Existence and Multiplicity of Solutions to a Class of Fractional p -Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ? N

Yun-Ho Kim
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Yun-Ho Kim: Department of Mathematics Education, Sangmyung University, Seoul 03016, Korea

Mathematics, 2020, vol. 8, issue 10, 1-17

Abstract: We are concerned with the following elliptic equations: ( − Δ ) p s v + V ( x ) | v | p − 2 v = λ a ( x ) | v | r − 2 v + g ( x , v ) i n R N , where ( − Δ ) p s is the fractional p -Laplacian operator with 0 < s < 1 < r < p < + ∞ , s p < N , the potential function V : R N → ( 0 , ∞ ) is a continuous potential function, and g : R N × R → R satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem.

Keywords: fractional p -Laplacian; variational methods; critical point theory (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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