Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPositive solutions for a coupled nonlinear Kirchhoff-type system with vanishing potentials
70
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we consider the strongly coupled nonlinear Kirchhoff-type system with vanshing potentials: [begin{cases} -left(a_1+b_1int_{mathbb{R}^3}| abla u|^2dxright)Delta u+lambda V(x)u=frac{alpha}{alpha+beta}|u|^{alpha-2}u|v|^{beta},&xinmathbb{R}^3, -left(a_2+b_2int_{mathbb{R}^3}| abla v|^2dxright)Delta v+lambda W(x)v=frac{beta}{alpha+beta}|u|^{alpha}|v|^{beta-2}v,&xinmathbb{R}^3, u,vin mathcal{D}^{1,2}(mathbb{R}^3), end{cases}] where $a_i>0$ are constants, $lambda,b_i>0$ are parameters for $i=1,2$, $alpha,beta>1$ satisfy $alpha+betale4$, the nonlinear term $F(x,u,v)=|u|^alpha|v|^beta$ is not 4-superlinear at infinity, $V(x)$, $W(x)$ are nonnegative continuous potentials. By establishing some new estimates and truncation technique, we obtain the existence of positive vector solutions for the above system when $b_1+b_2$ small and $lambda$ large. Moreover, the asymptotic behavior of these vector solutions is also explored as $textbf{b}=(b_1,b_2)to bf{0}$ and $lambdatoinfty$. In particular, our results extend some known ones in previous papers that only deals with the case where $alpha,beta>2$ with $alpha+beta<6$.
rate research
Read More
We study the coupled Hartree system $$ left{begin{array}{ll} -Delta u+ V_1(x)u =alpha_1big(|x|^{-4}ast u^{2}big)u+beta big(|x|^{-4}ast v^{2}big)u &mbox{in} mathbb{R}^N,[1mm] -Delta v+ V_2(x)v =alpha_2big(|x|^{-4}ast v^{2}big)v +betabig(|x|^{-4}ast u^{2}big)v &mbox{in} mathbb{R}^N, end{array}right. $$ where $Ngeq 5$, $beta>max{alpha_1,alpha_2}geqmin{alpha_1,alpha_2}>0$, and $V_1,,V_2in L^{N/2}(mathbb{R}^N)cap L_{text{loc}}^{infty}(mathbb{R}^N)$ are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with $V_1=V_2=0$ we employ moving sphere arguments in integral form to classify positive solutions and to prove the uniqueness of positive solutions up to translation and dilation, which is of independent interest. Then using the uniqueness property, we establish a nonlocal version of the global compactness lemma and prove the existence of a high energy positive solution for the system assuming that $|V_1|_{L^{N/2}(mathbb{R}^N)}+|V_2|_{L^{N/2}(mathbb{R}^N)}>0$ is suitably small.
In this article, we establish the existence of solutions to the fractional $p-$Kirchhoff type equations with a generalized Choquard nonlinearities without assuming the Ambrosetti-Rabinowitz condition.
This paper deals with the existence of positive solutions for the nonlinear system q(t)phi(p(t)u_{i}(t)))+f^{i}(t,textbf{u})=0,quad 0<t<1,quad i=1,2,...,n. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here $textbf{u}=(u_{1},...,u_{n})$ and $f^{i}, i=1,2,...,n$ are continuous and nonnegative functions, $p(t), q(t)hbox{rm :} [0,1]to (0,oo)$ are continuous functions. Moreover, we characterize the eigenvalue intervals for (q(t)phi(p(t)u_{i}(t)))+lambda h_{i}(t)g^{i} (textbf{u})=0, quad 0<t<1,quad i=1,2,...,n. The proof is based on a well-known fixed point theorem in cones.
In this paper, we study a class of Schr{o}dinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the `charge function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when `charge function is radially symmetric, we establish the existence of three positive solutions and the symmetry breaking of ground state solutions.
In this paper, we consider the following Kirchhoff type equation $$ -left(a+ bint_{R^3}| abla u|^2right)triangle {u}+V(x)u=f(u),,,xinR^3, $$ where $a,b>0$ and $fin C(R,R)$, and the potential $Vin C^1(R^3,R)$ is positive, bounded and satisfies suitable decay assumptions. By using a new perturbation approach together with a new version of global compactness lemma of Kirchhoff type, we prove the existence and multiplicity of bound state solutions for the above problem with a general nonlinearity. We especially point out that neither the corresponding Ambrosetti-Rabinowitz condition nor any monotonicity assumption is required for $f$. Moreover, the potential $V$ may not be radially symmetry or coercive. As a prototype, the nonlinear term involves the power-type nonlinearity $f(u) = |u|^{p-2}u$ for $pin (2, 6)$. In particular, our results generalize and improve the results by Li and Ye (J.Differential Equations, 257(2014): 566-600), in the sense that the case $pin(2,3]$ is left open there.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
