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

TL;DR

This paper proves the existence of arbitrarily many positive solutions for a nonlinear fractional Laplacian problem, showing how oscillations in the nonlinearity influence solution multiplicity using variational methods.

Contribution

It introduces new results on multiple positive solutions for fractional Laplacian problems with oscillating nonlinearities, expanding understanding of solution multiplicity in such equations.

Findings

Arbitrarily many positive solutions exist under oscillating nonlinearities.

Solution count is influenced by the parameters p and λ.

Various properties of solutions are characterized in specific norms.

Abstract

We establish the existence and multiplicity of positive solutions to the problems involving the fractional Laplacian: \begin{equation*} \left\{\begin{array}{lll} &(-\Delta)^{s}u=\lambda u^{p}+f(u),\,\,u>0 \quad &\mbox{in}\,\,\Omega,\\ &u=0\quad &\mbox{in}\,\,\mathbb{R}^{N}\setminus\Omega,\\ \end{array}\right. \end{equation*} where $\Omega\subset \mathbb{R}^{N}$ $(N\geq 2)$ 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 $X^{s}_{0}(\Omega)$-norms.