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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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