Critical point localization and multiplicity results in Banach spaces via Nehari manifold technique

In the paper, results on the existence of critical points in annular subsets of a cone are obtained with the additional goal of obtaining multiplicity results. Compared to other approaches in the literature based on the use of Krasnoselskii’s compression-extension theorem or topological index methods, our approach uses the Nehari manifold technique in a surprising combination with the cone version of Birkhoff-Kellogg’s invariant-direction theorem. This yields a simpler alternative to traditional methods involving deformation arguments or Ekeland variational principle. The new method is illustrated on a boundary value problem for p-Laplacian equations, and we believe that it will be useful for proving the existence, localization, and multiplicity of solutions for other classes of problems with variational structure.

💡 Research Summary

The paper introduces a novel method for locating critical points of a C¹ functional E in a Banach space X within annular subsets of a cone K. The authors combine the Nehari manifold technique with the cone version of the Birkhoff‑Kellogg invariant‑direction theorem, thereby avoiding the usual reliance on Krasnoselskii’s compression‑extension theorem, topological index arguments, or Ekeland’s variational principle.

The setting assumes a single‑valued, strictly monotone duality mapping J:X→X* with a continuous inverse J⁻¹. The cone K⊂X is non‑degenerate, and for given radii 0<r<R the annular set K_{r,R} = {u∈K : r≤‖u‖≤R} is considered. The Nehari manifold is defined as N_{r,R}= {u∈K_{r,R} : ⟨E′(u),u⟩=0}. Three hypotheses are imposed:

(H1) T:=J⁻¹∘N, with N(u)=J(u)−E′(u), is completely continuous and maps K into itself.
(H2) T stays away from zero on N_{r,R}, i.e. inf_{u∈N_{r,R}}‖T(u)‖>0.
(H3) For each u∈K_{r,R}, the scalar function α_u(s)=E(su) has a unique zero of its derivative in the interval (r‖u‖,R‖u‖), with the derivative positive at the left endpoint and negative at the right.

Under (H3) the Nehari manifold can be parametrised as N_{r,R}={s(u)u : u∈K_{r,R}}={u∈K_{r,R}: s(u)=1}, and every element satisfies r<‖u‖<R. Defining V={σu : u∈N_{r,R}, 0≤σ<1}, one shows that V is open in K and its relative boundary equals N_{r,R}. Using Dugundji’s retraction theorem, an open set U⊂X with K∩∂U=N_{r,R} is constructed.

Applying the Krasnoselskii‑Ladyzhenskaya invariant‑direction theorem to T on K∩U yields a point u*∈N_{r,R} and a scalar λ₀>0 with J⁻¹N(u*)=λ₀u*. Strict monotonicity of J forces λ₀=1, so N(u*)=J(u*) and consequently E′(u*)=0. Thus a critical point exists inside the prescribed annulus.

Multiplicities follow by taking several disjoint annuli (r_k,R_k) that each satisfy (H2)–(H3). For a finite family one obtains m distinct critical points; for an infinite increasing (or decreasing) sequence of radii one obtains an unbounded (or vanishing) sequence of solutions.

The abstract theory is illustrated on the Dirichlet problem for the p‑Laplacian
 −( |u′|^{p‑2}u′ )′ = f(u), u(0)=u(1)=0,
with f continuous, non‑negative and non‑decreasing on ℝ₊. The energy functional is
 E(u)= (1/p)‖u‖_{1,p}^p −∫₀¹F(u)dt, F′=f.
Choosing a cone K of non‑negative, symmetric, non‑decreasing functions that satisfy a Harnack‑type lower bound, the authors verify (H1)–(H3) and obtain at least one positive solution. By selecting appropriate sequences of radii, infinitely many positive solutions with prescribed norms are produced.

Overall, the paper provides a clean, C¹‑only framework that merges variational Nehari methods with fixed‑point invariant‑direction theory, yielding existence and multiplicity results without deformation arguments or higher regularity assumptions. The approach is expected to be applicable to a broad class of nonlinear elliptic problems and eigenvalue equations in Banach spaces.