Infinitely many radial positive solutions for nonlocal problems with lack of compactness

• Main Authors: Fen Zhou, Zifei Shen, Vicenţiu Rădulescu
• Format: Article
• Language: English
• Published: University of Szeged 2021-04-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: fractional laplacian; radial solution; lack of compactness; lyapunov–schmidt reduction method
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=9000

We are concerned with the qualitative and asymptotic analysis of solutions to the nonlocal equation $$ (-\Delta)^su+V(|z|)u=Q(|z|)u^p\quad \text{in} \ \mathbb{R}^{N},$$ where $N\geq 3,\ 0<s<1$, and $1<p<\frac{2N}{N-2s}$. As $r\to\infty$, we assume that the potentials $V(r)$ and $Q(r)$ behave as \begin{align*} V(r)&=V_0+\frac{a_1}{r^\alpha}+O\left(\frac{1}{r^{\alpha+\theta_1}}\right)\\ Q(r)&=Q_0+\frac{a_2}{r^\beta}+O\left(\frac{1}{r^{\beta+\theta_2}}\right) \end{align*} where $a_1,\ a_2 \in{\mathbb{R}}$, $\alpha,\, \beta>\frac{N+2s}{N+2s+1}$, and $\theta_1,\ \theta_2>0,\ V_0,\ Q_0>0$. Under various hypotheses on $a_1,\, a_2,\, \alpha,\, \beta$, we establish the existence of infinitely many radial solutions. A key role in our arguments is played by the Lyapunov–Schmidt reduction method.