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Existence of solutions to some fractional equations involving the Bessel operator in $\mathbb R^N$

Volume 122 / 2019

Nguyen Van Thin Annales Polonici Mathematici 122 (2019), 267-300 MSC: Primary 35J60, 35R11, 35S05. DOI: 10.4064/ap181013-13-2 Published online: 19 July 2019

Abstract

The aim of this paper is to study the existence of solutions to concave-convex nonlinear equations involving the Bessel operator in $\mathbb R^N$: \begin{gather*} M\Bigl(\int_{\mathbb R^{N}}|(I-\varDelta)^{\alpha/2}u|^2\,dx\Bigr)(I-\varDelta)^{\alpha}u+ \lambda V(x)u =\gamma f(x,u)+\mu \xi(x)|u|^{p-2}u,\\ M\Bigl(\int_{\mathbb R^{N}}|(I-\varDelta)^{\alpha/2}u|^2\,dx+\lambda \int_{\mathbb R^N}V(x)|u|^2dx\Bigr)((I-\varDelta)^{\alpha}u+\lambda V(x)u) =\gamma f(x,u)+\mu \xi(x)|u|^{p-2}u,\end{gather*} where $\lambda, \gamma, \mu$ are positive parameters, $\xi:\mathbb R^N\to (0,\infty)$ belongs to $L^{2/(2-p)}(\mathbb R^N),$ $1 \lt p \lt 2$, $M:[0, \infty)\to (0, \infty)$ is a continuous function, $V:\mathbb R^N\to \mathbb R^{+}$ is a continuous function, $0 \lt \alpha \lt 1$ with $2\alpha \lt N,$ and $f$ is a continuous function on $\mathbb R^N\times \mathbb R$ which does not satisfy the Ambrosetti–Rabinowitz condition. By using the Mountain Pass Theorem and the variational method, we obtain the existence of solutions to the above equations. Furthermore, if $M$ is degenerate ($M(0)=0$) and $f$ satisfies the Ambrosetti–Rabinowitz condition, we investigate the existence of solutions of that equation without the concave-convex nonlinearity in case its right side contains the critical exponent $2_{\alpha}^{*}=2N/(N-2\alpha).$ The difficulty lies in the lack of compactness and the degeneracy of $M$.

Authors

  • Nguyen Van ThinShandong University
    Department of Mathematics
    Jinan City, Shandong, P.R. China
    and
    Thai Nguyen University of Education
    Department of Mathematics
    Thai Nguyen City, Thai Nguyen, Viet Nam
    e-mail

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