Existence of positive ground state solutions for the coupled Choquard system with potential

In this paper, we study the following coupled Choquard system in ${\mathbb{R}}^N$: where $\alpha \in (0,N)$ and , in which $2_{\ast }^{\alpha }$ denotes $\frac{N+\alpha }{N-2}$ if $N\ge 3$ and $2_{\ast }^{\alpha }:=\infty$ if $N=1,2$. The function $I_{\alpha }$ is a Riesz potential. By using Nehari manifold method, we obtain the existence of a positive ground state solution in the case of bounded potential and periodic potential, respectively. In particular, the nonlinear term includes the well-studied case $p=q$ and $u(x)=v(x)$, and the less-studied case $p\ne q$ and $u(x)\ne v(x)$. Moreover, it seems to be the first existence result for the case $p\ne q$.     

Existence of ground state sign‐changing solutions for p‐Laplacian equations of Kirchhoff type

In this paper, we prove the existence of ground state sign‐changing solutions for the following class of elliptic equation: where , and K(x) are positive continuous functions. Firstly, we obtain one ground state sign‐changing solution u_b by using some new analytical skills and non‐Nehari manifold method. Furthermore, the energy of u_b is strictly larger than twice that of the ground state solutions of Nehari type. We also establish the convergence property of u_b as the parameter b↘0. Copyright © 2017 John Wiley & Sons, Ltd.     

Existence of the non-radially symmetric ground state for p-Laplacian equations involving Choquard type

We consider some p-Laplacian type equations with sum of nonlocal term and subcritical nonlinearities. We prove the existence of the ground states, which are positive. Because of including p=2, these results extend the results of Li, Ma and Zhang [Nonlinear Analysis: Real World Application 45(2019) 1-25]. When p=2, N=3, by a variant variational identity and a constraint set, we can prove the existence of a non-radially symmetric solution. Moreover, this solution u(x₁, x₂, x₃) is radially symmetric with respect to (x₁, x₂) and odd with respect to x₃.     

Existence of ground state solutions for Kirchhoff‐type problems involving critical Sobolev exponents

In this paper, we study the existence of ground state solutions for a Kirchhoff‐type problem in involving critical Sobolev exponent. With the help of Nehari manifold and the concentration‐compactness principle, we prove that problem admits at least one ground state solution.     

Ground state solutions for an indefinite Kirchhoff type problem with steep potential well

In this paper we study an indefinite Kirchhoff type equation with steep potential well. Under some suitable conditions, the existence and the non-existence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is also explored.     

带有Neumann边界的方程组多个正解存在性

研究带有Neumann边界条件的拟线性方程组的正解问题.在不同参数条件下,主要利用特征值理论和Nehari流形给出了方程组正解的存在性和多解性.这一结论很好的推广了已有的结果.     

Ground state solutions for a non-autonomous nonlinear Schrödinger-KdV system

We study the Schrödinger-KdV system\{\begin{array}{ll}-\mathrm{\Delta }u+{\lambda }_{\mathrm{1}}(x)u=u^{\mathrm{3}}+\beta uv,& u\in H^{\mathrm{1}}({ℝ}^N),\\ -\mathrm{\Delta }v+{\lambda }_{\mathrm{2}}(x)v={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{v}^{\mathrm{2}}+{\displaystyle \frac{\beta }{\mathrm{2}}}{u}^{\mathrm{2}},& v\in H^{\mathrm{1}}({ℝ}^N),\end{array}where N=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{}{\lambda }_i(x)\in C({ℝ}^N,ℝ),{\lim }_{\left|x\right|\to \infty }{\lambda }_i(x)={\lambda }_i(\infty ), and {\lambda }_i(x)\le {\lambda }_i(\infty ),i= 1,2,a.e. x\in {ℝ}^N.We obtain the existence of nontrivial ground state solutions for the above system by variational methods and the Nehari manifold.     

Ground state solutions for a semilinear elliptic problem with a convection term

By a sub-supersolution method and a perturbed argument, we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem −Δu+p(x)q|∇u|=f(x,u), u>0, x∈RN, , where q∈(1,2], for some α∈(0,1), p(x)?0, ∀x∈RN, and f:RN×(0,∞)→[0,∞) is a locally Hölder continuous function which may be singular at zero.     

Ground state solution on a Kirchhoff type equation involving two potentials

With the help of the Nehari manifold, a Kirchhoff type equation involving two potentials was considered. Under new assumptions, a ground state solution was obtained.     

Existence and multiplicity of normalized solutions for a class of fractional Choquard equations

In this paper, we study the existence and multiplicity of solutions with a prescribed L2-norm for a class of nonlinear fractional Choquard equations in RN:(-△)su-λu =(κα*|u|p)|u|p-2u,where N≥3,s∈(0,1),α∈(0,N),p∈(max{1 +(α+2s)/N,2},(N+α)/(N-2s)) and κα(x)=|x|α-N. To get such solutions,we look for critical points of the energy functional I(u) =1/2∫RN|(-△)s/2u|2-1/(2p)∫RN(κα*|u|p)|u|p on the constraints S(c)={u∈Hs(RN):‖u‖L2(RN)2=c},c >0.For the value p∈(max{1+(α+2s)/N,2},(N+α)/(N-2s)) considered, the functional I is unbounded from below on S(c). By using the constrained minimization method on a suitable submanifold of S(c), we prove that for any c>0, I has a critical point on S(c) with the least energy among all critical points of I restricted on S(c). After that,we describe a limiting behavior of the constrained critical point as c vanishes and tends to infinity. Moreover,by using a minimax procedure, we prove that for any c>0, there are infinitely many radial critical points of I restricted on S(c).     

Existence and multiplicity of nodal solutions for Dirichlet problems in upper half strip with holes

In this paper, we consider the existence and multiplicity of nodal solutions of semilinear elliptic equations. We prove that a semilinear elliptic equation in large domains does not admit any least energy nodal (sign-changing) solution and in an upper half strip with m-holes has at least m² 2-nodal solutions.     

Multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent

In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the Ljusternik-Schnirelmann category to prove that the existence of multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent.     

Existence of ground state solutions to Hamiltonian elliptic system with potentials

In this paper, we investigate nonlinear Hamiltonian elliptic system{-?u + b(向量)(x) · ?u +(V(x) + τ)u = K(x)g(v) in R~N,-?v-(向量)b(x)·?v +(V(x) + τ)v = K(x)f(u) in R~N,u(x) → 0 and v(x) → 0 as |x| →∞,where N ≥ 3, τ 0 is a positive parameter and V, K are nonnegative continuous functions,f and g are both superlinear at 0 with a quasicritical growth at infinity. By establishing a variational setting, the existence of ground state solutions is obtained.     

Multiplicity results for a Kirchhoff singular problem involving the fractional p-Laplacian

The aim of this paper is to study the multiplicity of solutions for a Kirchhoff singular problem involving the fractional p-Laplacian operator. Using the concentration compactness principle and Ekeland"s variational principle, we obtain two positive weak solutions.     

Existence of solutions for a nonlinear elliptic problem on a Riemannian manifold

Let (M,g) be a noncompact, connected, orientable smooth N-dimensional Riemannian manifold without boundary. We consider the existence of solutions of problem

(P)