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In this paper, we prove new existence and multiplicity results for critical points of lower semicontinuous functionals in Banach spaces, complementing the nonsmooth critical point theory set forth by Szulkin. In particular, we get general deformation lemmas avoiding the Palais-Smale condition, and we adapt a monotonicity trick suited to our problems. We apply our results to study entire solutions with finite energy to Born-Infeld type autonomous equations with continuous nonlinearity $f$. Under nearly optimal conditions on $f$, we construct solutions including for the first time, nonradial ones. A companion nonexistence theorem shows the sharpness of our assumptions., Comment: 58 pages. Comments are welcome

Given a measure $\rho$ on a domain $\Omega \subset \mathbb{R}^m$, we study spacelike graphs over $\Omega$ in Minkowski space with Lorentzian mean curvature $\rho$ and Dirichlet boundary condition on $\partial \Omega$. The graph function $u_\rho : \Omega \rightarrow \mathbb{R}$ also represents the electric potential generated by a charge $\rho$ in electrostatic Born-Infeld theory. While $u_\rho$ minimizes the action $$ I_\rho(\psi) = \int_{\Omega} \Big( 1 - \sqrt{1-|D\psi|^2} \Big) \mathrm{d} x - \langle \rho, \psi \rangle $$ among competitors with $|D\psi| \le 1$, because of a lack of smoothness of the Lagrangian density when $|D\psi| = 1$ a direct approach via minimization may not produce a solution to the Euler-Lagrange equation (BI). In this paper, we study existence and regularity of $u_\rho$ for general $\rho$, in a bounded domain and in the entire $\mathbb{R}^m$. In particular, we find sufficient conditions to guarantee that $u_\rho$ solves (BI) and enjoys log-improved $W^{2,2}_{\mathrm{loc}}$ estimates, and we construct examples helping to identify sharp thresholds for the regularity of $\rho$ to ensure the validity of (BI). One of the main difficulties is the possible presence of light segments in the graph of $u_\rho$, which will be discussed in detail., Comment: 72 pages. Final version, to appear in Annals of PDE. More explanations added (especially in the Introduction and in Subsection 4.4). Package axessibility included to make the paper available to visually impaired people

In this paper, we study Hardy's inequality in a limiting case: $$ \int_{\Omega} |\nabla u |^N dx \ge C_N(\Omega) \int_{\Omega} \frac{|u(x)|^N}{|x|^N \left(\log \frac{R}{|x|} \right)^N} dx $$ for functions $u \in W^{1,N}_0(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with $R = \sup_{x \in \Omega} |x|$. We study the (non-)attainability of the best constant $C_N(\Omega)$ in several cases. We provide sufficient conditions that assure $C_N(\Omega) > C_N(B_R)$ and $C_N(\Omega)$ is attained, here $B_R$ is the $N$-dimensional ball with center the origin and radius $R$. Also we provide an example of $\Omega \subset \mathbb{R}^2$ such that $C_2(\Omega) > C_2(B_R) = 1/4$ and $C_2(\Omega)$ is not attained.