Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space

Abstract

In this paper, by using Leray-Schauder degree arguments and critical point theory for convex, lower semicontinuous perturbations of C 1 -functionals, we obtain existence of classical positive radial solutions for Dirichlet problems of type div ∇vHere, B(R) = {x ∈ R N : |x| < R} and f : [0, R] × [0, α) → R is a continuous function, which is positive on (0, R] × (0, α).

Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space

“…Our aim here is indeed to extend to a genuine PDE setting what has been obtained in [6], for the one-dimensional problem, and in [3,4,7], for the radially symmetric problem in a ball. Namely, we will discuss the existence and the multiplicity of positive solutions of (1), assuming that the function f = f (x, s) is sublinear, or superlinear, or sub-superlinear near s = 0.…”

Existence and multiplicity of positive solutions of a one-dimensional mean curvature equation in Minkowski space

“…By Theorem 2.1, the problem (3.1) has at least one positive solution. We note that Theorem 1 of [2] cannot guarantee this conclusion since f ( 1 2 , s) = 0, ∀s ∈ [0, 1).…”

“…Moreover, it is easy to show by a standard argument that T is compact on K ∩ B ρ for all ρ ∈ (0, 1) (see [2]). In addition, it can easily be verified that u is a positive solution of problem (1.1) if u ∈ K ∩ B 1 is a fixed point of the nonlinear operator T. Now, we state and prove the existence and multiplicity of positive solutions of problem (1.1) and (1.2) by using a fixed point theorem of cone expansion and compression of norm type.…”

Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions

“…The nonlinear differential operator appearing in (1.1) is usually meant as a mean curvature operator in the Lorentz-Minkowski space and it is of interest in Differential Geometry and General Relativity [3,26,27]; it also appears in the nonlinear theory of Electromagnetism, where it is usually referred to as Born-Infeld operator [9,10,13]. In recent years, it has become very popular among specialists in Nonlinear Analysis, and various existence/multiplicity results for the associated boundary value problems are available, both in the ODE and in the PDE case, possibly in a non-radial setting (see, among others, [1,2,4,5,6,7,17,18,19,20,23,24,30] and the references therein).…”