Geodesic
Counter-example for Liouville theorems for indefinite problems on half spaces
Acta mathematica Universitatis Comenianae, Tome 81 (2012) no. 1, pp. 105-116 
J. Földes; J. Földes. Counter-example for Liouville theorems for indefinite problems on half spaces. Acta mathematica Universitatis Comenianae, Tome 81 (2012) no. 1, pp. 105-116. http://geodesic.mathdoc.fr/item/AMUC_2012_81_1_a9/
@article{AMUC_2012_81_1_a9,
     author = {J. F\"oldes and J. F\"oldes},
     title = { Counter-example for {Liouville} theorems for indefinite problems on half spaces},
     journal = {Acta mathematica Universitatis Comenianae},
     pages = {105--116},
     year = {2012},
     volume = {81},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2012_81_1_a9/}
}
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Voir la notice de l'article provenant de la source Comenius University

The goal of this paper is a construction of an counter-example to Liouville theorems for indefinite problems on half spaces. Since Liouville theorems are closely related to the scaling method for elliptic and parabolic problems, our counter-example indicates that one has to impose additional assumptions on the nodal set of nonlinearity in order to obtain a priori estimates for indefinite elliptic problems. The counter-example is constructed by shooting method in one-dimensional case and then extended to higher dimensions.