Energy estimates for seminodal solutions to an elliptic system with mixed couplings. (English) Zbl 1511.35144

Some semilinear elliptic systems with mixed couplings on the whole space \(\mathbb{R}^N\) are studied in this paper. Sufficient conditions are given in order to obtain fully nontrivial (i.e., with all nonzero components) solutions in the Sobolev space \(H^1(\mathbb{R}^N)\). Upper estimates to the energy of the system are also obtained, they are a useful auxiliary tool all along the paper. The so-called unimodal or semi-positive solutions are especially interesting. The method of proof uses symmetries combined with concentration-compactness arguments. Also, solutions with positive and non-radial signs changing components to an associated singularly perturbed system on the unit ball are exhibited.


35J47 Second-order elliptic systems
35J61 Semilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
Full Text: DOI arXiv


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