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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 \(S\).

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
Full Text: DOI
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