Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations

Abstract

In this paper, we consider the problem of the existence of non-negative weak solution u of

$$\left\{ \begin{gathered} \Delta u + u^p = 0 in \Omega \hfill \\ u = 0 on \partial \Omega \hfill \\ \end{gathered} \right.$$

having a given closed set S as its singular set. We prove that when$\frac{n}{{n - 2}}< p< \frac{{n + 2\sqrt {n - 1} }}{{n - 4 + 2\sqrt {n - 1} }}$ and S is a closed subset of Ω, then there are infinite many positive weak solutions with S as their singular set. Applying this method to the conformal scalar curvature equation for n ≥ 9, we construct a weak solution$u \in L^{\frac{{n + 2}}{{n - 2}}} \left( {S^n } \right)$ of$L_0 u + L^{\frac{{n + 2}}{{n - 2}}} = 0$ such that Sⁿ is the singular set of u where L₀ is the conformal Laplacian with respect to the standard metric of Sⁿ. When n = 4 or 6, this kind of solution has been constructed by Pacard.