Existence of solutions for discontinuous p(x)-Laplacian problems with critical exponents
In this article, we study the existence of solutions to the problem $$displaylines{ -hbox{div}(|abla u|^{p(x)-2}abla u) =lambda |u|^{p^{*}(x)-2}u + f(u)quad x in Omega ,cr u = 0 quad x in partialOmega, }$$ where $Omega$ is a smooth bounded domain in ${mathbb{R}}^{N}$, $p(x)$ is a continuous...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
2012-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2012/25/abstr.html |
| Summary: | In this article, we study the existence of solutions to the problem $$displaylines{ -hbox{div}(|abla u|^{p(x)-2}abla u) =lambda |u|^{p^{*}(x)-2}u + f(u)quad x in Omega ,cr u = 0 quad x in partialOmega, }$$ where $Omega$ is a smooth bounded domain in ${mathbb{R}}^{N}$, $p(x)$ is a continuous function with $1<p(x)<N$ 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. |
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| ISSN: | 1072-6691 |