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Existence and convergence of positive weak solutions of $$-\Delta u=u^{n\over {n-2}}$$ in bounded domains of $$\mathbb{R}^ n$$, $$n\geq 3$$. (English) Zbl 0822.35051
The main result of this paper can be summarized as follows: For any closed subset $$S$$ of the open domain $$\Omega\subset \mathbb{R}^ n$$ there exist infinitely many positive weak solutions $$u\in L^{n/(n-2)} (\Omega)$$ of the equation $-\Delta u= u^{n/ (n-2)} \quad \text{in } \Omega, \qquad u=0 \quad \text{on } \partial \Omega \tag $$*$$$ whose singular set is given by $$S$$. To construct these solutions, the author uses the notion of quasi-solutions, that is a function $$\overline {u}$$ which satisfies $$(*)$$ up to a function $$f$$ which can be made arbitrarily small. Then, given any sequence $$\{x_ i\}$$ of points in $$\Omega$$ which is dense in $$S$$, a sequence of quasi solutions $$\{\overline {u}_ i\}$$ is constructed, where the singular set of $$\overline {u}_ i$$ is equal to $$\{x_ i, \dots, x_ i\}$$. By a careful analysis it is shown that this sequence leads to a weak solution of $$(*)$$ with singular set $$S$$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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##### References:
 [1] Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal.14, 349-381 (1973) · Zbl 0273.49063 [2] Aviles, P.: On isolated singularities in some nonlinear partial differential equations. Indiana Univ. Math. J.35, 773-791 (1983) · Zbl 0548.35042 [3] Aviles, P.: Local behavior of solutions of some elliptic equations. Commun. Math. Phys.108, 177-192 (1987) · Zbl 0617.35040 [4] Brezis, H., Bethuel, F., Coron, J.M.: Relaxed energies for harmonic maps. Variational Problems. Paris Juin 1988. Basel, Boston: Birkhäuser 1989 [5] Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functional. Proc. Am. Math. Soc.88, 486-490 (1983) · Zbl 0526.46037 [6] Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs. In: Brezis, H., Lions, J.L. (eds) Research Notes in Mathematics, vol. 60. London: Pitman 1982 · Zbl 0496.35030 [7] DiPerna, R.J., Majda, V.G.: Reduced Hausdorff dimension and concentration cancellation for two dimensional incompressible flow. J. Am. Math. Soc.1, 59-95 (1988) · Zbl 0707.76026 [8] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. (Grundlehren Math. Wiss., vol.224) Berlin Heidelberg New York: Springer 1977 · Zbl 0361.35003 [9] Mazzeo, R., Smale, N.: Conformally fiat metrics of constant positive scalar curvature on subdomains of the sphere. J. Differ. Geom.34, 581-621 (1991) · Zbl 0759.53029 [10] Pacard, F.: A note on the regularity of weak solutions of ??u=u ? in ? n ,n?3. Houston J. Math.18 (4), 621-632 (1993) · Zbl 0819.35045 [11] Pacard, F.: A regularity criterion for positive weak solutions of ??u=u ?. Comment. Math. Helv.68, 73-84 (1993) · Zbl 0799.35070 [12] Pacard, F.: Existence et convergence de solutions faibles positives de ??u= $$u^{\frac{n}{{n - 2}}}$$ dans des ouverts bornés de ? n ,n?3. C. R. Acad. Sci. Paris, Ser. 1314, 729-734 (1992) · Zbl 0777.35019 [13] Riviere, T.: Applications harmoniques deB 3 dansS 2 ayant une ligne de singularités. C. R. Acad. Sci. Paris, Ser. 1313, 583-587 (1991) [14] Riviere, T.: Applications harmoniques de B3 dans S2 partout discontinues. C. R. Acad. Sci. Paris, Ser. 1314, 719-724 (1992) [15] Schoen, R.: The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Commun. Pure Appl. Math.41, 317-392 (1988) · Zbl 0674.35027 [16] Schoen, R., Yau, S.T.: Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math.92, 47-72 (1988) · Zbl 0658.53038 [17] Zheng, Y.: Concentration in sequences of solutions to the nonlinear Klein-Gordon equation. Indiana Univ. Math. J.40, 201-235 (1991) · Zbl 0725.35102 [18] Ziemer, W.P.: Weakly differentiable functions. Berlin Heidelberg New York: Springer 1989 · Zbl 0692.46022
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