• G A Afrouzi

Articles written in Proceedings – Mathematical Sciences

• On Critical Exponent for the Existence and Stability Properties of Positive Weak Solutions for some Nonlinear Elliptic Systems Involving the $(p, q)$-Laplacian and Indefinite Weight Function

This paper deals with the existence and stability properties of positive weak solutions to classes of nonlinear systems involving the $(p,q)$-Laplacian of the form

\begin{equation*}\begin{cases}-\Delta_p u=\lambda a(x)v^\alpha-c, &lt; x\in\Omega,\\ -\Delta_qv=\lambda b(x)u^\beta-c, &lt; x\in\Omega,\\ u=0=v, &lt; x\in\partial\Omega,\end{cases}\end{equation*}

where $\Delta_p$ denotes the 𝑝-Laplacian operator defined by $\Delta_pz=\mathrm{div}(|\nabla z|^{p-2}\nabla z),p&gt;1,\lambda$ and 𝑐 are positive parameters, 𝛺 is a bounded domain in $R^N(N\geq 1)$ with smooth boundary, $\alpha, \beta &gt; 0$ and the weights $a(x),b(x)$ satisfying $a(x)\in C(\Omega),b(x)\in C(\Omega)$ and $a(x)&gt;a_0&gt;0,b(x)&gt;b_0&gt;0$, for $x\in\Omega$. We first study the existence of positive weak solution by using the method of sub-super solution and then we study the stability properties of positive weak solution.

• Remark on an Infinite Semipositone Problem with Indefinite Weight and Falling Zeros

In this work, we consider the positive solutions to the singular problem

\begin{equation*}\begin{cases}-\Delta u=am(x)u-f(u)-\frac{c}{u^\alpha} &amp; \text{in}\quad\Omega,\\ u=0 &amp; \text{on}\quad\partial\Omega,\end{cases}\end{equation*}

where $0 &lt; \alpha &lt; 1,a&gt;0$ and $c&gt;0$ are constants, 𝛺 is a bounded domain with smooth boundary $\partial\Omega,\Delta$ is a Laplacian operator, and $f:[0,\infty]\longrightarrow\mathbb{R}$ is a continuous function. The weight functions $m(x)$ satisfies $m(x)\in C(\Omega)$ and $m(x)&gt;m_0&gt;0$ for $x\in\Omega$ and also $\|m\|_\infty=l &lt; \infty$. We assume that there exist $A&gt;0, M&gt;0,p&gt;1$ such that $alu-M\leq f(u)\leq Au^p$ for all $u\in[0,\infty)$. We prove the existence of a positive solution via the method of sub-supersolutions when $m_0 a&gt;\frac{2\lambda_1}{1+\alpha}$ and 𝑐 is small. Here $\lambda_1$ is the first eigenvalue of operator $-\Delta$ with Dirichlet boundary conditions.

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

We mainly consider the existence of a positive weak solution of the following system\begin{equation*}\left\{\begin{matrix}-\Delta_p u + |u|^{p-2} u = \gamma [g (x) a(u)+ c(x) f (v)], \quad \text{ in } \Omega,\\-\Delta_q v + |v|^{q-2} v = \mu [g (x) b(v)+ c(x) h (u)], \quad \text{ in } \Omega,\\\hspace{3cm} u = v = 0, \hspace{3.8cm} \text{ on } \partial \, \Omega,\end{matrix}\right.\end{equation*}where $\Delta_p u = \text{ div}(|\nabla_u|^{p-2} \nabla_u), p, q &gt; 1$ and $\lambda, \, \mu$ are positive parameters, and $\Omega \subset R^N$ is a bounded domain with smooth boundary $\partial \Omega$ and $g, \, c$ are nonnegative and continuous functions and $f, h, a, b$ are $C^1$ nondecreasing functions satisfying $a(0), b(0) \geq 0$. We have proved the existence of a positive weak solution for $\lambda$, $\mu$ large when$$\lim\limits_{x \to \infty} \frac{f[M (h(x))^{\frac{1}{q-1}}]}{x^{p-1}} = 0$$for every $M &gt; 0$.

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