No Arabic abstract
This paper is devoted to study the existence and multiplicity solutions for the nonlinear Schrodinger-Poisson systems involving fractional Laplacian operator: begin{equation}label{eq*} left{ aligned &(-Delta)^{s} u+V(x)u+ phi u=f(x,u), quad &text{in }mathbb{R}^3, &(-Delta)^{t} phi=u^2, quad &text{in }mathbb{R}^3, endaligned right. end{equation} where $(-Delta)^{alpha}$ stands for the fractional Laplacian of order $alphain (0,,,1)$. Under certain assumptions on $V$ and $f$, we obtain infinitely many high energy solutions for eqref{eq*} without assuming the Ambrosetti-Rabinowitz condition by using the fountain theorem.
We prove a nonlinear Poisson type formula for the Schrodinger group. Such a formula had been derived in a previous paper by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, energy-critical nonlinearities are allowed.
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.
We study positive solutions to the fractional Lane-Emden system begin{equation*} tag{S}label{S} left{ begin{aligned} (-Delta)^s u &= v^p+mu quad &&text{in } Omega (-Delta)^s v &= u^q+ u quad &&text{in } Omega u = v &= 0 quad &&text{in } Omega^c={mathbb R}^N setminus Omega, end{aligned} right. end{equation*} where $Omega$ is a $C^2$ bounded domains in ${mathbb R}^N$, $sin(0,1)$, $N>2s$, $p>0$, $q>0$ and $mu,, u$ are positive measures in $Omega$. We prove the existence of the minimal positive solution of the above system under a smallness condition on the total mass of $mu$ and $ u$. Furthermore, if $p,q in (1,frac{N+s}{N-s})$ and $0 leq mu,, uin L^r(Omega)$ for some $r>frac{N}{2s}$ then we show the existence of at least two positive solutions of the above system. We also discuss the regularity of the solutions.
We are concerned with existence results for a critical problem of Brezis-Nirenberg Type involving an integro-differential operator. Our study includes the fractional Laplacian. Our approach still applies when adding small singular terms. It hinges on appropriate choices of parameters in the mountain-pass theorem
We study the existence and multiplicity of nonnegative solutions, as well as the behaviour of corresponding parameter-dependent branches, to the equation $-Delta u = (1-u) u^m - lambda u^n$ in a bounded domain $Omega subset mathbb{R}^N$ endowed with the zero Dirichlet boundary data, where $0<m leq 1$ and $n>0$. When $lambda > 0$, the obtained solutions can be seen as steady states of the corresponding reaction-diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination. In addition to the main new results, we formulate a few relevant conjectures.