Multiplicity of positive solutions for semilinear elliptic systems
Summary: 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 prove that for sufficiently small \(\lambda, \mu > 0\), there are at least \(k + 1\) positive solutions of the semilinear elliptic systems \(-\Delta u = \lambda f(x)|u|^{q-2}u + (\alpha/(\alpha + \beta))h(x)|u|^{\alpha -2}u|v|^\beta\), \(-\Delta v = \mu g(x)|v|^{q-2}v + (\beta/(\alpha + \beta))h(x)|u|^\alpha|v|^{\beta-2}v\), where \(0 \in \Omega \subset \mathbb R^N\) is a bounded domain, \(\alpha > 1\), \(\beta > 1\), and \(N/(N - 2) < q < 2 < \alpha + \beta = 2^\ast\) for \(N > 4\).
scientific article; zbMATH DE number 6209424
scientific article