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.


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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