2026/07/17 by Alberto Boscaggin, Francesca Colasuonno, Ricardo Ziegele
#math.AP
We consider the Dirichlet problem for the mean curvature operator in Minkowski space, -div((∇ u)/(√(1-|∇ u|2))) = λu + μh(x,u) in Ω, u = 0 on ∂Ω, in a bounded domain Ω⊂ ℝN, where λ, μ are real parameters, and the nonlinearity h is superlinear at u = 0. In particular, we study the combined effect of the parameters λ, μ on the multiplicity of solutions. In the general setting, following Szulkin's approach for nonsmooth functionals, we prove the existence, for λ not belonging to the spectrum of the Dirichlet Laplacian and μ sufficiently large, of a global minimizing solution (with negative action level) and of a min-max solution (with positive action level). Moreover, we characterize the limiting profiles of these solutions as μ→ +∞. More precisely, when the global minimizer is positive, its limit profile is dist(⋅,∂Ω), thus saturating, in the limit, the geometric constraint |∇ u|≤1, while min-max solutions collapse uniformly to zero as μ→+∞. A nonexistence criterion is also given for suitable values of λ and μ. Finally, when the domain Ω is a ball, using a shooting approach, we establish the existence of arbitrarily many nodal radial solutions for every λ≥ 0 and for μ sufficiently large.