Existence and nonexistence results for quasilinear elliptic equations involving the $p$-Laplacian

Abdellaoui, Boumediene ; Felli, Veronica ; Peral, Ireneo

Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 2, pp. 445-484.

Abstract

The paper deals with the study of a quasilinear elliptic equation involving the p-laplacian with a Hardy-type singular potential and a critical nonlinearity. Existence and nonexistence results are first proved for the equation with a concave singular term. Then we study the critical case related to Hardy inequality, providing a description of the behavior of radial solutions of the limiting problem and obtaining existence and multiplicity results for perturbed problems through variational and topological arguments.

Résumé

L’articolo riguarda lo studio di un’equazione ellittica quasi-lineare con il p-laplaciano, caratterizzata dalla presenza di un termine singolare di tipo Hardy ed una nonlinearità critica. Si dimostrano dapprima risultati di esistenza e non esistenza per l’equazione con un termine singolare concavo. Quindi si passa a studiare il caso critico legato alla disuguaglianza di Hardy, fornendo una descrizione del comportamento delle soluzioni radiali del problema limite e ottenendo risultati di esistenza e molteplicità mediante metodi variazionali e topologici.

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