Construction of Solutions for Hénon-Type Equation with Critical Growth

Abstract

Abstract We consider the following Hénon-type problem with critical growth: (H) { - Δ ⁢ u = K ⁢ ( | y ′ | , y ′′ ) ⁢ u 2 * - 1 , u > 0 in  ⁢ B 1 , u = 0 on  ⁢ ∂ ⁡ B 1 , \displaystyle{}\left\{\begin{aligned} \displaystyle{}-\Delta u&\displaystyle=K% (|y^{\prime}|,y^{\prime\prime})u^{2^{*}-1},\quad u>0&&\displaystyle\text{in }B% _{1},\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial B_{1},\end{% aligned}\right. where 2 * = 2 ⁢ N N - 2 {2^{*}\kern-1.0pt=\kern-1.0pt\frac{2N}{N-2}} , N ≥ 5 {N\kern-1.0pt\geq\kern-1.0pt5} , B 1 {B_{1}} is the unit sphere in ℝ N {\mathbb{R}^{N}} , y = ( y ′ , y ′′ ) ∈ ℝ 2 × ℝ N - 2 y\kern-1.0pt=\kern-1.0pt(y^{\prime},y^{\prime\prime})\kern-1.0pt\in\kern-1.0pt% \mathbb{R}^{2}\times\mathbb{R}^{N-2} , r = | y ′ | {r\kern-1.0pt=\kern-1.0pt|y^{\prime}|} and K ⁢ ( y ) = K ⁢ ( r , y ′′ ) ∈ C 2 ⁢ ( B 1 ) {K(y)\kern-1.0pt=\kern-1.0ptK(r,y^{\prime\prime})\kern-1.0pt\in\kern-1.0ptC^{2% }(B_{1})} is a bounded non-negative function. By using a finite reduction argument and local Pohozaev-type identities, we prove that if N ≥ 5 {N\geq 5} and K ⁢ ( r , y ′′ ) {K(r,y^{\prime\prime})} has a stable critical point y 0 = ( r 0 , y 0 ′′ ) ∈ ∂ ⁡ B 1 {y_{0}=(r_{0},y_{0}^{\prime\prime})\in\partial B_{1}} , then the above problem has infinitely many solutions, whose energy can be arbitrarily large.