Multiple solutions to multi-critical Schrödinger equations

Abstract

Abstract In this article, we investigate the existence of multiple positive solutions to the following multi-critical Schrödinger equation: (0.1) − Δ u + λ V ( x ) u = μ ∣ u ∣ p − 2 u + ∑ i = 1 k ( ∣ x ∣ − ( N − α i ) ∗ ∣ u ∣ 2 i ∗ ) ∣ u ∣ 2 i ∗ − 2 u in R N , u ∈ H 1 ( R N ) , \left\{\begin{array}{l}-\Delta u+\lambda V\left(x)u=\mu | u{| }^{p-2}u+\mathop{\displaystyle \sum }\limits_{i=1}^{k}\left(| x{| }^{-\left(N-{\alpha }_{i})}\ast | u{| }^{{2}_{i}^{\ast }})| u{| }^{{2}_{i}^{\ast }-2}u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\hspace{0.33em}\in {H}^{1}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}\end{array}\right. where λ , μ ∈ R + , N ≥ 4 \lambda ,\mu \in {{\mathbb{R}}}^{+},N\ge 4 , and 2 i ∗ = N + α i N − 2 {2}_{i}^{\ast }=\frac{N+{\alpha }_{i}}{N-2} with N − 4 < α i < N N-4\lt {\alpha }_{i}\lt N , i = 1 , 2 , … , k i=1,2,\ldots ,k are critical exponents and 2 < p < 2 min ∗ = min { 2 i ∗ : i = 1 , 2 , … , k } 2\lt p\lt {2}_{\min }^{\ast }={\rm{\min }}\left\{{2}_{i}^{\ast }:i=1,2,\ldots ,k\right\} . Suppose that Ω = int V − 1 ( 0 ) ⊂ R N \Omega ={\rm{int}}\hspace{0.33em}{V}^{-1}\left(0)\subset {{\mathbb{R}}}^{N} is a bounded domain, we show that for λ \lambda large, problem (0.1) possesses at least cat Ω ( Ω ) {{\rm{cat}}}_{\Omega }\left(\Omega ) positive solutions.