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In this paper, we consider the existence of nodal solutions with two bubbles to the slightly subcritical problem with the fractional Laplacian begin{equation*} left{aligned &(-Delta)^su=|u|^{p-1-varepsilon}u mbox{in} Omega &u=0 mbox{on} partialOmega, endaligned right. end{equation*} where $Omega$ is a smooth bounded domain in $mathbb R^N$, $N>2s$, $0<s<1$, $ p=frac{N+2s}{N-2s}$ and $varepsilon>0$ is a small parameter, which can be seen as a nonlocal analog of the results of Bartsch, Micheletti and Pistoia (2006) cite{Bartsch1}.
The paper is concerned with the slightly subcritical elliptic problem with Hardy term [ left{ begin{aligned} -Delta u-mufrac{u}{|x|^2} &= |u|^{2^{ast}-2-epsilon}u &&quad text{in } Omega, u &= 0&&quad text{on } partialOmega, end{aligned} right. ] in
In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation begin{equation*} (-Delta)^{frac{alpha}{2}} u=lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{alpha}-2}u, quadtext{in},,Omega, u=0,text{on},,partialOmega, end
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=0quad &mbox{in},,mathbb{R}^
Nodal solutions of a parametric (p_1,p_2)-Laplacian system, with Neumann boundary conditions, are obtained by chiefly constructing appropriate sub-super-solution pairs.
It is well known that a single nonlinear fractional Schrodinger equation with a potential $V(x)$ and a small parameter $varepsilon $ may have a positive solution that is concentrated at the nondegenerate minimum point of $V(x)$. In this paper, we can