Nonlinear Elliptic Equations with Neumann Boundary Conditions

This chapter aims to present relevant knowledge regarding recent progress on nonlinear elliptic equations with Neumann boundary conditions. In fact, all the results presented here bring novelties with respect to the available literature. We emphasize the

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Nonlinear Elliptic Equations with Neumann Boundary Conditions

Abstract This chapter aims to present relevant knowledge regarding recent progress on nonlinear elliptic equations with Neumann boundary conditions. In fact, all the results presented here bring novelties with respect to the available literature. We emphasize the specific functional setting and techniques involved in handling the Neumann problems, which are distinct in comparison with those for the Dirichlet problems. The first section of the chapter discusses the multiple solutions that arise at near resonance, from the left and from the right, in the Neumann problems depending on parameters. The second section focuses on nonlinear Neumann problems whose differential part is described by a general nonhomogeneous operator. The third section builds a common approach for both sublinear and superlinear cases of semilinear Neumann problems. Related comments and references are given in a remarks section.

12.1 Nonlinear Neumann Problems Using Variational Methods Let Ω ⊂ RN be a bounded domain with a C2 -boundary ∂ Ω . In this section, we study the following nonlinear parametric Neumann problem: ⎧ ⎨ −Δ p u(x) = λ |u(x)| p−2 u(x) + f (x, u(x)) in Ω , ∂u =0 on ∂ Ω , ⎩ ∂ np

(12.1)

where λ ∈ R is a parameter, 1 < p < +∞, and f : Ω × R → R is a Carathéodory function. We recall that Δ p denotes the p-Laplace differential operator defined by Δ p u = div (|∇u| p−2 ∇u) for all u ∈ W 1,p (Ω ) (where |·| stands for the Euclidean norm − 1 ,p

of RN ) and ∂∂nup := γn (|∇u| p−2 ∇u) ∈ W p (∂ Ω ) denotes the generalized outward normal derivative (Theorem 1.39). We recall that the notion of solution to Neumann problems such as (12.1) is given in Definition 8.2 [see also Remark 8.3(b)]. D. Motreanu et al., Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, DOI 10.1007/978-1-4614-9323-5__12, © Springer Science+Business Media, LLC 2014

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12 Nonlinear Elliptic Equations with Neumann Boundary Conditions

We examine the existence and multiplicity of nontrivial solutions when the parameter λ ∈ R is near 0 (that is, when near resonance occurs with respect to the first eigenvalue λ0 = 0 of the negative Neumann p-Laplacian). We consider two distinct cases depending on whether the parameter λ approaches 0 from below or from above. In the first case (near resonance from the left) we establish the existence of three nontrivial, smooth solutions of problem (12.1); moreover, in the semilinear case (i.e., for p = 2), by strengthening the regularity conditions on f (x, ·), we are able to produce four nontrivial, smooth solutions. In the second case (near resonance from the right), we produce two nontrivial, smooth solutions. In both cases, we use variational methods based on critical point theory (Chap. 5) and also on Morse theory (Chap. 6). To start with, for later use, we state the following result on local C1 (Ω )- versus local W 1,p (Ω )-minimizers, similar to Proposition 11.4. The proof of this result will be given in wider generalit

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