Abstract
This paper deals with some recent results for scalar field equations in the case of a potential which converges from above to a limit at infinity. In particular, we shall collect some of the results of the authors which have contributed to partially answer a conjecture of Wei and Yan and we shall announce some results in a forthcoming paper to the aim of giving a short survey which collects the main ideas based on the use of symmetry assumptions and of symmetry methods introduced for the above problem.
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