Large number of bubble solutions for a perturbed fractional Laplacian equation
Large number of bubble solutions for a perturbed fractional Laplacian equation
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发表时间：
2019-08
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通讯作者：
Chunhua Wang;Suting Wei
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作者：
Chunhua Wang;Suting Wei
This paper deals with the following nonlinear perturbed fractional Laplacian equation $$(-\Delta)^s u = K(|y'|,y'')u^{\frac{N+2s}{N-2s}\pm\epsilon},\,\,u>0,\,\,u\in D^{1,s}(\mathbb{R}^N),$$ where $0 0$ is a small parameter and $K(y)$ is nonnegative and bounded. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\geq 4,\max\{\frac{N+1-\sqrt{N^{2}-2N+9}}{4},\frac{3-\sqrt{N^{2}-6N+13}}{2}\} 0$ and $K(r_0, y_0'')>0,$ then the above problem has large number of bubble solutions if $\epsilon>0$ is small enough. Also there exist solutions whose functional energy is in the order $\epsilon^{-\frac{N-2s-2}{(N-2s)^{2}}}$. Here, instead of estimating directly the derivatives of the reduced functional, we apply some local Pohozaev identities to locate the concentration points of the bubble solutions. Moreover, the concentration points of the bubble solutions include a saddle point of $K(y)$.
