Abstract
AbstractThis paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem:−div∇v1+|∇v|2=f(x,v,∇v)inΩ,a0v+a1∂v∂ν=0on∂Ω,\left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{Ω},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{Ω},\end{array}\right.withΩ\text{Ω}an open ball inℝN{{\mathbb{R}}}^{N}, in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the functionfallow us to complement or improve several results in the literature.
Highlights
Introduction and main resultsThis paper deals with the existence of classical solutions of the mean curvature problem: − t N−1u′ 1 + u′2 ′t N−1f (t, u, u′) in ]0, R[, (1.1)u′(0) = 0, a0u(R) + a1u′(R) = 0, in the presence of couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition, that is, the sub-solution is smaller than the super-solution
U′(0) = 0, a0u(R) + a1u′(R) = 0, in the presence of couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition, that is, the sub-solution is smaller than the super-solution
Definition 1.1. (Notion of solution) By a classical solution of (1.1) we mean a function u ∈ C1([0, R]) ∩ C2(]0, R[) which satisfies the differential equation for all t ∈ ]0, R[ and the boundary condition
Summary
This paper deals with the existence of classical solutions of the mean curvature problem:. A complementary result for the same problem was obtained by a different approach – monotone iteration versus variational techniques – in [19, Theorem 3.1], assuming that f is gradient independent, continuous, but non-decreasing with respect to the state variable, and again the sub- and the super-solutions are regular and fulfill the Dirichlet boundary conditions. Since these results require that f satisfies some monotonicity property, it seems interesting to investigate the situation where such assumptions fail. It is clear that the complementary nature of the assumptions introduced in this paper allows us to deal with manifold situations, which can be treated by suitably combining our statements, but are not covered by other results available in the existing literature, such as those in [29,34,37,40,44,45,46,47,48,49]
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