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Let $Omegasubset mathbb{R}^n$ be a bounded $C^1$ domain and $p>1$. For $alpha>0$, define the quantity [ Lambda(alpha)=inf_{uin W^{1,p}(Omega),, u otequiv 0} Big(int_Omega | abla u|^p,mathrm{d}x - alpha int_{partialOmega} |u|^p ,mathrm{d} sBig)Big/ int_Omega |u|^p,mathrm{d} x ] with $mathrm{d} s$ being the hypersurface measure, which is the lowest eigenvalue of the $p$-laplacian in $Omega$ with a non-linear $alpha$-dependent Robin boundary condition. We show the asymptotics $Lambda(alpha) =(1-p)alpha^{p/(p-1)}+o(alpha^{p/(p-1)})$ as $alpha$ tends to $+infty$. The result was only known for the linear case $p=2$ or under stronger smoothness assumptions. Our proof is much shorter and is based on completely different and elementary arguments, and it allows for an improved remainder estimate for $C^{1,lambda}$ domains.
We consider a nonlinear elliptic equation driven by the Robin $p$-Laplacian plus an indefinite potential. In the reaction we have the competing effects of a strictly $(p-1)$-sublinear parametric term and of a $(p-1)$-linear and nonuniformly nonresona
In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary conditions are
We study the discrete spectrum of the Robin Laplacian $Q^{Omega}_alpha$ in $L^2(Omega)$, [ umapsto -Delta u, quad dfrac{partial u}{partial n}=alpha u text{ on }partialOmega, ] where $Omegasubset mathbb{R}^{3}$ is a conical domain with a regular cross
We consider quasimodes on planar domains with a partially rectangular boundary. We prove that for any $epsilon_0>0$, an $O(lambda^{-epsilon_0})$ quasimode must have $L^2$ mass in the wings bounded below by $lambda^{-2-delta}$ for any $delta>0$. The p
We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $Omega subset mathbb{R}^N$. By means of topological argumen