JournalsrlmVol. 22, No. 1pp. 89–111

Variational methods for nonlinear perturbations of singular <i>ϕ</i>-Laplacians

  • Cristian Bereanu

    Romanian Academy, Bucharest, Romania
  • Petru Jebelean

    West University of Timisoara, Romania
  • Jean Mawhin

    Université Catholique de Louvain, Belgium
Variational methods for nonlinear perturbations of singular <i>ϕ</i>-Laplacians cover
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Abstract

Motivated by the existence of radial solutions to the Neumann problem involving the mean extrinsic curvature operator in Minkowski space \begin{eqnarray*} \mbox{div}\left(\frac{\nabla v}{\sqrt{1-|\nabla v|^2}}\right)=g(|x|,v) \quad \mbox{in} \quad \mathcal A, \quad \frac{\partial v}{\partial \nu}=0 \quad \mbox{on} \quad \partial\mathcal A, \end{eqnarray*} where 0R1<R20 \leq R_1 < R_2, A={xRN:R1xR2}\mathcal A=\{ x\in \mathbb{R}^N: R_1\leq |x|\leq R_2\} and g:[R1,R2]×RRg:[R_1,R_2]\times \mathbb{R}\to\mathbb{R} is continuous, we study the more general problem \begin{eqnarray*} [r^{N-1} \phi(u')]' =r^{N-1}g(r,u), \quad u'(R_1)=0=u'(R_2), \end{eqnarray*} where ϕ:=Φ:(a,a)R\phi:=\Phi':(-a,a) \to \mathbb{R} is an increasing homeomorphism with ϕ(0)=0\phi(0)=0 and the continuous function Φ:[a,a]R\Phi:[-a,a] \to \mathbb{R} is of class C1C^1 on (a,a)(-a,a). The associated functional in the space of continuous functions over [R1,R2][R_1,R_2] is the sum of a convex lower semicontinuous functional and of a functional of class C1C^1. Using the critical point theory of Szulkin, we obtain various existence and multiplicity results for several classes of nonlinearities. We also discuss the case of the periodic problem.

Cite this article

Cristian Bereanu, Petru Jebelean, Jean Mawhin, Variational methods for nonlinear perturbations of singular <i>ϕ</i>-Laplacians. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 1, pp. 89–111

DOI 10.4171/RLM/589