We study the Dirichlet boundary value problem for the $p$-Laplacian of the form $$-{\Delta }_{p}u-{\lambda }_1{\left|u\right|}^{p-2}u=f\text{in}\Omega ,u=0\text{on}\partial \Omega ,$$ where $\Omega \subset {ℝ}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $N\ge 1$, $p>1$, $f\in C\left(\overline{\Omega }\right)$ and ${\lambda }_1>0$ is the first eigenvalue of ${\Delta }_p$. We study the geometry of the energy functional $$E_p\left(u\right)=\frac{1}{p}{\int }_{\Omega }{\left|\nabla u\right|}^p-\frac{{\lambda }_1}{p}{\int }_{\Omega }{\left|u\right|}^p-{\int }_{\Omega }fu$$ and show the difference between the case $1<p<2$ and the case $p>2$. We also give the characterization of the right hand sides $f$ for which the above Dirichlet problem is solvable and has multiple solutions.

We discuss how the choice of the functional setting and the definition of the weak solution affect the existence and uniqueness of the solution to the equation $$-{\Delta }_{p}u=f\text{in}\Omega ,$$ where $\Omega$ is a very general domain in ${ℝ}^N$, including the case $\Omega ={ℝ}^N$.

We discuss the interior Hölder everywhere regularity for minimizers of quasilinear functionals of the type $$𝒜(u;\Omega )={\int }_{\Omega }{A}_{ij}^{\alpha \beta }(x,u)D_{\alpha }{u}^i{D}_{\beta }{u}^j\mathrm{d}x$$ whose gradients belong to the Morrey space $L^{2,n-2}(\Omega ,{ℝ}^{nN})$.

Interior ${ℒ}_{loc}^{2,n}$-regularity for the gradient of a weak solution to nonlinear second order elliptic systems is investigated.

The aim of this paper is to show that the Liouville-type property is a sufficient and necessary condition for the regularity of weak solutions of quasilinear elliptic systems of higher orders.

The aim of this paper is to show that Liouville type property is a sufficient and necessary condition for the regularity of weak solutions of nonlinear elliptic systems of the higher order.

We study properties of Lipschitz truncations of Sobolev functions with constant and variable exponent. As non-trivial applications we use the Lipschitz truncations to provide a simplified proof of an existence result for incompressible power-law like fluids presented in [Frehse et al., SIAM J. Math. Anal34 (2003) 1064–1083]. We also establish new existence results to a class of incompressible electro-rheological fluids.