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This paper is concerned with a degenerate \(p(x)\)-Laplacian equation with a nonsmooth potential. We establish a compact embedding \(W^{1,p(x)}(\omega,\Omega )\hookrightarrow L^{q(x)}(\alpha(x),\Omega)\) under suitable conditions and obtain the existence and multiplicity of solutions to the degenerate \(p(x)\)-Laplacian equation by the theories of nonsmooth critical point and the variable exponent Lebesgue-Sobolev spaces. Some recent results in the literature are generalized and improved.

Solutions for a degenerate p ( x ) $p(x)$ -Laplacian equation with a nonsmooth potential