The Nehari Manifold for p-Laplacian Equation with Dirichlet Boundary Condition

Abstract

The Nehari manifold for the equation −∆pu(x) = λu(x)|u(x)|p−2 + b(x)|u(x)|γ−2u(x) for x ∈ Ω together with Dirichlet boundary condition is investigated in the case where 0 < γ < p. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form of t → J(tu) where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as λ changes, and show how existence results for positive solutions of the equation are linked to the properties of Nehari manifold.

Nehari manifold for non-local elliptic operator with concave–convex nonlinearities and sign-changing weight functions

“…Moreover multiplicity results with polynomial type nonlinearity with sign-changing weight functions using Nehari manifold and fibering map analysis is also studied in many papers(see refs. [22,4,9,11,12,13,14,15,5]. To the best of our knowledge there is no work for non-local operator with convex-concave type nonlinearity and sign changing weight functions.…”

Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function

“…The approach is not new but the results that we obtained are new. Our work is motivated by the work of Servadei and Valdinoci [18], Brown and Zhang [16] and Afrouzi et al [4]. First we define the space X 0 = u| u : R n → R is measurable, u| Ω ∈ L p (Ω), (u(x) − u(y)) p K(x − y) ∈ L p (Q), u = 0 on R n \ Ω , p-fractional Laplacian with sign changing weight function 3 where Q = R 2n \ (CΩ × CΩ).…”

“…Moreover multiplicity results with polynomial type nonlinearity with sign-changing weight functions using Nehari manifold and fibering map analysis is also studied in many papers ( see refs. [23,4,9,11,12,13,14,15,5]). In this work we use fibering map analysis and Nehari manifold approach to solve the problem (1.1).…”

THE NEHARI MANIFOLD APPROACH FOR DIRICHLET PROBLEM INVOLVING THE p(x)-LAPLACIAN EQUATION

“…and a (x) ≡ a, b (x) ≡ b > 0, using Ekeland's variational principle and the mountain-pass lemma Mihȃilescu proved that, if a and b small enough then there are two distinct solutions for the problem; in [22] under the assumptions 1 Mihȃilescu and Rȃdulescu showed that there exists λ * such that any λ ∈ (0, λ * ) is an eigenvalue for the problem by using Ekeland's variational principle and the mountain-pass lemma; in [14] for the case p (x) = q (x) > 1, h (x) = 0, where p (x) is continuous on Ω, and a (x) ≡ 1, b (x) = 0, Fan, Zhang, and Zhao obtained that, Λ = Λ p(x) , the set of eigenvalues, is a nonempty infinite set such that sup Λ = +∞. In addition, they present some sufficient conditions for inf Λ = 0 and for inf Λ > 0, respectively; in [4] under the conditions p (x) = 2, q (x) = 2 and 1 < h < N +2 N −2 , where h is constant, and a (x) , b (x) : Ω ⊂ R N → R are smooth functions which may change sign in Ω, Brown and Zhang used the relationship between the Nehari manifold and fibrering maps to show how existence and nonexistence results for positive solutions of the equation are linked to properties of the Nehari manifold; in [2] Afrouzi, Mahdavi, and Naghizadeh dealt with the similar problem for the case p (x) = q (x) = p, h (x) = h, 1 < h < p, a (x) = 1 and b (x) : Ω ⊂ R N → R is a smooth function which may change sign and they discussed the existence and multiplicity of non-negative solutions of the problem from a variational viewpoint by making use of the Nehari manifold. Under the conditions p…”