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POSITIVE SOLUTIONS FOR A CLASS OF p(x)-LAPLACIAN PROBLEMS

Published online by Cambridge University Press:  01 September 2009

G. A. AFROUZI and
H. GHORBANI
G. A. AFROUZI
Affiliation:
Department of Mathematics, Faculty of Basic Sciences, Mazandaran University, Babolsar, Iran e-mail: afrouzi@umz.ac.ir
H. GHORBANI
Affiliation:
Department of Mathematics, Faculty of Basic Sciences, Mazandaran University, Babolsar, Iran e-mail: afrouzi@umz.ac.ir
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Abstract

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We consider the system$ \begin{eqnarray*} \left\{\begin{array}{ll} -\Delta_{p(x)} u=\lambda_1 f(v) + \mu_1 h(u) &~{\rm in}\ \Omega\\ -\Delta_{q(x)} v=\lambda_2 g(u) + \mu_2 \gamma(v) & ~{\rm in}\ \Omega,\\ u=v= 0 & ~{\rm on}\ \partial \Omega \end{array}\right. \end{eqnarray*} $where p(x), q(x) ∈ C1(RN) are radial symmetric functions such that sup|∇ p(x)| < ∞, sup|∇ q(x)| < ∞ and 1 < inf p(x) ≤ sup p(x) < ∞, 1 < inf q(x) ≤ sup q(x) < ∞, where −Δp(x)u = −div(|∇u|p(x)−2∇u), −Δq(x)v = −div(|∇v|q(x)−2∇v), respectively are called p(x)-Laplacian and q(x)-Laplacian, λ1, λ2, μ1 and μ2 are positive parameters and Ω = B(0, R) ⊂ RN is a bounded radial symmetric domain, where R is sufficiently large. We prove the existence of a positive solution when$ \lim_{u \rightarrow +\infty}\frac{f(M(g(u))^\frac{1}{q^--1})}{u^{p^--1}} =0, $for every M > 0, $\lim_{u \rightarrow +\infty} \frac{h(u)}{u^{p^--1}} = 0$ and $\lim_{u \rightarrow +\infty} \frac{\gamma(u)}{u^{q^--1}} = 0$. In particular, we do not assume any sign conditions on f(0), g(0), h(0) or γ(0).

Keywords

35J6035B3035B40

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 3 , September 2009, pp. 571 - 578
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

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