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Existence, multiplicity and concentration for a class of fractional $ p \& q $ Laplacian problems in $ \mathbb{R} ^{N} $

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  • In this work we consider the following class of fractional $p \& q$ Laplacian problems

    $ \begin{equation*} (-\Delta)_{p}^{s}u+ (-\Delta)_{q}^{s}u + V( \varepsilon x) (|u|^{p-2}u + |u|^{q-2}u) = f(u) \mbox{ in } \mathbb{R} ^{N}, \end{equation*} $

    where $ \varepsilon >0 $ is a parameter, $ s\in (0, 1) $, $ 1< p<q<\frac{N}{s} $, $ (-\Delta)^{s}_{t} $, with $ t\in \{p,q\} $, is the fractional $ t $-Laplacian operator, $ V: \mathbb{R} ^{N}\rightarrow \mathbb{R} $ is a continuous potential and $ f: \mathbb{R} \rightarrow \mathbb{R} $ is a $ \mathcal{C} ^{1} $-function with subcritical growth. Applying minimax theorems and the Ljusternik-Schnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that $ \varepsilon $ is sufficiently small.

    Mathematics Subject Classification: Primary: 35R11, 35A15; Secondary: 47G20, 58E05.

    Citation:

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