On a bi-nonlocal fourth order elliptic problem
Abstract: This paper is aiming at obtaining weak solution for a bi-nonlocal fourth order elliptic problem with Navier boundary condition. Our approach is based on variational methods and critical point theory.
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Cited by 3 publications
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p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators
We investigate the existence of non-trivial weak solutions for the following p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators { M ( σ ) ( Δ p ( x ) 2 u - Δ p ( x ) u ) = λ ϑ ( x ) | u | q ( x ) - 2 u ( ∫ Ω ϑ ( x ) q ( x ) | u | q ( x ) d x ) r in Ω , u ∈ W 2 , p ( . ) ( Ω ) ∩ W 0 1 , p ( . ) ( Ω ) , \left\{ {\matrix{ {M\left( \sigma \right)\left( {\Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u} \right) = \lambda \vartheta \left( x \right){{\left| u \right|}^{q\left( x \right) - 2}}u{{\left( {\int_\Omega {{{\vartheta \left( x \right)} \over {q\left( x \right)}}{{\left| u \right|}^{q\left( x \right)}}dx} } \right)}^r}\,{\rm{in}}\,\Omega ,} \hfill \cr {u \in {W^{2,p\left( . \right)}}\left( \Omega \right) \cap W_0^{1,p\left( . \right)}\left( \Omega \right),} \hfill \cr } } \right. under some suitable conditions on the continuous functions p, q, the non-negative function ϑ and M(σ), where σ : = ∫ Ω | Δ u | p ( x ) p ( x ) + | ∇ u | p ( x ) p ( x ) d x . \sigma : = \int_\Omega {{{{{\left| {\Delta u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}} + {{{{\left| {\nabla u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}}dx.} Our main results is obtained by employing variational techniques and the well-known symmetric mountain pass lemma.
p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators
We investigate the existence of non-trivial weak solutions for the following p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators { M ( σ ) ( Δ p ( x ) 2 u - Δ p ( x ) u ) = λ ϑ ( x ) | u | q ( x ) - 2 u ( ∫ Ω ϑ ( x ) q ( x ) | u | q ( x ) d x ) r in Ω , u ∈ W 2 , p ( . ) ( Ω ) ∩ W 0 1 , p ( . ) ( Ω ) , \left\{ {\matrix{ {M\left( \sigma \right)\left( {\Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u} \right) = \lambda \vartheta \left( x \right){{\left| u \right|}^{q\left( x \right) - 2}}u{{\left( {\int_\Omega {{{\vartheta \left( x \right)} \over {q\left( x \right)}}{{\left| u \right|}^{q\left( x \right)}}dx} } \right)}^r}\,{\rm{in}}\,\Omega ,} \hfill \cr {u \in {W^{2,p\left( . \right)}}\left( \Omega \right) \cap W_0^{1,p\left( . \right)}\left( \Omega \right),} \hfill \cr } } \right. under some suitable conditions on the continuous functions p, q, the non-negative function ϑ and M(σ), where σ : = ∫ Ω | Δ u | p ( x ) p ( x ) + | ∇ u | p ( x ) p ( x ) d x . \sigma : = \int_\Omega {{{{{\left| {\Delta u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}} + {{{{\left| {\nabla u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}}dx.} Our main results is obtained by employing variational techniques and the well-known symmetric mountain pass lemma.
Existence and multiplicity of solutions for a class of nonlocal elliptic transmission systems
By using the approach based on variationnel methods and critical point theory, more precisely, the symmetric mountain pass theorem, we study the existence and multiplicity of weak solutions for a class of elliptic transmision system with nonlocal term.
A new class of multiple nonlocal problems with two parameters and variable-order fractional $ p(\cdot) $-Laplacian
<abstract><p>In the present manuscript, we focus on a novel tri-nonlocal Kirchhoff problem, which involves the $ p(x) $-fractional Laplacian equations of variable order. The problem is stated as follows:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{ll} M\Big(\sigma_{p(x, y)}(u)\Big)(-\Delta)^{s(\cdot)}_{p(\cdot)}u(x) = \lambda |u|^{q(x)-2}u\left(\int_{\Omega}\frac{1}{q(x)} |u|^{q(x)}dx \right)^{k_1}+\beta|u|^{r(x)-2}u\left(\int_{\Omega}\frac{1}{r(x)} |u|^{r(x)}dx \right)^{k_2} \quad \mbox{in }\Omega, \\ \ u = 0 \quad \mbox{on }\partial\Omega, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where the nonlocal term is defined as</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \sigma_{p(x, y)}(u) = \int_{\Omega\times \Omega}\frac{1}{p(x, y)}\frac{|u(x)-u(y)|^{p(x, y)}}{|x-y|^{N+s(x, y)p(x, y)}} \, dx\, dy. $\end{document} </tex-math></disp-formula></p> <p>Here, $ \Omega\subset\mathbb{R}^{N} $ represents a bounded smooth domain with at least $ N\geq2 $. The function $ M(s) $ is given by $ M(s) = a - bs^\gamma $, where $ a\geq 0 $, $ b > 0 $, and $ \gamma > 0 $. The parameters $ k_1 $, $ k_2 $, $ \lambda $ and $ \beta $ are real parameters, while the variables $ p(x) $, $ s(\cdot) $, $ q(x) $, and $ r(x) $ are continuous and can change with respect to $ x $. To tackle this problem, we employ some new methods and variational approaches along with two specific methods, namely the Fountain theorem and the symmetric Mountain Pass theorem. By utilizing these techniques, we establish the existence and multiplicity of solutions for this problem separately in two distinct cases: when $ a > 0 $ and when $ a = 0 $. To the best of our knowledge, these results are the first contributions to research on the variable-order $ p(x) $-fractional Laplacian operator.</p></abstract>
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