Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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MULTIPLICITY RESULTS OF CRITICAL LOCAL EQUATION RELATED TO THE GENUS THEORY

  • Mohsen Alimohammady (Department of Mathematics Faculty of Mathematical Sciences University of Mazandarn) ;
  • Asieh Rezvani (Department of Mathematics Qaemshahr Branch, Islamic Azad University) ;
  • Cemil Tunc (Department of Mathematics Faculty of sciences Van Yuzuncu Yil university)
  • Received : 2022.10.12
  • Accepted : 2023.04.26
  • Published : 2023.10.31
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Abstract

Using variational methods, Krasnoselskii's genus theory and symmetric mountain pass theorem, we introduce the existence and multiplicity of solutions of a parameteric local equation. At first, we consider the following equation $$\{-div[a(x,{\mid}{\nabla}u{\mid}){\nabla}u]\,=\,{\mu}(b(x){\mid}u{\mid}^{s(x)-2}-{\mid}u{\mid}^{r(x)-2})u\;in\;{\Omega},\\u\,=0\,on\;{\partial}{\Omega},$$ where Ω⊆ ℝN is a bounded domain, µ is a positive real parameter, p, r and s are continuous real functions on ${\bar{\Omega}}$ and a(x, ξ) is of type |ξ|p(x)-2. Next, we study boundedness and simplicity of eigenfunction for the case a(x, |∇u|)∇u = g(x)|∇u|p(x)-2∇u, where g ∈ L∞(Ω) and g(x) ≥ 0 and the case $a(x,\,{\mid}{\nabla}u{\mid}){{\nabla}u}\,=\,(1\,+\,{\nabla}u{\mid}^2)^{\frac{p(x)-2}{2}}{\nabla}u$ such that p(x) ≡ p.

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Acknowledgement

This work was financially supported by KRF 2003-041-C20009.

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