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Prescribed energy solutions of concave-convex type problems involving sign-changing or vanishing weights

Kanishka Perera, Humberto Ramos Quoirin, Kaye Silva

TL;DR

The paper develops a unified abstract framework to find pairs (λ,u) solving a prescribed-energy problem for a family Φ_λ = I1 − λ I2, including concave–convex elliptic problems with sign-changing or vanishing weights. It leverages Nehari-type manifolds and Krasnoselskii genus to construct multiple critical points at a fixed energy level c by selecting λ = λ(c,u) and studying the reduced energy Λ on constrained sets. The authors establish general conditions (H1)-(H2) ensuring the existence of sequences of energy levels λ_{c,k}^± and corresponding solutions, yielding infinite families of solutions and energy-curve bifurcations for fixed λ. Applications to p-Laplacian models with sign-changing weights demonstrate new infinite families of solutions at prescribed energies, including bifurcation from 0 and ∞, and extend existing results in the literature to more general weight configurations and, in some cases, to nonlocal operators.

Abstract

We provide an abstract approach to find couples $(λ,u) \in \mathbb{R} \times X$ satisfying $$Φ_λ(u)=c \quad \mbox{and} \quad Φ'_λ(u)=0,$$ for some suitable values of $c \in \mathbb{R}$. Here $Φ_λ$ is a $C^1$ functional (set on a Banach space $X$) whose main prototype is the energy functional associated to a concave-convex problem with sign-changing or vanishing weights. This approach allows us to derive several existence, multiplicity and bifurcation type results for the equation $Φ'_λ(u)=0$ with $λ$ fixed.

Prescribed energy solutions of concave-convex type problems involving sign-changing or vanishing weights

TL;DR

The paper develops a unified abstract framework to find pairs (λ,u) solving a prescribed-energy problem for a family Φ_λ = I1 − λ I2, including concave–convex elliptic problems with sign-changing or vanishing weights. It leverages Nehari-type manifolds and Krasnoselskii genus to construct multiple critical points at a fixed energy level c by selecting λ = λ(c,u) and studying the reduced energy Λ on constrained sets. The authors establish general conditions (H1)-(H2) ensuring the existence of sequences of energy levels λ_{c,k}^± and corresponding solutions, yielding infinite families of solutions and energy-curve bifurcations for fixed λ. Applications to p-Laplacian models with sign-changing weights demonstrate new infinite families of solutions at prescribed energies, including bifurcation from 0 and ∞, and extend existing results in the literature to more general weight configurations and, in some cases, to nonlocal operators.

Abstract

We provide an abstract approach to find couples satisfying for some suitable values of . Here is a functional (set on a Banach space ) whose main prototype is the energy functional associated to a concave-convex problem with sign-changing or vanishing weights. This approach allows us to derive several existence, multiplicity and bifurcation type results for the equation with fixed.

Paper Structure

This paper contains 13 sections, 35 theorems, 92 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Concave-convex problems with sign-changing or vanishing weights
  3. Main abstract result - the prescribed energy problem
  4. Abstract concave-convex problems with sign-changing or vanishing weights
  5. Applications
  6. Proof of main results
  7. Proof of Theorem \ref{['thm1']}
  8. Conditions $(C1)$-$(C3)$ imply $(H1)$ and $(H2)$
  9. Behavior of the energy curves
  10. Proof of Theorems \ref{['thm2']}, \ref{['thm3']}, \ref{['thm4']} and \ref{['thm5']}
  11. Applications
  12. Further examples and discussions
  13. A deformation lemma

Key Result

Theorem 1.1

Suppose $(H1)$ and $(H2)$, and let $\lambda_{c,k}$ be given by 50. Then for any $c \in I$ and $1 \le k\le \gamma(\mathcal{S}_\mathcal{C})$ there exists $u_{c,k}\in \mathcal{C}$ such that Moreover, if $\gamma(\mathcal{S}_\mathcal{C})=\infty$ and $\widetilde{\Lambda}(c,\cdot)$ satisfies the Palais--Smale condition at any level, then $(\lambda_{c,k})$ is a nondecreasing unbounded sequence.

Figures (6)

  • Figure 1: Energy curves for Theorem \ref{['thm2']}. The red curves correspond to $(\lambda_{c,k}^+,c)$, $c\in (c^*,0)$ and the blue ones to $(\lambda_{c,k}^-,c)$, $c\in (c^*,c^{**})$.
  • Figure 2: Energy curves for Theorem \ref{['thm3']}. Red curves correspond to $(\lambda_{c,k}^-,c)$, with $c\in (\overline{c}^*,0)$. Blue curves correspond to $(\lambda_{c,k}^+,c)$, with $c\in (\overline{c}^*,\overline{c}^{**})$.
  • Figure 3: Energy curves for Theorem \ref{['thm4']}. Red curves corresponds to $(\lambda_{c,k}^+,c)$, $c\in (c^*,0)$ and blue curves are $(\lambda_{c,k}^-,c)$, $c\in (c^*,\infty)$.
  • Figure 4: Energy curves for Theorem \ref{['thm5']}. Red curves corresponds to $(\lambda_{c,k}^-,c)$, $c\in (c^*,0)$ and blue curves are $(\lambda_{c,k}^+,c)$, $c\in (c^*,\infty)$.
  • Figure 5: Energy curves from Theorems \ref{['thmapp1']} and \ref{['thmapp1.1']} under the conditions $\mathcal{A}^+\cap \mathcal{B}^+ \neq \emptyset$ and $\mathcal{A}^-\cap \mathcal{B}^+ \neq \emptyset$. We assume here that $c^*=\overline{c}^*$ and $c^{**}=\overline{c}^{**}$ for the sake of simplicity.
  • ...and 1 more figures

Theorems & Definitions (72)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 62 more