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http://dx.doi.org/10.4134/JKMS.j190693

EXISTENCE, MULTIPLICITY AND REGULARITY OF SOLUTIONS FOR THE FRACTIONAL p-LAPLACIAN EQUATION  

Kim, Yun-Ho (Department of Mathematics Education Sangmyung University)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.6, 2020 , pp. 1451-1470 More about this Journal
Abstract
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.
Keywords
Fractional p-Laplacian; weak solution; critical points; variational method;
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