We prove the existence of a renormalized solution for a nonlinear non coercive divergence problem with lower order terms and measure data. In a particular case we also give a uniqueness result.

In this article we are interested in the existence and uniqueness of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations $$\begin{array}{cc}\hfill \Delta \left(v\left(x\right){\left|\Delta u\right|}^{p-2}\Delta u\right)& -\sum _{j=1}^n{D}_j\left[\omega \left(x\right){𝒜}_j(x,u,\nabla u)\right]\hfill \\ \hfill =& f_0\left(x\right)-\sum _{j=1}^n{D}_j{f}_j\left(x\right),in\Omega \hfill \end{array}$$ in the setting of the weighted Sobolev spaces.

In this paper we study the nonlinear Dirichlet problem involving p(x)-Laplacian (hemivariational inequality) with nonsmooth potential. By using nonsmooth critical point theory for locally Lipschitz functionals due to Chang [6] and the properties of variational Sobolev spaces, we establish conditions which ensure the existence of solution for our problem.

In this article, we prove the existence of entropy solutions for the Dirichlet problem $$\left(P\right)\left\{\begin{array}{cc}-\mathrm{div}\left[\omega \right(x\left)𝒜\right(x,u,\nabla u\left)\right]=f\left(x\right)-\mathrm{div}\left(G\right),\hfill & \text{in}\Omega \hfill \\ u\left(x\right)=0,\hfill & \text{on}\partial \Omega \hfill \end{array}\right.$$ where $\Omega$ is a bounded open set of ${}^N$, $N\ge 2$, $f\in L^1\left(\Omega \right)$ and $G/\omega \in {\left[L^{p^{\text{'}}}(\Omega ,\omega )\right]}^N$.

In the present paper, we prove existence results of entropy solutions to a class of nonlinear degenerate parabolic $p(·)$-Laplacian problem with Dirichlet-type boundary conditions and $L^1$ data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.