[Paper Review] Abstract multiplicity theorems and applications to critical growth problems

Kanishka Perera|arXiv (Cornell University)|Aug 15, 2023

Nonlinear Partial Differential Equations4 citations

TL;DR

This paper establishes abstract multiplicity theorems using the $Η_2$-cohomological index to prove that critical growth $p$-Laplacian and $(p,q)$-Laplacian problems possess arbitrarily many nontrivial solutions for all sufficiently large parameter values $\lambda > 0$. The key contribution is an explicit lower bound on $\lambda$ ensuring $m$ distinct pairs of solutions, derived from a sequence of eigenvalues based on the cohomological index, with the number of solutions tending to infinity as $\lambda \to \infty$. The results are new even in the semilinear case $p=2$.

ABSTRACT

We prove some abstract multiplicity theorems that can be used to obtain multiple nontrivial solutions of critical growth $p$-Laplacian and $(p,q)$-Laplacian type problems. We show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter $λ> 0$. In particular, the number of solutions goes to infinity as $λ o \infty$. Moreover, we give an explicit lower bound on $λ$ in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues constructed using the ${\mathbb Z}_2$-cohomological index. This is a consequence of the fact that our abstract multiplicity results make essential use of the piercing property of the cohomological index, which is not shared by the genus. Our result for the $p$-Laplacian is new even in the semilinear case $p = 2$.

Motivation & Objective

• Address the long-standing open question of whether critical growth $p$-Laplacian and $(p,q)$-Laplacian problems admit multiple solutions for large parameter values $\lambda > 0$.
• Develop abstract multiplicity theorems that leverage the $\mathbb{Z}_2$-cohomological index to ensure multiple nontrivial solutions.
• Establish an explicit lower bound on $\lambda$ to guarantee a given number $m$ of distinct pairs of nontrivial solutions.
• Extend previous results on the Brézis-Nirenberg and García Azorero-Peral problems by proving the number of solutions tends to infinity as $\lambda \to \infty$.
• Provide a new framework using the piercing property of the cohomological index, which is stronger than the genus-based approach.

Proposed method

• Introduce abstract multiplicity theorems (Theorems 2.1, 2.4, 2.5) based on the $\mathbb{Z}_2$-cohomological index, which ensures the existence of multiple critical points via the piercing property.
• Define a sequence of eigenvalues $\lambda_m$ using the $\mathbb{Z}_2$-cohomological index, which is unbounded and increases with $m$, serving as a key parameter in the lower bound for $\lambda$.
• Apply the abstract theorems to the $p$-Laplacian problem $-\Delta_p u = \lambda |u|^{r-2}u + |u|^{p^* - 2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, with $1 < p < N$, $p < r < p^*$.
• Use Sobolev and Hölder inequalities to control the nonlinear terms in the energy functional, ensuring the $(\text{PS})_c$ condition holds for $c < \frac{1}{N} S^{N/p}$.
• Derive an explicit lower bound for $\lambda$ in terms of $\lambda_m$, the volume $|\Omega|$, the best Sobolev constant $S$, and the exponents $p, r, p^*$, ensuring $m$ distinct pairs of solutions.
• Apply the same framework to the $(p,q)$-Laplacian problem $-\Delta_p u - \Delta_q u = \lambda |u|^{r-2}u + |u|^{p^* - 2}u$, with $1 < q < p < N$, $p \leq r < p^*$, and obtain analogous multiplicity results.

Experimental results

Research questions

• RQ1 Can abstract multiplicity theorems based on the $\mathbb{Z}_2$-cohomological index be used to prove the existence of arbitrarily many nontrivial solutions for critical growth $p$-Laplacian problems?
• RQ2 Does the number of solutions of the critical $p$-Laplacian problem $-\Delta_p u = \lambda |u|^{r-2}u + |u|^{p^* - 2}u$ in $\Omega$ tend to infinity as $\lambda \to \infty$?
• RQ3 What is an explicit lower bound on $\lambda$ that guarantees $m$ distinct pairs of nontrivial solutions for such problems?
• RQ4 Can the abstract framework be extended to $(p,q)$-Laplacian problems with critical growth, and does it yield the same multiplicity behavior?
• RQ5 Is the cohomological index approach superior to the genus-based method in ensuring multiplicity, particularly due to its piercing property?

Key findings

• The number of nontrivial solutions for the critical $p$-Laplacian problem $-\Delta_p u = \lambda |u|^{r-2}u + |u|^{p^* - 2}u$ in $\Omega$ tends to infinity as $\lambda \to \infty$, even in the semilinear case $p=2$.
• For any $m \in \mathbb{N}$, the $p$-Laplacian problem has $m$ distinct pairs of nontrivial solutions for all $\lambda$ greater than the explicit bound $\lambda > r|\Omega|^{r/p - 1} \sup_{\tau > 0} \left[ \frac{\lambda_m}{p\tau^{r-p}} - \frac{S^{N/p}}{N\tau^r} - \frac{1}{p^* |\Omega|^{p/(N-p)}} \tau^{p^* - r} \right]$.
• The same multiplicity result holds for the $(p,q)$-Laplacian problem $-\Delta_p u - \Delta_q u = \lambda |u|^{r-2}u + |u|^{p^* - 2}u$, with $1 < q < p < N$, $p \leq r < p^*$, ensuring $m$ distinct pairs of solutions for large $\lambda$.
• The lower bound on $\lambda$ increases with $m$ because $\lambda_m \to \infty$ as $m \to \infty$, confirming that the number of solutions grows without bound as $\lambda$ increases.
• The cohomological index-based eigenvalue sequence $\lambda_m$ is essential to the proof, as its piercing property enables the construction of multiple critical points, a feature not shared by the genus.
• The abstract multiplicity theorems are new and applicable beyond the specific problems, providing a general framework for proving multiple solutions in critical growth problems via topological index theory.