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<p>We use three different techniques to show the existence of positive solutions for the sinh-Gordon equation with a concave term, namely −Δ<i>υ</i> = <i>λυ</i><sup><i>q</i></sup> + sinh(<i>α υ</i><sup><i>m</i></sup>) in some bounded domain Ω with <i>υ</i> = 0 on <i>∂</i>Ω, where <i>λ, α, m</i> &gt; 0 and 0 &lt; <i>q</i> &lt; 1. By means of a topological fixed point method we get a solution <i>υ</i> even though Ω has dimension bigger than two and <i>m</i> &gt; 1. For two-dimensional domains and 0 &lt; <i>m</i> ≤ 2, another solution <i>u</i><sub>0</sub> is a consequence of a Galerkin convergence scheme. In our third approach, for <i>m</i> = 2, we use variational methods together with critical level estimates applied to an energy functional shifted by <i>u</i><sub>0</sub>, allowing us to find a function <i>w</i> such that <i>u</i><sub>0</sub> + <i>w</i> is a true second solution.</p>

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Multiple solutions for a sinh-Gordon equation with a concave term

  • 期刊论文

  • 发表日期：

    2025年9月7日

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  • Anderson L. A. de Araujo,
  • Marcelo Montenegro

摘要

We use three different techniques to show the existence of positive solutions for the sinh-Gordon equation with a concave term, namely −Δυ = λυq + sinh(α υm) in some bounded domain Ω with υ = 0 on ∂Ω, where λ, α, m > 0 and 0 < q < 1. By means of a topological fixed point method we get a solution υ even though Ω has dimension bigger than two and m > 1. For two-dimensional domains and 0 < m ≤ 2, another solution u0 is a consequence of a Galerkin convergence scheme. In our third approach, for m = 2, we use variational methods together with critical level estimates applied to an energy functional shifted by u0, allowing us to find a function w such that u0 + w is a true second solution.