The paper surveys recent results obtained for the existence and multiplicity of radial solutions of Dirichlet problems of the type $$\nabla ·\left(\frac{\nabla v}{\sqrt{1-{\left|\nabla v\right|}^2}}\right)=f\left(\right|x|,v)\text{in}{B}_R,u=0\text{on}\partial B_R,$$ where $B_R$ is the open ball of center $0$ and radius $R$ in ${ℝ}^n$, and $f$ is continuous. Comparison is made with similar results for the Laplacian. Topological and variational methods are used and the case of positive solutions is emphasized. The paper ends with the case of a general domain.

In this paper we prove existence results for some nonlinear degenerate elliptic equations with data in the space of bounded Radon measures and we improve the results already obtained in Cirmi G.R., On the existence of solutions to non-linear degenerate elliptic equations with measure data, Ricerche Mat. 42 (1993), no. 2, 315–329.

The paper is devoted to the solvability of a nonlinear elliptic problem in a plane multiply connected domain. On the inner components of its boundary Dirichlet conditions are known up to additive constants which have to be determined together with the sought solution so that the so-called trailing stagnation conditions are satisfied. The results have applications in the stream function solution of subsonic flows past groups of profiles or cascades of profiles.

We prove the existence of solutions of the unilateral problem for equations of the type Au - divϕ(u) = μ in Orlicz spaces, where A is a Leray-Lions operator defined on $\left(A\right)\subset W₀¹L_M\left(\Omega \right)$, $\mu \in L¹\left(\Omega \right)+W^{-1}{E}_{M̅}\left(\Omega \right)$ and $\varphi \in C⁰(ℝ,{ℝ}^N)$.

We investigate the behavior of weak solutions to the nonlocal Robin problem for linear elliptic divergence second order equations in a neighborhood of a boundary corner point. We find an exponent of the solution's decreasing rate under minimal assumptions on the problem coefficients.

In this article, we study the existence of nontrivial weak solutions for the following boundary value problem: $$-{\Delta }_{\gamma }u=f(x,u)\text{in}\Omega ,u=0\text{on}\partial \Omega ,$$ where $\Omega$ is a bounded domain with smooth boundary in ${ℝ}^N$, $\Omega \cap \{x_j=0\}\ne \varnothing$ for some $j$, ${\Delta }_{\gamma }$ is a subelliptic linear operator of the type $${\Delta }_{\gamma }:=\sum _{j=1}^N{\partial }_{x_j}\left({\gamma }_j^2{\partial }_{x_j}\right),{\partial }_{x_j}:=\frac{\partial }{\partial x_j},N\ge 2,$$ where $\gamma \left(x\right)=({\gamma }_1\left(x\right),{\gamma }_2\left(x\right),\cdots ,{\gamma }_N\left(x\right))$ satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity $f(x,\xi )$ is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.

It is well-known that the “standard” oblique derivative problem, $\Delta u=0$ in $\Omega$, $\partial u/\partial \nu -u=0$ on $\partial \Omega$ ($\nu$ is the unit inner normal) has a unique solution even when the boundary condition is not assumed to hold on the entire boundary. When the boundary condition is modified to satisfy an obliqueness condition, the behavior at a single boundary point can change the uniqueness result. We give two simple examples to demonstrate what can happen.