Under appropriate assumptions on $f$ and $h$, we prove the existence of as least two weak solutions for the problem by using the Ekeland variational principle and the mountain pass theorem in critical point theory.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we consider a class of anisotropic elliptic equations of Kirchhoff type $$\begin{cases}- M\left(\sum\limits_{i=1}^N\int_\Omega\frac{1}{p_i(x)}|\partial_{x_i}u|^{p_i(x)}\,dx\right)\sum\limits_{i=1}^N\partial_{x_i} \Big(|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u\Big) = f(x,u) + h(x), \quad x\in \Omega,\\u = 0, \quad x\in \partial\Omega,\end{cases}$$where $\Omega \subset \R^N$ ($N \geq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $M(t) = a+bt^\tau$, $\tau>0$ is a positive constant, and $p_i$, $i = 1, 2, ..., N$ are continuous functions on $\overline\Omega$ such that $2 \leq p_i(x)<N$, $a>0$, $b\geq 0$.
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Key Findings & Data:In this paper, we consider a class of anisotropic elliptic equations of Kirchhoff type $$\begin{cases}- M\left(\sum\limits_{i=1}^N\int_\Omega\frac{1}{p_i(x)}|\partial_{x_i}u|^{p_i(x)}\,dx\right)\sum\limits_{i=1}^N\partial_{x_i} \Big(|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u\Big) = f(x,u) + h(x), \quad x\in \Omega,\\u = 0, \quad x\in \partial\Omega,\end{cases}$$where $\Omega \subset \R^N$ ($N \geq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $M(t) = a+bt^\tau$, $\tau>0$ is a positive constant, and $p_i$, $i = 1, 2, ..., N$ are continuous functions on $\overline\Omega$ such that $2 \leq p_i(x)<N$, $a>0$, $b\geq 0$.
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Conclusion & Impact:Under appropriate assumptions on $f$ and $h$, we prove the existence of as least two weak solutions for the problem by using the Ekeland variational principle and the mountain pass theorem in critical point theory.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Anisotropic elliptic equations · Kirchhoff type equations · Variable exponents
Abstract
In this paper, we consider a class of anisotropic elliptic equations of Kirchhoff type $$\begin{cases}- M\left(\sum\limits_{i=1}^N\int_\Omega\frac{1}{p_i(x)}|\partial_{x_i}u|^{p_i(x)}\,dx\right)\sum\limits_{i=1}^N\partial_{x_i} \Big(|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u\Big) = f(x,u) + h(x), \quad x\in \Omega,\\u = 0, \quad x\in \partial\Omega,\end{cases}$$where $\Omega \subset \R^N$ ($N \geq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $M(t) = a+bt^\tau$, $\tau>0$ is a positive constant, and $p_i$, $i = 1, 2, ..., N$ are continuous functions on $\overline\Omega$ such that $2 \leq p_i(x)<N$, $a>0$, $b\geq 0$. Under appropriate assumptions on $f$ and $h$, we prove the existence of as least two weak solutions for the problem by using the Ekeland variational principle and the mountain pass theorem in critical point theory.
CHUNG, N. T. (2018). Multiple solutions for an anisotropic elliptic equation of Kirchhoff type in bounded domain. Results in Nonlinear Analysis, Cilt: 1 - Sayı: 3, 116–127. https://ajindex.com/articles/797785
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