Geodesic
Solutions of \(p(x)\)-Laplacian equations with critical exponent and perturbations in \(R^N\)
Electronic journal of differential equations, Tome 2012 (2012)
Based on the theory of variable exponent Sobolev spaces, we study a class of $p(x)$-Laplacian equations in $\mathbb{R}^{N}$ involving the critical exponent. Firstly, we modify the principle of concentration compactness in $W^{1,p(x)}(\mathbb{R}^{N})$ and obtain a new type of Sobolev inequalities involving the atoms. Then, by using variational method, we obtain the existence of weak solutions when the perturbation is small enough.
Classification : 35J60, 46E35
Keywords: variable exponent Sobolev space, critical exponent, weak solution 
@article{EJDE_2012__2012__a90,
     author = {Zhang,  Xia and Fu,  Yongqiang},
     title = {Solutions of {\(p(x)\)-Laplacian} equations with critical exponent and perturbations in {\(R^N\)}},
     journal = {Electronic journal of differential equations},
     year = {2012},
     volume = {2012},
     zbl = {1259.35098},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a90/}
}
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AU  - Fu,  Yongqiang
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JO  - Electronic journal of differential equations
PY  - 2012
VL  - 2012
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ID  - EJDE_2012__2012__a90
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%A Fu,  Yongqiang
%T Solutions of \(p(x)\)-Laplacian equations with critical exponent and perturbations in \(R^N\)
%J Electronic journal of differential equations
%D 2012
%V 2012
%U http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a90/
%G en
%F EJDE_2012__2012__a90
Zhang,  Xia; Fu,  Yongqiang. Solutions of \(p(x)\)-Laplacian equations with critical exponent and perturbations in \(R^N\). Electronic journal of differential equations, Tome 2012 (2012). http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a90/