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- W4299374611 abstract "We study the non-linear minimization problem on $H^1_0(Omega)subset L^q$ with $q=frac{2n}{n-2}$, $alpha>0$ and $ngeq4$~: [inf_{substack{uin H^1_0(Omega) |u|_{L^q}=1}}int_Omega a(x,u)|nabla u|^2 - lambda int_{Omega} |u|^2.] where $a(x,s)$ presents a global minimum $alpha$ at $(x_0,0)$ with $x_0inOmega$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behaviour of $a(x,s)$ with respect to $s$. The model case is [inf_{substack{uin H^1_0(Omega) |u|_{L^q}=1}}int_Omega (alpha+|x|^beta |u|^k)|nabla u|^2 - lambda int_{Omega} |u|^2.] In a previous paper dedicated to the same problem with $lambda=0$, we showed that minimizers exist only in the range $beta<kn/q$, which corresponds to a dominant non-linear term. On the contrary, the linear influence for $betageq kn/q$ prevented their existence. The goal of this present paper is to show that for $0<lambdaleq alphalambda_1(Omega)$, $0leq kleq q-2$ and $beta > kn/q + 2$, minimizers do exist." @default.
- W4299374611 created "2022-10-02" @default.
- W4299374611 creator A5007718903 @default.
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- W4299374611 date "2017-10-16" @default.
- W4299374611 modified "2023-10-17" @default.
- W4299374611 title "Existence of Solutions of a Non-Linear Eigenvalue Problem with a Variable Weight" @default.
- W4299374611 doi "https://doi.org/10.48550/arxiv.1710.05653" @default.
- W4299374611 hasPublicationYear "2017" @default.
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