Arbitrary many positive solutions for a nonlinear problem involving the fractional Laplacian

Abstract

We establish the existence and multiplicity of positive solutions to the problems involving the fractional Laplacian: \begin{equation*} \left\{\begin{array}{lll} &(-Δ)^{s}u=λu^{p}+f(u),\,\,u>0 \quad &\mbox{in}\,\,Ω,\\ &u=0\quad &\mbox{in}\,\,\mathbb{R}^{N}\setminusΩ,\\ \end{array}\right. \end{equation*} where $\Omega \subset {\mathbb{R}}^N$ $(N\ge 2)$ is a bounded smooth domain, $s\in (0,1)$, $p>0$, $\lambda \in \mathbb{R}$ and $(-\Delta {)}^s$ stands for the fractional Laplacian. When $f$ oscillates near the origin or at infinity, via the variational argument we prove that the problem has arbitrarily many positive solutions and the number of solutions to problem is strongly influenced by $u^p$ and $\lambda$. Moreover, various properties of the solutions are also described in $L^{\infty }$- and $X_0^s(\Omega )$-norms.

29 pages

References in corpus (4)

• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian
• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian
• Positive solutions to some asymptotically linear fractional Schrodinger equations
• Three solutions for a fractional elliptic problems with critical and supercritical growth