Multiple solutions to multi-critical Schrödinger equations
نویسندگان
چکیده
Abstract In this article, we investigate the existence of multiple positive solutions to following multi-critical Schrödinger equation: (0.1) − Δ u + λ V ( x ) = μ ∣ p 2 ∑ i 1 k N α ∗ width="1.0em" width="0.1em" in width="0.33em" mathvariant="double-struck">R , ∈ H \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 }^{{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 xmlns:m="http://www.w3.org/1998/Math/MathML"> ≥ 4 \lambda ,\mu \in {{\mathbb{R}}}^{+},N\ge 4 , and {2}_{i}^{\ast }=\frac{N+{\alpha }_{i}}{N-2} with < N-4\lt {\alpha }_{i}\lt N , … i=1,2,\ldots ,k are critical exponents min mathvariant="normal">min { : } 2\lt p\lt {2}_{\min }^{\ast }={\rm{\min }}\left\{{2}_{i}^{\ast }:i=1,2,\ldots ,k\right\} . Suppose that mathvariant="normal">Ω mathvariant="normal">int 0 ⊂ \Omega ={\rm{int}}\hspace{0.33em}{V}^{-1}\left(0)\subset {{\mathbb{R}}}^{N} is a bounded domain, show for large, problem possesses at least mathvariant="normal">cat {{\rm{cat}}}_{\Omega }\left(\Omega ) solutions.
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ژورنال
عنوان ژورنال: Advanced Nonlinear Studies
سال: 2022
ISSN: ['1536-1365', '2169-0375']
DOI: https://doi.org/10.1515/ans-2022-0014