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

Nguyen Van Thin Annales Polonici Mathematici MSC: Primary 35J60, 35R11, 35S05. DOI: 10.4064/ap181013-13-2 Opublikowany online: 19 July 2019


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 $M$.


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

Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Przepisz kod z obrazka

Odśwież obrazek

Odśwież obrazek