Large Energy Bubble Solutions for Schrödinger Equation with Supercritical Growth

We consider the following nonlinear Schrödinger equation involving supercritical growth: { - Δ ⁢ u + V ⁢ ( y ) ⁢ u = Q ⁢ ( y ) ⁢ u 2 * - 1 + ε in ⁢ ℝ N , u > 0 , u ∈ H 1 ⁢ ( ℝ N ) , \left\{\begin{aligned} &\displaystyle{-}\Delta u+V(y)u=Q(y)u^{2^{*}-1+% \varepsilon}&&\displaystyle\phantom{}\text{in }\mathbb{R}^{N},\\ &\displaystyle u>0,\quad u\in H^{1}(\mathbb{R}^{N}),\end{aligned}\right.{} where 2 * = 2 ⁢ N N - 2 {2^{*}=\frac{2N}{N-2}} is the critical Sobolev exponent, N ≥ 5 {N\geq 5} , and V ⁢ ( y ) {V(y)} and Q ⁢ ( y ) {Q(y)} are bounded nonnegative functions in ℝ N {\mathbb{R}^{N}} . By using the finite reduction argument and local Pohozaev-type identities, under some suitable assumptions on the functions V and Q, we prove that for ε > 0 {\varepsilon>0} is small enough, problem ( * ) {(*)} has large number of bubble solutions whose functional energy is in the order ε - N - 4 ( N - 2 ) 2 . {\varepsilon^{-\frac{N-4}{(N-2)^{2}}}.}