Existence results for elliptic equation involving polyharmonic operator and a critical growth

Abstract

AbstractIn this work, we study the two following minimization problems for $$r \in \mathbb {N}^{*}$$ r ∈ N ∗ , $$\begin{aligned} \begin{array}{ccc} S_{0,r}(\varphi )=\displaystyle \inf _{u\in H_{0}^{r}(\Omega ),\,\Vert u+\varphi \Vert _{L^{2^{*r}}}=1}\Vert u\Vert _{r}^{2}&\text { and }\,\,&S_{\theta ,r}(\varphi )=\displaystyle \inf _{u\in H_{\theta }^{r}(\Omega ),\,\Vert u+\varphi \Vert _{L^{2^{*r}}}=1}\Vert u\Vert _{r}^{2}, \end{array} \end{aligned}$$ S 0 , r ( φ ) = inf u ∈ H 0 r ( Ω ) , ‖ u + φ ‖ L 2 ∗ r = 1 ‖ u ‖ r 2 and S θ , r ( φ ) = inf u ∈ H θ r ( Ω ) , ‖ u + φ ‖ L 2 ∗ r = 1 ‖ u ‖ r 2 , where $$\Omega \subset \mathbb {R}^{N}, $$ Ω ⊂ R N , $$N > 2r$$ N > 2 r , is a smooth bounded domain, $$2^{*r}=\frac{2N}{N-2 r}$$ 2 ∗ r = 2 N N - 2 r , $$\varphi \in L^{2^{*r}} (\Omega ) \cap C(\Omega )$$ φ ∈ L 2 ∗ r ( Ω ) ∩ C ( Ω ) and the norm $$\Vert . \Vert _{r}=\displaystyle { \int _{\Omega } |(-\Delta )^{\alpha }.|^{2}dx}$$ ‖ . ‖ r = ∫ Ω | ( - Δ ) α . | 2 d x where $$ \alpha =\frac{r}{2} $$ α = r 2 if r is even and $$\Vert . \Vert _{r}=\displaystyle { \int _{\Omega } |\nabla (-\Delta )^{\alpha }. |^{2}dx }$$ ‖ . ‖ r = ∫ Ω | ∇ ( - Δ ) α . | 2 d x where $$\alpha = \frac{r-1}{2}$$ α = r - 1 2 if r is odd. Firstly, we prove that, when $$\varphi \not \equiv 0, $$ φ ≢ 0 , the infimum in $$S_{0,r}(\varphi )$$ S 0 , r ( φ ) and $$S_{\theta ,r}(\varphi )$$ S θ , r ( φ ) are achieved. Secondly, we show that $$ S_{\theta ,r}(\varphi )< S_{0,r}(\varphi ) $$ S θ , r ( φ ) < S 0 , r ( φ ) for a large class of $$\varphi $$ φ .