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Existence of three nontrivial solutions for an elliptic system. (English) Zbl 1159.35334

Summary: We consider the existence of nontrivial solutions for an elliptic system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones \(K_1,K_2\subset C(\overline{\Omega})\) and computing the fixed point index in \(K_1\), \(K_2\) and \(K_1\times K_2\), we obtain that the elliptic system has three nontrivial solutions \((u,0)\), \((0,v)\) and \((u^*,v^*)\). It is remarkable that the third nontrivial solution \((u^*,v^*)\) is established on the Cartesian product of two cones, in which the feature of two equations can be exploited better.

MSC:

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J50 Variational methods for elliptic systems
35D05 Existence of generalized solutions of PDE (MSC2000)
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