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- W2901842011 abstract "In this paper we are proving the existence of a nontrivial solution of the ${p}(x)$- Laplacian equation with Dirichlet boundary condition. We will use the variational method and concentration compactness principle involving positive radon measure $mu$. begin{align*} begin{split} -Delta_{p(x)}u & = |u|^{q(x)-2}u+f(x,u)+mu,,mbox{in},,Omega, u & = 0,, mbox{on},, partialOmega, end{split} end{align*} where $Omega subset mathbb{R}^N$ is a smooth bounded domain, $mu > 0$ and $1 < p^{-}:=underset{xin Omega}{text{inf}};p(x) leq p^{+}:= underset{xin Omega}{text{sup}};p(x) < q^{-}:=underset{xin Omega}{text{inf}};q(x)leq q(x) leq p^{ast}(x) < N$. The function $f$ satisfies certain conditions. Here, $q^{prime}(x)=frac{q(x)}{q(x)-1}$ is the conjugate of $q(x)$ and $p^{ast}(x)=frac{Np(x)}{N-p(x)}$ is the Sobolev conjugate of $p(x)$." @default.
- W2901842011 created "2018-11-29" @default.
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- W2901842011 creator A5063540430 @default.
- W2901842011 date "2018-11-15" @default.
- W2901842011 modified "2023-09-27" @default.
- W2901842011 title "Existence of solution of the $p(x)$-Laplacian problem involving critical exponent and Radon measure" @default.
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