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Register a new userTwo-bubble nodal solutions for slightly subcritical Fractional Laplacian
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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}.
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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 a bounded domain $Omegasubsetmathbb{R}^N$ with $0inOmega$, in dimensions $Nge7$. We prove the existence of multi-bubble nodal solutions that blow up positively at the origin and negatively at a different point as $epsilonto0$ and $mu=epsilon^alpha$ with $alpha>frac{N-4}{N-2}$. In the case of $Omega$ being a ball centered at the origin we can obtain solutions with up to $5$ bubbles of different signs. We also obtain nodal bubble tower solutions, i.e. superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order. The asymptotic shape of the solutions is determined in detail.
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{equation*} where $Omegasubset mathbb{R}^{N}(Ngeq 2)$ is a bounded domain with smooth boundary, $0<alpha<2$, $(-Delta)^{frac{alpha}{2}}$ stands for the fractional Laplacian operator, $fin C(Omegatimesmathbb{R},mathbb{R})$ may be sign changing and $lambda$ is a positive parameter. We will prove that there exists $lambda_{*}>0$ such that the problem has at least two positive solutions for each $lambdain (0,,,lambda_{*})$. In addition, the concentration behavior of the solutions are investigated.
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}^{N}setminusOmega, end{array}right. end{equation*} where $Omegasubset mathbb{R}^{N}$ $(Ngeq 2)$ is a bounded smooth domain, $sin (0,1)$, $p>0$, $lambdain mathbb{R}$ 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.
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 find two different positive solutions for two weakly coupled fractional Schrodinger systems with a small parameter $varepsilon $ and two potentials $V_{1}(x)$ and $V_{2}(x)$ having the same minimum point are concentrated at the same point minimum point of $V_{1}(x)$ and $V_{2}left(xright) $. In fact that by using the energy estimates, Nehari manifold technique and the Lusternik-Schnirelmann theory of critical points, we obtain the multiplicity results for a class of fractional Laplacian system. Furthermore, the existence and nonexistence of least energy positive solutions are also explored.
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