On the multiplicity of nodal solutions of a prescribed mean curvature problem

Abstract

We prove existence and multiplicity results for sign‐changing solutions of a prescribed mean curvature equation with Dirichlet boundary conditions. Our arguments involve a perturbation of the degenerate part \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$1/\sqrt{1+s}$\end{document}, which allows us to use classical variational techniques and to localize small regular solutions.

Existence of solution for quasilinear equations involving local conditions

“…The following lemma is a variant of the well-known Moser iterative scheme, see for instance [2,18]. Lemma 2.4.…”

“…In this paper, we deal with the question of the existence of weak solutions of the problem −div(a(|∇u| 2 )∇u) = λf (x, u) in Ω, u = 0 on ∂Ω, (1.1) where Ω is a bounded domain of R N with smooth boundary ∂Ω, N > 2, a : R → [0, ∞) is a continuous function, λ is a positive parameter, and f : Ω × R → R is a Carathéodory function which grows as |u| p−2 u near zero for 2 < p < 2 * . Since quasilinear equations serve as model of a wide class of differential operators, there has been a considerable amount of works on this subject (see [2][3][4]16] and references therein). For our purpose we consider two principal operators: a(t) ≡ 1 which reduces (1.1) to the Laplacian case, that is the scalar equation −Δu = λf (x, u); and a(t) = (1 + t) −1/2 where we obtain the mean curvature operator.…”

Existence of nodal solutions for quasilinear elliptic problems in ℝN

“…This approach originated in a series of works on the Dirichlet problem for semilinear equations, a review of which can be found in [4]. It has also been used more recently in the context of a prescribed mean-curvature problem in Bonheure et al [6], from which some of our arguments are inspired. Note that the Nehari approach stronlgy relies on the monotonicity assumption (g3), which is not needed in [3].…”

Existence and multiplicity of solutions for a prescribed mean-curvature problem with critical growth

“…In the work [6], the authors studied a related subcritical problem in which they obtained positive solutions. In the recent work [3], Bonheure, Derlet and Valeriola have studied a purely subcritical version of (P λ ), where they proved the existence and multiplicity of nodal H 1 0 (Ω) solutions, to sufficiently large values of λ. They overcame the difficulty in working in the BV (Ω) space, which is the natural functional space to treat (P λ ), by doing a truncation in the degenerate part of the mean-curvature operator in order to make possible construct a variational framework in the Sobolev space H 1 0 (Ω).…”

“…where • is the Sobolev norm in H 1 0 (Ω). Our approach follows the main ideas of Bonheure et al in [3], in order to make possible consider a related modified problem in H 1 0 (Ω). Afterwards, to get solutions of the modified problem we apply Krasnoselskii genus theory in the same way that Azorero and Alonso in [1].…”