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In this paper, we prove that the second eigenfunctions of the p-Laplacian, p > 1, are not radial on the unit ball in R N , for any N ≥ 2. Our proof relies on the variational characterization of the second eigenvalue and a variant of the deformation lemma. We also construct an infinite sequence of eigenpairs {τ n , Ψ n } such that Ψ n is nonradial and has exactly 2n nodal domains. A few related open problems are also stated.

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Let B 1 be a ball in R N centred at the origin and B 0 be a smaller ball compactly contained in B 1 . For p ∈ (1, ∞), using the shape derivative method, we show that the first eigenvalue of the p-Laplacian in annulus B 1 \ B 0 strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as p → 1 and p → ∞ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the p-Laplacian on bounded radial domains. Mathematics Subject Classification (2010): 35J92, 35P30, 35B06, 49R05. Ω |∇u| p dx : u ∈ W 1,p 0 (Ω) \ {0} with u p = 1 .In this article we consider Ω of the formdenotes the open ball of radius r > 0 centred at z ∈ R N . Since the p-Laplacian is invariant under orthogonal transformations, it can be easily seen that λ 1 (B R 1 (x) \ B R 0 (y)) = λ 1 (B R 1 (0) \ B R 0 (se 1 ))

Payne-Weinberger showed that 'among the class of membranes with given area A, free along the interior boundaries and fixed along the outer boundary of given length L 0 , the annulus Ω # has the highest fundamental frequency,' where Ω # is a concentric annulus with the same area as Ω and the same outer boundary length as L 0 .We extend this result for the higher dimensional domains and p-Laplacian with p ∈ (1, ∞), under the additional assumption that the outer boundary is a sphere. As an application, we prove that the nodal set of the second eigenfunctions of p-Laplacian (with mixed boundary conditions) on a ball and a concentric annulus cannot be a concentric sphere.Keywords Reverse Faber-Krahn inequality · Interior parallels · Isoperimetric inequality · Nagy's inequalities · First eigenvalue of p-Laplacian · Mixed boundary value problems · Elasticity problems · Non-radiality IntroductionLet us first recall the famous conjecture by Lord Rayleigh from his book 'The theory of sound' [23] published in 1877. He conjectured that 'among all planar domains Ω of fixed area, the disk is the domain that minimises the first Dirichlet eigenvalue λ 1 (Ω) of the Laplacian.' This conjecture was open for a very long time. The first proof of this conjecture was published in 1923 by Faber [8]. In 1925, Krahn [14] gave an independent proof for this conjecture, and later he extended the same to the higher dimension. Now this result is collectively known as the Faber-Krahn inequality, and it states that:with the equality if and only if Ω is a ball (up to a measure zero set), where Ω * is a ball of the same measure as Ω.For more on Faber-Krahn inequality and related results, we refer to [18] and [21].