Multiplicity of Positive Solutions for Semilinear Elliptic Systems

Abstract

We study the effect of the coefficient $h(x)$ of the critical nonlinearity on the number of positive solutions for semilinear elliptic systems. Under suitable assumptions for $f(x)$, $g(x)$, and $h(x)$, we should prove that for sufficiently small $\lambda ,\mathrm{}\mu >\mathrm{0}$, there are at least $k+\mathrm{1}$ positive solutions of the semilinear elliptic systems $-\mathrm{\Delta }u=\lambda f(x)|u{|}^{q-\mathrm{2}}u+\mathrm{(\alpha }/(\alpha +\mathrm{\beta )})h(x)|u{|}^{\alpha -\mathrm{2}}u|v{|}^{\beta }$, $\mathrm{}-\mathrm{\Delta }v=\mu g(x)|v{|}^{q-\mathrm{2}}v+\mathrm{(\beta }/(\alpha +\mathrm{\beta )})h(x)|u{|}^{\alpha }|v{|}^{\beta -\mathrm{2}}v$, where $\mathrm{0}\in \mathrm{\Omega }\subset {ℝ}^N$ is a bounded domain, $\alpha >\mathrm{1}$, $\beta >\mathrm{1}$, and for $N>\mathrm{4}$.