In this talk, I will overview the regularity theory for p(x)-Laplace equation. The p(x)-Laplace equation is denoted by
div(|Du|^{p(x)-2}Du)=0 in Omega,
where p(x):Omega to mr satisfies 1<p_-leq p(x)leq p_+<infty. This equation is a generalization of the p-Laplace equation div(|Du|^{p-2}Du)=0, where p is a constant in (1,infty).
One can expect that the regularity of solutions to p(x)-Laplace equation depends on the one of p(x). So, I will present the conditions on p(x) in order to obtain various regularities of solutions.
In addition, I will briefly introduce Calderon-Zygmund type estimates for elliptic equations in the setting of variable exponent Lebesgue space.

In this talk, we consider nonlinear elliptic equations involving the fractional Laplacian or the pseudo-relativistic Laplacian. We shall be concerned about existence and nonexistence results, and asymptotic profile of the solutions when a parameter of the equations is close to a critical value.