berlin · Zbl 0324.26002 |
| [21] | Tao, S.; Zhang, Z., On the existence of explosive solutions for semilinear elliptic problems, Nonlinear Anal., 48, 1043-1050 (2002) · Zbl 1033.35040 |
| [22] | Véron, L., Semilinear elliptic equations with uniform blowup on the boundary, J. Anal. Math., 59, 231-250 (1992) · Zbl 0802.35042 |
| [23] | Zhang, Z., A remark on the existence of explosive solutions for a class of semilinear elliptic equations, Nonlinear Anal., 41, 143-148 (2000) · Zbl 0964.35053 |
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