Effect of Non-linear Lower Order Terms in Quasilinear Equations Involving the $p(x)$-Laplacian

Abstract

In this work, we study the existence of $W_0^{1,p(\cdot )}$-solutions to the following boundary value problem involving the $p(\cdot )$-Laplacian operator: \begin{equation*} \left\lbrace \begin{array}{l} -Δ_{p(x)}u+|\nabla u|^{q(x)}=λg(x)u^{η(x)}+f(x), \quad\textnormal{ in } Ω, \\\qquad \,\,\,\,\,\quad \quad\qquad\quad u\geq 0, \quad\textnormal{ in } Ω \qquad \,\,\,\,\,\quad \quad\qquad\quad u= 0, \,\,\quad \text{on } \partialΩ. \end{array} \right. \end{equation*}under appropriate ranges on the variable exponents. We give assumptions on $f$ and $g$ in terms of the growth exponents $q$ and $\eta$ under which the above problem has a non-negative solution for all $\lambda >0$.

21 pages, no figures