2013
DOI: 10.1016/j.jfa.2013.04.006
|View full text |Cite
|
Sign up to set email alerts
|

Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
4
1

Citation Types

2
63
0

Year Published

2013
2013
2022
2022

Publication Types

Select...
8

Relationship

1
7

Authors

Journals

citations
Cited by 89 publications
(65 citation statements)
references
References 18 publications
2
63
0
Order By: Relevance
“…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.…”
Section: Introductionmentioning
confidence: 86%
“…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.…”
Section: Introductionmentioning
confidence: 86%
“…It is worth noticing that the above argument also allows us to easily recover the existence of four one-signed radial solutions (two positive ones and two negative ones) already proved in [6,11] with topological and variational techniques, respectively, see Remark 4.1. Incidentally, we mention that a shooting approach has been recently exploited to investigate the existence of radial ground-state solutions, as well [1,2].…”
Section: Introductionmentioning
confidence: 54%
“…To give a better insight into the statement of Theorem 1.1, we collect here below some numerical simulations obtained with MAPLE c software (see Figures 1, 2 and 3 below) for the equation div ∇u 15) choosing N = 2 and R = 10. The Dirichlet solutions shown are found for λ = 5; for completeness, we also depict the one-signed solutions already found in [6,11] (see Remark 4.1). The reader will certainly notice how the large solutions u l,j found above appear almost sharp-cornered, with slope approximately equal to ±1; we remark that, in principle, this is not a drawback of numerics but rather a consequence of the peculiar properties of the Minkowski curvature operator, as an elementary singular perturbation analysis shows.…”
Section: Some Numerical Simulationsmentioning
confidence: 86%
See 1 more Smart Citation
“…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).…”
Section: Introductionmentioning
confidence: 99%