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- W2334801086 abstract "In this paper, we consider the following Schrödinger equation with critical growth$$-Delta u+(lambda a(x)-delta)u=|u|^{2^*-2}u quad hbox{ in } mathbb{R}^N, $$ where $Ngeq 5$, $2^*$ is the critical Sobolev exponent, $delta>0$ is a constant, $a(x)geq 0$ and its zero set is not empty. We will show that if the zero set of $a(x)$ has several isolated connected components$Omega_1,cdots,Omega_k$ such that the interior of $Omega_i (i=1, 2, ...,k)$ is not empty and $partialOmega_i (i=1, 2, ..., k)$ is smooth, then for any non-emptysubset $Jsubset {1,2,cdots,k}$ and $lambda$ sufficiently large, the equation admits a solution which is trapped in aneighborhood of $bigcup_{jin J}Omega_j$. Our strategy to obtain the main results is as follows: By using local mountain pass method combining with penalization of the nonlinearities, we first prove the existence of single-bump solutions which are trapped in the neighborhood of only one isolated component of zero set. Then we construct the multi-bump solution by summing these one-bump solutions as the first approximation solution. The real solution will be obtained by delicate estimates of the error term, this last step is done by using Contraction Image Principle." @default.
- W2334801086 created "2016-06-24" @default.
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- W2334801086 date "2015-01-01" @default.
- W2334801086 modified "2023-10-16" @default.
- W2334801086 title "Multi-bump solutions for Schrödinger equation involving critical growth and potential wells" @default.
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- W2334801086 doi "https://doi.org/10.3934/dcds.2015.35.3393" @default.
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