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.

For $0<p-1<q$ and either $ϵ=1$ or $ϵ=-1$, we prove the existence of solutions of $-{\Delta }_{p}u=ϵu^q$ in a cone $C_S$, with vertex 0 and opening $S$, vanishing on $\partial C_S$, of the form $u\left(x\right)={\left|x\right|}^{-\beta }\omega (x/\left|x\right|)$. The problem reduces to a quasilinear elliptic equation on $S$ and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.

We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D\subset {ℝ}^d$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $𝒫$. We associate to Problem $𝒫$ an optimal control problem, denoted by $𝒬$. Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\tilde{K}$, we provide results concerning the well-posedness of problems...