Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth
In this article, we are interested in the existence of nontrivial solutions for the following nonhomogeneous Choquard equation involving the pp-biharmonic operator: M∫Ω∣Δu∣pdxΔp2u−Δpu=λ(∣x∣−μ⁎∣u∣q)∣u∣q−2u+∣u∣p*−2u+f,inΩ,u=Δu=0,on∂Ω,\left\{\begin{array}{l}M\left(\mathop{\displaystyle \int }\limits_{\...
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2025-03-01
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| description | In this article, we are interested in the existence of nontrivial solutions for the following nonhomogeneous Choquard equation involving the pp-biharmonic operator: M∫Ω∣Δu∣pdxΔp2u−Δpu=λ(∣x∣−μ⁎∣u∣q)∣u∣q−2u+∣u∣p*−2u+f,inΩ,u=Δu=0,on∂Ω,\left\{\begin{array}{l}M\left(\mathop{\displaystyle \int }\limits_{\Omega }{| \Delta u| }^{p}{\rm{d}}x\right){\Delta }_{p}^{2}u-{\Delta }_{p}u=\lambda \left({| x| }^{-\mu }\ast {| u| }^{q}){| u| }^{q-2}u+{| u| }^{{p}^{* }-2}u+f,\hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\\ u=\Delta u=0,\hspace{1em}{\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω⊂RN\Omega \subset {{\mathbb{R}}}^{N}, N≥3N\ge 3 is a smooth bounded domain, 1<p<N21\lt p\lt \frac{N}{2}, 0<μ<N0\lt \mu \lt N, p<2q<p*p\lt 2q\lt {p}^{* }, p*=NpN−2p{p}^{* }=\frac{Np}{N-2p} denotes the Sobolev conjugate of pp. MM is a nondecreasing and continuous function, and λ>0\lambda \gt 0 is a parameter. Δp2u≔Δ(∣Δu∣p−2Δu){\Delta }_{p}^{2}u:= \Delta \left({| \Delta u| }^{p-2}\Delta u) is the operator of fourth order called the p-biharmonic operator, Δpu≔div(∣∇u∣p−2∇u){\Delta }_{p}u:= \hspace{0.1em}\text{div}\hspace{0.1em}\left({| \nabla u| }^{p-2}\nabla u) is the pp-Laplacian operator. f≥0f\ge 0, f∈Lpp−1(Ω)f\in {L}^{\tfrac{p}{p-1}}\left(\Omega ), and ∣f∣pp−1{| f| }_{\tfrac{p}{p-1}} is sufficiently small. Using the concentration-compactness principle together with the mountain pass theorem, we obtain the existence of nontrivial solutions for the aforementioned problem in both nondegenerate and degenerate cases. |
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| spelling | doaj-art-9f24daaaccfc4b8aad842ebd06c5efde2025-08-20T02:10:50ZengDe GruyterDemonstratio Mathematica2391-46612025-03-01581619010.1515/dema-2025-0111Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growthHai Quan0Zhang Jing1College of Mathematics Science, Inner Mongolia Normal University, Hohhot, P. R. ChinaCollege of Mathematics Science, Inner Mongolia Normal University, Hohhot, P. R. ChinaIn this article, we are interested in the existence of nontrivial solutions for the following nonhomogeneous Choquard equation involving the pp-biharmonic operator: M∫Ω∣Δu∣pdxΔp2u−Δpu=λ(∣x∣−μ⁎∣u∣q)∣u∣q−2u+∣u∣p*−2u+f,inΩ,u=Δu=0,on∂Ω,\left\{\begin{array}{l}M\left(\mathop{\displaystyle \int }\limits_{\Omega }{| \Delta u| }^{p}{\rm{d}}x\right){\Delta }_{p}^{2}u-{\Delta }_{p}u=\lambda \left({| x| }^{-\mu }\ast {| u| }^{q}){| u| }^{q-2}u+{| u| }^{{p}^{* }-2}u+f,\hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\\ u=\Delta u=0,\hspace{1em}{\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω⊂RN\Omega \subset {{\mathbb{R}}}^{N}, N≥3N\ge 3 is a smooth bounded domain, 1<p<N21\lt p\lt \frac{N}{2}, 0<μ<N0\lt \mu \lt N, p<2q<p*p\lt 2q\lt {p}^{* }, p*=NpN−2p{p}^{* }=\frac{Np}{N-2p} denotes the Sobolev conjugate of pp. MM is a nondecreasing and continuous function, and λ>0\lambda \gt 0 is a parameter. Δp2u≔Δ(∣Δu∣p−2Δu){\Delta }_{p}^{2}u:= \Delta \left({| \Delta u| }^{p-2}\Delta u) is the operator of fourth order called the p-biharmonic operator, Δpu≔div(∣∇u∣p−2∇u){\Delta }_{p}u:= \hspace{0.1em}\text{div}\hspace{0.1em}\left({| \nabla u| }^{p-2}\nabla u) is the pp-Laplacian operator. f≥0f\ge 0, f∈Lpp−1(Ω)f\in {L}^{\tfrac{p}{p-1}}\left(\Omega ), and ∣f∣pp−1{| f| }_{\tfrac{p}{p-1}} is sufficiently small. Using the concentration-compactness principle together with the mountain pass theorem, we obtain the existence of nontrivial solutions for the aforementioned problem in both nondegenerate and degenerate cases.https://doi.org/10.1515/dema-2025-0111nonhomogeneousp-biharmonic operatornondegeneratedegenerate35a0135a1535b33 |
| spellingShingle | Hai Quan Zhang Jing Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth Demonstratio Mathematica nonhomogeneous p-biharmonic operator nondegenerate degenerate 35a01 35a15 35b33 |
| title | Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth |
| title_full | Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth |
| title_fullStr | Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth |
| title_full_unstemmed | Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth |
| title_short | Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth |
| title_sort | existence results for nonhomogeneous choquard equation involving p biharmonic operator and critical growth |
| topic | nonhomogeneous p-biharmonic operator nondegenerate degenerate 35a01 35a15 35b33 |
| url | https://doi.org/10.1515/dema-2025-0111 |
| work_keys_str_mv | AT haiquan existenceresultsfornonhomogeneouschoquardequationinvolvingpbiharmonicoperatorandcriticalgrowth AT zhangjing existenceresultsfornonhomogeneouschoquardequationinvolvingpbiharmonicoperatorandcriticalgrowth |