Existence of positive weak solutions for (𝑝, 𝑞)-Laplacian nonlinear systems

Proceedings Mathematical Sciences

Samira Ala³  G A Afrouzi²  A Niknam¹ 

[1] Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran$$;Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran$$;Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU), Tehran, Iran$$

关键词: 𝑝-Laplacian systems;    sub-supersolution;    positive weak solutions.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

We mainly consider the existence of a positive weak solution of the following systemegin{equation*}left{egin{matrix}-𝛥_p u + |u|^{p-2} u = 𝛾 [g (x) a(u)+ c(x) f (v)], quad ext{ in } 𝛺,-𝛥_q v + |v|^{q-2} v = 𝜇 [g (x) b(v)+ c(x) h (u)], quad ext{ in } 𝛺,hspace{3cm} u = v = 0, hspace{3.8cm} ext{ on } 𝜕 , 𝛺,end{matrix}ight.end{equation*}where $𝛥_p u = ext{ div}(|abla_u|^{p-2} abla_u), p, q > 1$ and $𝜆, , 𝜇$ are positive parameters, and $𝛺 subset R^N$ is a bounded domain with smooth boundary $𝜕 𝛺$ and $g, , c$ are nonnegative and continuous functions and $f, h, a, b$ are $C^1$ nondecreasing functions satisfying $a(0), b(0) ≥ 0$. We have proved the existence of a positive weak solution for 𝜆, 𝜇 large when$$limlimits_{x → ∞} frac{f[M (h(x))^{frac{1}{q-1}}]}{x^{p-1}} = 0$$for every 𝑀 > 0.

【 授权许可】

Unknown