On some multiple solutions for a $p(x)$-Laplace equation with supercritical growth

We consider the multiplicity of solutions for the $p(x)$-Laplacian problems involving the supercritical Sobolev growth via Ricceri’s principle. By means of the truncation combining with De Giorgi iteration, we can extend the result about subcritical and critical growth to the supercritical growth and obtain at least three solutions for the $p(x)$ Laplacian problem.