The Nehari manifold approach for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-Laplacian problem with Neumann boundary condition

Taghavi, Ali; Afrouzi, G. A.; Ghorbani, Horieh

doi:10.14232/ejqtde.2013.1.39

Electron. J. Qual. Theory Differ. Equ.

2013

DOI: 10.14232/ejqtde.2013.1.39

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The Nehari manifold approach forp(x)-Laplacian problem with Neumann boundary condition

Ali Taghavi¹,

G. A. Afrouzi²,

Horieh Ghorbani³

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Cited by 2 publications

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“…In variable exponent cases this method has some difficulties in comparison with the Nehari manifolds approach in p-Laplacian problems. This is due to the non-homogeneity of the variable exponent p. Nevertheless, in recent years, several authors have used the Nehari manifold and fibering maps to solve quasilinear problems with variable exponent (see [18,22,30]).…”

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The fibering map approach for a singular elliptic system involving the $p(x)$-Laplacian and nonlinear boundary conditions

Kratou¹,

Saoudi²

2021

Revista De La Unión Matemática Argentina

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The fibering map approach for a singular elliptic system involving the $p(x)$-Laplacian and nonlinear boundary conditions

Kratou¹,

Saoudi²

2021

Revista De La Unión Matemática Argentina

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Existence of a solution for a nonlocal elliptic system of (p(x),q(x))-Kirchhoff type

Taghavi

Ghorbani

2018

Advances in Pure and Applied Mathematics

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Abstract In this paper, we consider the system \left\{\begin{aligned} &\displaystyle{-}M_{1}\bigg{(}\int_{\Omega}\frac{\lvert% \nabla u\rvert^{p(x)}+\lvert u\rvert^{p(x)}}{p(x)}\,dx\biggr{)}(\Delta_{p(x)}u% -\lvert u\rvert^{p(x)-2}u)=\lambda a(x)\lvert u\rvert^{r_{1}(x)-2}u-\mu b(x)% \lvert u\rvert^{\alpha(x)-2}u&&\displaystyle\text{in }\Omega,\\ &\displaystyle{-}M_{2}\bigg{(}\int_{\Omega}\frac{\lvert\nabla v\rvert^{q(x)}+% \lvert v\rvert^{q(x)}}{q(x)}\,dx\biggr{)}(\Delta_{q(x)}v-\lvert v\rvert^{q(x)-% 2}v)=\lambda c(x)\lvert v\rvert^{r_{2}(x)-2}v-\mu d(x)\lvert v\rvert^{\beta(x)% -2}v&&\displaystyle\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where Ω is a bounded domain in {\mathbb{R}^{N}} ( {N\geq 2} ) with a smooth boundary {\partial\Omega} , {M_{1}(t),M_{2}(t)} are continuous functions and {\lambda,\mu>0} . We prove that for any {\mu>0} there exists {\lambda_{*}} sufficiently small such that for any {\lambda\in(0,\lambda_{*})} the above system has a nontrivial weak solution. The proof relies on some variational arguments based on Ekeland’s variational principle, and some adequate variational methods.

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The Nehari manifold approach forp(x)-Laplacian problem with Neumann boundary condition

Cited by 2 publications

References 16 publications

The fibering map approach for a singular elliptic system involving the $p(x)$-Laplacian and nonlinear boundary conditions

The fibering map approach for a singular elliptic system involving the $p(x)$-Laplacian and nonlinear boundary conditions

Existence of a solution for a nonlocal elliptic system of (p(x),q(x))-Kirchhoff type

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