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Multiplicity of positive solutions for semilinear elliptic systems. (English) Zbl 1277.35161

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 \(N > 4\).

MSC:

35J47 Second-order elliptic systems
35J61 Semilinear elliptic equations
35B09 Positive solutions to PDEs
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