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Convergence and partial regularity for weak solutions of some nonlinear elliptic equation: The supercritical case. (English) Zbl 0837.35026
Summary: We prove a partial regularity result for stationary weak solutions of $$- \Delta u= u^\alpha$$, when $$\alpha$$ is greater than the critical Sobolev exponent.

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations
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##### References:
 [1] F. Bethuel, Partial Regularity for Stationary Harmonic Maps, to appear in Manuscripta Math. [2] Campanato, S., Proprieta di inclusione per spazi di Morrey, Ricerche Mat., Vol. 12, 67-86, (1963) · Zbl 0192.22703 [3] Campanato, S., Equazioni ellittiche del II ordine e spazi $$\mathcal{L}^{(2, \operatorname{\lambda})}$$, Ann. Mat. Pura Appl., Vol. 4, 69, 321-381, (1965) · Zbl 0145.36603 [4] Evans, L. C., Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal., Vol. 116, 101-113, (1991) · Zbl 0754.58007 [5] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear analysis, Annals of Mathematical Studies, Vol. 105, (1989), Princeton Univ. Press [6] Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of the second order, (1977), Springer Berlin-Heidelberg-New York · Zbl 0361.35003 [7] Hardt, R.; Kinderlehrer, D.; Lin, F. H., Existence and partial regularity of static liquid cristal configurations, Comm. Math. Phys., Vol. 105, 547-570, (1986) · Zbl 0611.35077 [8] Morrey, C. B., Multiple integrals in the calculus of variations, (1966), Springer Berlin-Heidelberg-New York · Zbl 0142.38701 [9] Pacard, F., A note on the regularity of weak solutions of −δu = u^{α} in ℝ^{n}, n ≥ 3, Houston Journal of Math., Vol. 18, 4, 621-632, (1982) · Zbl 0819.35045 [10] F. Pacard, Partial Regularity for Weak Solutions of a Nonlinear Elliptic Equation, to appear in Manuscripta Mathematica. · Zbl 0811.35011 [11] Pohozaev, S. I., Eigenfunctions of the equation δu + λf(u) = 0, Soviet. Math. Doklady, Vol. 6, 1408-1411, (1965) · Zbl 0141.30202 [12] Schoen, R., Analytic aspects for the harmonic map problem, Math. Sei. Res. Inst. Publi., Vol. 2, (1984), Springer Berlin · Zbl 0551.58011
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