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

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