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- W4302590675 abstract "We study the following elliptic problem $-A(u) = lambda u^q$ with Dirichlet boundary conditions, where $A(u) (x) = Delta u (x) chi_{D_1} (x)+ Delta_p u(x) chi_{D_2}(x)$ is the Laplacian in one part of the domain, $D_1$, and the $p-$Laplacian (with $p>2$) in the rest of the domain, $D_2 $. We show that this problem exhibits a concave-convex nature for $1<q<p-1$. In fact, we prove that there exists a positive value $lambda^*$ such that the problem has no positive solution for $lambda > lambda^*$ and a minimal positive solution for $0<lambda < lambda^*$. If in addition we assume that $p$ is subcritical, that is, $p<2N/(N-2)$ then there are at least two positive solutions for almost every $0<lambda < lambda^*$, the first one (that exists for all $0<lambda < lambda^*$) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every $0<lambda < lambda^*$) comes from an appropriate (and delicate) mountain pass argument." @default.
- W4302590675 created "2022-10-06" @default.
- W4302590675 creator A5016268671 @default.
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- W4302590675 date "2017-03-09" @default.
- W4302590675 modified "2023-09-27" @default.
- W4302590675 title "A concave-convex problem with a variable operator" @default.
- W4302590675 doi "https://doi.org/10.48550/arxiv.1703.03376" @default.
- W4302590675 hasPublicationYear "2017" @default.
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