Geodesic
Existence of solutions for discontinuous \(p(x)\)-Laplacian problems with critical exponents
Electronic journal of differential equations, Tome 2012 (2012)
In this article, we study the existence of solutions to the problem

$\displaylines{ -\hbox{div}(|\nabla u|^{p(x)-2}\nabla u) =\lambda |u|^{p^{*}(x)-2}u + f(u)\quad x \in \Omega ,\cr u = 0 \quad x \in \partial\Omega, }$

where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^{N}, p(x)$ is a continuous function with $1$ and $p^{*}(x) = \frac{Np(x)}{N-p(x)}$. Applying nonsmooth critical point theory for locally Lipschitz functionals, we show that there is at least one nontrivial solution when $\lambda$ less than a certain number, and $f$ maybe discontinuous.
Classification : 35J92, 35J70, 35R70
Keywords: $p(x)$-Laplacian problem, critical Sobolev exponents, discontinuous nonlinearities 
@article{EJDE_2012__2012__a34,
     author = {Shang,  Xudong and Wang,  Zhigang},
     title = {Existence of solutions for discontinuous {\(p(x)\)-Laplacian} problems with critical exponents},
     journal = {Electronic journal of differential equations},
     year = {2012},
     volume = {2012},
     zbl = {1241.35114},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a34/}
}
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%A Wang,  Zhigang
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Shang,  Xudong; Wang,  Zhigang. Existence of solutions for discontinuous \(p(x)\)-Laplacian problems with critical exponents. Electronic journal of differential equations, Tome 2012 (2012). http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a34/