Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball

Abstract

We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation 8 > < > :where B R is a ball in R N (N ≥ 2). According to the behaviour of f = f (r, s) near s = 0, we prove the existence of either one, two or three positive solutions. All results are obtained by reduction to an equivalent non-singular one-dimensional problem, to which variational methods can be applied in a standard way.

Pairs of nodal solutions for a Minkowski-curvature boundary value problem in a ball

“…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].…”

“…where q : [0, R] → R is a continuous function, p > 1 and λ is a positive parameter. As shown in [6,11], the role of λ is crucial for the existence of positive solutions of (1.2). In particular, it was proved therein that, when q + ≡ 0 and λ > 0 is large enough, two positive (and two negative) solutions appear, while in general nonexistence holds for λ → 0 + (see also [10] for a previous achievement in a one-dimensional setting).…”

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

“…By bifurcation and topological methods, the author determined the interval of parameter λ in which the above problem has zero/one/two nontrivial nonnegative solutions according to sublinear/linear/superlinear nonlinearity at zero. We refer the reader to [7][8][9][10][11][12][13][14][15] for the N -dimensional mean curvature equation in Minkowski space. In particular, for one-dimensional mean curvature equation with Dirichlet/Neumann/periodic/mixed type boundary conditions in Minkowski space, we refer the reader to [16][17][18][19][20][21][22][23][24] and the references therein.…”

Multiple periodic solutions for perturbed relativistic pendulum systems

“…The main idea we use to treat the singular problem (1.1) is to reduce it to an equivalent non-singular one to which classical variational methods can be applied. Similarly, although technically different, ideas have been recently used by Coelho et al [2], [3] in the study of positive solutions of some problems involving curvature operator in Minkowski space.…”