In this paper we prove that every weak and strong local minimizer $u\in W^{1,2}(\Omega ,{ℝ}^3)$ of the functional $I\left(u\right)={\int }_{\Omega }{\left|Du\right|}^2+f\left(\mathrm{Adj}Du\right)+g\left(\det Du\right),$ where $u:\Omega \subset {ℝ}^3\to {ℝ}^3$, f grows like ${\left|\mathrm{Adj}Du\right|}^p$, g grows like ${\left|\det Du\right|}^q$ and 1<q<p<2, is $C^{1,\alpha }$ on an open subset ${\Omega }_0$ of Ω such that $\mathrm{𝑚𝑒𝑎𝑠}(\Omega \setminus {\Omega }_0)=0$. Such functionals naturally arise from nonlinear elasticity problems. The key point in order to obtain the partial regularity result is to establish an energy estimate of Caccioppoli type, which is based on an appropriate choice of the test functions. The limit case $p=q\le 2$ is also treated for weak local minimizers. ...

We obtain in this paper a multiplicity result for strongly indefinite semilinear elliptic systems in bounded domains as well as in ${ℝ}^N$.

Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $-{\Delta }_{p}u+{\left|u\right|}^{p-2}u=f_{1\lambda ₁}{\left(x\right)\left|u\right|}^{q-2}u+2\alpha /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha -2}u{\left|v\right|}^{\beta }$, x ∈ Ω, ⎨ $-{\Delta }_{p}v+{\left|v\right|}^{p-2}v=f_{2\lambda ₂}{\left(x\right)\left|v\right|}^{q-2}v+2\beta /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v$, x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $\Omega \subset {ℝ}^N$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and $f_{i{\lambda }_i}\left(x\right)={\lambda }_i{f}_{i+}\left(x\right)+f_{i-}\left(x\right)$ (i = 1,2) are sign-changing functions, where $f_{i\pm }\left(x\right)=max\pm f_i\left(x\right),0$, $g_{\mu }\left(x\right)=a\left(x\right)+\mu b\left(x\right)$, and ${\Delta }_{p}u={div\left(\right|\nabla u|}^{p-2}\nabla u)$ denotes the p-Laplace operator. We use variational methods.