Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth - Hai Quan; Zhang Jing (Demonstratio Mathematica)
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.