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<p>We prove some abstract multiplicity theorems that can be used to obtain multiple nontrivial solutions of critical growth <i>p</i>-Laplacian and (<i>p, q</i>)-Laplacian type problems. We show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter <i>λ</i> &gt; 0. In particular, the number of solutions goes to infinity as <i>λ</i> → ∞. Moreover, we give an explicit lower bound on <i>λ</i> in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues constructed using the ℤ<sub>2</sub>-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.</p>

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Abstract multiplicity theorems and applications to critical growth problems

  • 期刊论文

  • 发表日期：

    2025年9月7日

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  • Kanishka Perera

摘要

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 → ∞. 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 ℤ2-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.