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Further applications of the Nehari manifold method to functionals in $C^1(X \setminus \{0\})$

Edir J. F. Leite, Humberto Ramos Quoirin, Kaye Silva

TL;DR

This work develops a two-point Nehari manifold framework for functionals in $C^1(X\setminus\{0\})$ whose fibering maps admit two critical points, producing ground-state and two unbounded sequences of critical values via restricted Ljusternik–Schnirelman theory. The authors establish the continuity of the fibering maps, the sphere-to-Nehari-homeomorphisms, and PS conditions on the restricted manifolds, yielding two families of critical values $\lambda_n^{+}$ and $\lambda_n^{-}$ with $\lambda_1^{+}$ as the ground state and $\lambda_n^{+}\le\lambda_n^{-}$; under extra hypotheses, they prove $\lambda_{n}^{+}<\lambda_{n}^{-}$ for all $n$. They apply the abstract results to prescribed energy problems and to the affine $p$-Laplacian with concave–convex nonlinearities, obtaining multiplicity results and detailed energy properties for small negative energy levels and small parameters $\lambda$. The paper thereby provides a robust, general variational framework for obtaining multiple solutions in nonstandard growth and nonlocal-type settings, including the affine operator context.

Abstract

We proceed with the study of the Nehari manifold method for functionals in $C^1(X \setminus \{0\})$, where $X$ is a Banach space. We deal now with functionals whose fibering maps have two critical points (a minimiser followed by a maximiser). Under some additional conditions we show that the Nehari manifold method provides us with the ground state level and two sequences of critical values for these functionals. These results are applied to the class of {\it prescribed energy problems} as well as to the concave-convex problem for the {\it affine} $p$-Laplacian operator.

Further applications of the Nehari manifold method to functionals in $C^1(X \setminus \{0\})$

TL;DR

This work develops a two-point Nehari manifold framework for functionals in whose fibering maps admit two critical points, producing ground-state and two unbounded sequences of critical values via restricted Ljusternik–Schnirelman theory. The authors establish the continuity of the fibering maps, the sphere-to-Nehari-homeomorphisms, and PS conditions on the restricted manifolds, yielding two families of critical values and with as the ground state and ; under extra hypotheses, they prove for all . They apply the abstract results to prescribed energy problems and to the affine -Laplacian with concave–convex nonlinearities, obtaining multiplicity results and detailed energy properties for small negative energy levels and small parameters . The paper thereby provides a robust, general variational framework for obtaining multiple solutions in nonstandard growth and nonlocal-type settings, including the affine operator context.

Abstract

We proceed with the study of the Nehari manifold method for functionals in , where is a Banach space. We deal now with functionals whose fibering maps have two critical points (a minimiser followed by a maximiser). Under some additional conditions we show that the Nehari manifold method provides us with the ground state level and two sequences of critical values for these functionals. These results are applied to the class of {\it prescribed energy problems} as well as to the concave-convex problem for the {\it affine} -Laplacian operator.

Paper Structure

This paper contains 10 sections, 25 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The prescribed energy problem
  3. The concave-convex problem for the affine $p$-Laplacian
  4. Proofs of main results
  5. Proof of Theorem \ref{['tn']}
  6. Proof of Theorem \ref{['c1']}
  7. On the inequality $\lambda_{n,c}^+< \lambda_{n,c}^-$
  8. Applications of Corollary \ref{['cc1']} to some boundary value problems
  9. Proof of Theorem \ref{['taf']}
  10. A class of functionals with homogeneous terms

Key Result

Theorem 1.1

Assume (H1) and $(H2)^{\pm}$. Then $(\lambda_n^{\pm})$ is a nondecreasing unbounded sequence of critical values of $\Phi$. Moreover $\lambda_1^+$ is the ground state level of $\Phi$, i.e. its least critical level, $\lambda_1^+<\lambda_1^-$, and $\lambda_n^+ \le \lambda_n^-$ for any $n \ge 2$.

Figures (1)

  • Figure 1: Some possible behaviors of the map $t \mapsto \Phi(tu)$ under (H1).

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Remark 2.2
  • proof : Proof of Theorem \ref{['tn']}
  • ...and 36 more