The effect of the domain topology on the multiplicity of solutions for a semilinear elliptic system with critical Sobolev exponent is discussed. We show that if the coupling term is sufficiently small, then there exist at least cat $\Omega$ positive solutions, where cat denotes the Ljusternik-Schnirelman category.
Michinori Ishiwata. "Multiple solutions for semilinear elliptic systems involving critical Sobolev exponent." Differential Integral Equations 20 (11) 1237 - 1252, 2007. https://doi.org/10.57262/die/1356039287