Singular Extremal Solutions on Thin Ellipsoids with Varying Nonlinearities

Let $N=m+1$ and consider the thin ellipsoid $${\mathrm{\Omega }}_{\epsilon }=\{(y,x_N)\in {\mathbb{R}}^m\times \mathbb{R}:\mathrm{\mid }y{\mathrm{\mid }}^2+{\epsilon }^{-2}{x}_N^2<1\}\mathrm{.}$$ We prove that in every sufficiently large dimension, for every sufficiently small $\epsilon >0$, there exists a smooth positive, strictly increasing, strictly convex, superlinear nonlinearity $f_{\epsilon }$ for which the extremal solution in ${\mathrm{\Omega }}_{\epsilon }$ is an unbounded $H_0^1$ solution. Combining this result with Dancer’s thin-domain regularity theorem for the Gelfand nonlinearity $f(t)=e^t$, we obtain on the same sufficiently thin ellipsoids a bounded Gelfand extremal solution and an unbounded extremal solution for another nonlinearity. Thus, in this two-part sense, Brézis’ Open Problem~6.1 is resolved in every sufficiently large dimension.