A Study of Biharmonic Equation Involving Nonlocal Terms and Critical Sobolev Exponent

Abstract

AbstractIn this paper, we investigate the existence of ground state solutions and non-existence of non-trivial weak solution of the equation $$\begin{aligned} \Delta ^{2} u= \Big (|x|^{-\theta }*|u|^{p_{\theta }}\Big )|u|^{p_{\theta }-2}u +\alpha \Big (|x|^{-\gamma }*|u|^{p}\Big )|u|^{p-2}u \quad \text{ in } {{\mathbb {R}}}^{N}, \end{aligned}$$ Δ 2 u = ( | x | - θ ∗ | u | p θ ) | u | p θ - 2 u + α ( | x | - γ ∗ | u | p ) | u | p - 2 u in R N , where $$0<p\le p_{\gamma }^{*}$$ 0 < p ≤ p γ ∗ , $$\alpha >0$$ α > 0 , $$\theta , \gamma \in (0,N)$$ θ , γ ∈ ( 0 , N ) , $$p_{\theta }=\frac{2(N-\theta )}{N-4}$$ p θ = 2 ( N - θ ) N - 4 and $$N\ge 5$$ N ≥ 5 . Firstly, we prove the non-existence by establishing Pohozaev type of identity. Next, we study the existence of ground state solutions by using the minimization method on the associated Nehari manifold.