We study the quasilinear equation [(P)qquad - { m div} (A(x,u) | abla u|^{p-2} abla u) + rac1p A_t(x,u) | abla u|^p + |u|^{p-2}u = g(x,u) qquad hbox{in $R^N$,} ] with $Nge 3$, $p > 1$, where $A(x,t)$, $A_t(x,t) = rac{partial A}{partial t}(x,t)$ and $g(x,t)$ are Carath'eodory functions on $R^N imes R$. Suitable assumptions on $A(x,t)$ and $g(x,t)$ set off the variational structure of $(P)$ and its related functional $J$ is $C^1$ on the Banach space $X = W^{1,p}(R^N) cap L^infty(R^N)$. In order to overcome the lack of compactness, we assume that the problem has radial symmetry, then we look for critical points of $J$ restricted to $X_r$, subspace of the radial functions in $X$. Following an approach which exploits the interaction between $|cdot|_X$ and the norm on $W^{1,p}(R^N)$, we prove the existence of at least one weak bounded radial solution of $(P)$ by applying a generalized version of the Ambrosetti-Rabinowitz Mountain Pass Theorem.

Existence of radial bounded solutions for some quasilinear elliptic equations in R^N

Anna Maria Candela;Addolorata Salvatore
2020-01-01

Abstract

We study the quasilinear equation [(P)qquad - { m div} (A(x,u) | abla u|^{p-2} abla u) + rac1p A_t(x,u) | abla u|^p + |u|^{p-2}u = g(x,u) qquad hbox{in $R^N$,} ] with $Nge 3$, $p > 1$, where $A(x,t)$, $A_t(x,t) = rac{partial A}{partial t}(x,t)$ and $g(x,t)$ are Carath'eodory functions on $R^N imes R$. Suitable assumptions on $A(x,t)$ and $g(x,t)$ set off the variational structure of $(P)$ and its related functional $J$ is $C^1$ on the Banach space $X = W^{1,p}(R^N) cap L^infty(R^N)$. In order to overcome the lack of compactness, we assume that the problem has radial symmetry, then we look for critical points of $J$ restricted to $X_r$, subspace of the radial functions in $X$. Following an approach which exploits the interaction between $|cdot|_X$ and the norm on $W^{1,p}(R^N)$, we prove the existence of at least one weak bounded radial solution of $(P)$ by applying a generalized version of the Ambrosetti-Rabinowitz Mountain Pass Theorem.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11586/231287
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 7
  • ???jsp.display-item.citation.isi??? 6
social impact