Affiliations: [a] Université de Pau et des Pays de l’Adour, LMAP (UMR E2S-UPPA CNRS 5142), Bat. IPRA, Avenue de l’Université F-64013 Pau, France | [b] Institute of Mathematics of UFRC RAS, 112, Chernyshevsky str., 450008, Ufa, Russia | [c] Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970, Goiania, Brazil | [d] Research Institute of Mathematics, Seoul National University, Seoul 08826, South Korea
Abstract: We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: −Δu=λup−uq in RN+1, where 0<q<p<1 and λ∈R. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter λ, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p, q. Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space RN+1.
