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.
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 ))
We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form:Here u = div(∇u) is the Laplacian of u, 1 λ is the diffusion coefficient, K and c are positive constants and Ω ⊂ R N is a smooth bounded region with ∂Ω in C 2 . This model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. In this paper we discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve. We prove our results by the method of sub-supersolutions.
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