JournalsrlmVol. 22, No. 1pp. 29–50

An obstacle problem with gradient term and asymptotically linear reaction

  • Boumediene Abdellaoui

    Université Aboubekr Belkaïd, Tlemcen, Algeria
  • Sidi Mohamed Bouguima

    Université Aboubekr Belkaïd, Tlemcen, Algeria
  • Ireneo Peral

    Universidad Autónoma de Madrid, Spain
An obstacle problem with gradient term and asymptotically linear reaction cover
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Abstract

We will consider the following obstacle problem

ΩuTk(vu)dx+Ωh(u)uqTk(vu)dxΩ(g(x,u)+f)Tk(vu)dx,\int_{\Omega}\nabla u\nabla T_{k}(v-u)dx +\int_{\Omega }h(u)\left\vert \nabla u\right\vert ^{q}T_{k}(v-u)dx\geq \int_{\Omega }\left(g(x,u)+f\right) T_{k}(v-u)dx,

with the condition that uψu\geq\psi a.e in Ω.\Omega . Under suitable condition relating gg,hh and qq, we show the existence of a solution for all fL1(Ω)f \in L^1(\Omega) .

The main feature is, assuming that g(x,s)g(x,s) is asymptotically linear as s±|s|\to \pm\infty and independently of the values of

lims±g(x,s)s,\lim\limits_{s\to \pm\infty }\dfrac{g(x,s)}{s},

to obtain a solution for all λ>0\lambda>0 and fL1(Ω)f\in L^1(\Omega). In this sense we could say that the first order term break down any resonant effect.

Cite this article

Boumediene Abdellaoui, Sidi Mohamed Bouguima, Ireneo Peral, An obstacle problem with gradient term and asymptotically linear reaction. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 1, pp. 29–50

DOI 10.4171/RLM/586