대한수학회지 (Journal of the Korean Mathematical Society)

  • 제57권6호
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  • Pages.1451-1470
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  • 2020
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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EXISTENCE, MULTIPLICITY AND REGULARITY OF SOLUTIONS FOR THE FRACTIONAL p-LAPLACIAN EQUATION

  • Kim, Yun-Ho (Department of Mathematics Education Sangmyung University)
  • 투고 : 2019.10.15
  • 심사 : 2020.02.13
  • 발행 : 2020.11.01
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초록

We are concerned with the following elliptic equations: $$\{(-{\Delta})^s_pu={\lambda}f(x,u)\;{\text{in {\Omega}}},\\u=0\;{\text{on {\mathbb{R}}^N{\backslash}{\Omega}},$$ where λ are real parameters, (-∆)sp is the fractional p-Laplacian operator, 0 < s < 1 < p < + ∞, sp < N, and f : Ω × ℝ → ℝ satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters λ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity f has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in L∞(Ω) of any possible weak solution by applying the bootstrap argument.

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