Liouville theorem for Choquard equation with finite Morse indices

In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation-△u=∫RN|u(y)|p|x-y|αdy|u(x)|p-2u(x) in RN where N ≥ 3, 0 α min{4, N}. Suppose that 2 p (2 N-α)/(N-2),we will show that this problem does not possess nontrivial solution with finite Morse index. While for p=(2 N-α)/(N-2),if i(u) ∞, then we have ∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~α dxdy ∞ and ∫_RN|▽u|~2 dx=∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~αdxdy.     

Existence of multiple solutions for a fractional p-Laplacian system with concave-convex term

In this article, we study the existence of multiple solutions for the following system driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions{(-?)_p~su = a(x)|u|~(q-2) u +2α/α + βc(x)|u|~(α-2) u|v|~β, in ?,(-?)_p~sv = b(x)|v|~(q-2) v +2β/α + βc(x)|u|α|v|~(β-2) v, in ?,u = v = 0, in Rn\?,(0.1) where Ω is a smooth bounded domain in Rn, n ps with s ∈(0,1) fixed, a(x), b(x), c(x) ≥ 0 and a(x),b(x),c(x) ∈L∞(Ω), 1 q p and α,β 1 satisfy pα + βp*,p* =np/n-ps.By Nehari manifold and fibering maps with proper conditions, we obtain the multiplicity of solutions to problem(0.1).?????     

Multiplicity and concentration behaviour of positive solutions for Schrödinger-Kirchhoff type equations involving the p-Laplacian in RN

In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of Schrödinger-Kirchhoff type

$-{\in }^{p}M\left({\in }^{p-N}{\int }_{{\mathbb{R}}^N}{\left|?u\right|}^p\right){\Delta }_{p}u+V\left(x\right){\left|u\right|}^{p-2}u=f\left(u\right)$

in ${\mathbb{R}}^N$, where Δ_p is the p-Laplacian operator, 1 < p < N, M: ${\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ and V: ${\mathbb{R}}^{\text{N}}\to {\mathbb{R}}^{+}$ are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik-Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.     

Quasilinear elliptic problems with critical exponents and Hardy terms

Let $\Omega \ni 0$ be an open bounded domain, $\Omega \subset R^N(N>p^2)$. We are concerned with the multiplicity of positive solutions of $-{\mathrm{\Delta }}_{p}u-\mu \frac{|u{|}^{p-2}u}{|x{|}^p}=\lambda |u{|}^{p-2}u+Q(x)|u{|}^{p^{*}-2}u,u\in W_0^{1,p}(\Omega )\text{,}$where $-{\mathrm{\Delta }}_{p}u=-\mathrm{div}(|\nabla u{|}^{p-2}\nabla u),1<p<N,p^{*}=\frac{\mathit{Np}}{N-p},0<\mu <{\left(\frac{N-p}{p}\right)}^p,\lambda >0$and $Q(x)$ is a nonnegative function on $\overline{\Omega }$. By investigating the effect of the coefficient of the critical nonlinearity, we, by means of variational method, prove the existence of multiple positive solutions.     

EXISTENCE AND BLOW-UP BEHAVIOR OF CONSTRAINED MINIMIZERS FOR SCHRDINGER-POISSON-SLATER SYSTEM

In this article,we study constrained minimizers of the following variational problem e(p):=inf{u∈H1(R3),||u||22=p}E(u),p〉0,where E(u)is the Schrdinger-Poisson-Slater(SPS)energy functional E(u):=1/2∫R3︱▽u(x)︱2dx-1/4∫R3∫R3u2(y)u2(x)/︱x-y︱dydx-1/p∫R3︱u(x)︱pdx in R3 and p∈(2,6).We prove the existence of minimizers for the cases 2p10/3,ρ0,and p=10/3,0ρρ~*,and show that e(ρ)=-∞for the other cases,whereρ~*=||φ||_2~2 andφ(x)is the unique(up to translations)positive radially symmetric solution of-△u+u=u~(7/3)in R~3.Moreover,when e(ρ~*)=-∞,the blow-up behavior of minimizers asρ↗ρ~*is also analyzed rigorously.     

Multiplicity of positive solutions for a class of concave-convex elliptic equations with critical growth

In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,{-△u=g(x)|u|2*-2u+λf(x)|u|q-2u,x∈Ω,u=0,x∈■Ω where Ω■R~N(N≥3) is an open bounded domain with smooth boundary, 1 q 2, λ 0.2*=2 N/(N-2)is the critical Sobolev exponent,f∈L2~*/(2~*-q)(Ω)is nonzero and nonnegative,and g ∈ C(■) is a positive function with k local maximum points. By the Nehari method and variational method,k+1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75(2012) 2660-2671].     

Bound states of schrödinger equations with electromagnetic fields and vanishing potentials

We study the bound states to nonlinear Schrödinger equations with electro-magnetic fields $\text{i}h\frac{?\psi }{?t}={(\frac{h}{i}?-A(x))}^2\psi +V(x)\psi -K(x){|}^{p-1}\psi =0,\text{on}{?}_{+}\times {?}^N$. Let $G(x)={[V(x)]}^{\frac{p+1}{p-1}-\frac{N}{2}}{[K(x)]}^{-\frac{2}{p-1}}$ and suppose that G(x) has k local minimum points. For h > 0 small, we find multi-bump bound states ψh(x,t) = _e^?lEt/^h_Uh(χ) with Uh concentrating at the local minimum points of G(x) simultaneously as h → 0. The potentials V(x) and K(x) are allowed to be either compactly supported or unbounded at infinity.     

Existence of standing waves of nonlinear Schrödinger equations with potentials vanishing at infinity

For a singularly perturbed nonlinear elliptic equation ${\epsilon }^2\mathrm{\Delta }u?V(x)u+u^p=0$, $x\in {\mathbb{R}}^N$, we prove the existence of bump solutions concentrating around positive critical points of V when nonnegative V is not identically zero for $p\in (\frac{N}{N?2},\frac{N+2}{N?2})$ or nonnegative V satisfies ${\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}_{|x|\to \infty }V(x){|x|}^2\log |x|>0$ for $p=\frac{N}{N?2}$.     

Global well-posedness and blow-up for the hartree equation

For 2 γ min{4, n}, we consider the focusing Hartree equation iu_t+ △u +(|x|~(-γ)* |u|~2)u = 0, x ∈ R~n.(0.1)Let M [u] and E [u] denote the mass and energy, respectively, of a solution u, and Q be the ground state of-△ Q + Q =(|x|~(-γ)* |Q|~2)Q. Guo and Wang [Z. Angew. Math.Phy.,2014] established a dichotomy for scattering versus blow-up for the Cauchy problem of(0.1) if M [u]~(1-s_c)E [u]~(s_c) M [Q]~(1-s_c)E [Q]~(s_c)(s_c=(γ-2)/2). In this paper, we consider the complementary case M [u]~(1-s_c)E [u]~(s_c)≥ M [Q]~(1-s_c)E [Q]~(s_c) and obtain a criteria on blow-up and global existence for the Hartree equation(0.1).     

Sharp well-posedness of the cauchy problem for the higher-order dispersive equation

This current paper is devoted to the Cauchy problem for higher order dispersive equation u_t+ ?_x~(2n+1)u = ?_x(u?_x~nu) + ?_x~(n-1)(u_x~2), n ≥ 2, n ∈ N~+.By using Besov-type spaces, we prove that the associated problem is locally well-posed in H~(-n/2+3/4,-1/(2n))(R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H~(s,a)(R) with s -n/2+3/4 and all a∈R.     

Positive solutions for a weighted fractional system

In this article, we study positive solutions to the system{A_αu(x) = C_(n,α)PV∫_(Rn)(a1(x-y)(u(x)-u(y)))/(|x-y|~(n+α))dy = f(u(x), B_βv(x) = C_(n,β)PV ∫_(Rn)(a2(x-y)(v(x)-v(y))/(|x-y|~(n+β))dy = g(u(x),v(x)).To reach our aim, by using the method of moving planes, we prove a narrow region principle and a decay at infinity by the iteration method. On the basis of these results, we conclude radial symmetry and monotonicity of positive solutions for the problems involving the weighted fractional system on an unit ball and the whole space. Furthermore, non-existence of nonnegative solutions on a half space is given.     

Uniqueness of degree-one Ginzburg–Landau vortex in the unit ball in dimensions N ≥ 7

For $\epsilon >0$, we consider the Ginzburg–Landau functional for ${\mathbb{R}}^N$-valued maps defined in the unit ball $B^N?{\mathbb{R}}^N$ with the vortex boundary data x on $?B^N$. In dimensions $N\ge 7$, we prove that, for every $\epsilon >0$, there exists a unique global minimizer $u_{\epsilon }$ of this problem; moreover, $u_{\epsilon }$ is symmetric and of the form $u_{\epsilon }(x)=f_{\epsilon }(|x|)\frac{x}{|x|}$ for $x\in B^N$.     

Multiplicity of solutions of weighted (p,q)-Laplacian with small source

In this article, we study the existence of infinitely many solutions to the degenerate quasilinear elliptic system-div(h_1(x)|▽u|~(p-2)▽u)=d(x)|u|~(r-2)u+G_u(x,u,v) in Ω,-div(h_2(x)|▽u|~(p-2)▽v)=f(x)|v|~(s-2)v + G_u(x,u,v) in Ω,u=v=0 on ■Ω where Ω is a bonded domain in R~N with smooth boundary ■Ω,N≥2,1 r p ∞,1 s q ∞; h_1(x) and h_2(x) are allowed to have "essential" zeroes at some points inΩ; d(x)|u|~(r-2)u and f(x)|v|~(s-2)v are small sources with Gu(x,u,v), Gv(x,u,v) being their high-order perturbations with respect to(u,v) near the origin, respectively.     

Odd symmetry of least energy nodal solutions for the Choquard equation

We consider the Choquard equation (also known as the stationary Hartree equation or Schrödinger–Newton equation)

$?\mathrm{\Delta }u+u=(I_{\alpha }?|u{|}^p)|u{|}^{p?2}u.$

Here $I_{\alpha }$ stands for the Riesz potential of order $\alpha \in (0,N)$, and $\frac{N?2}{N+\alpha }<\frac{1}{p}\le \frac{1}{2}$. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when α is either close to 0 or close to N.     

Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents in RN

Let $N\ge 3,0\le s<2,0\le \mu <{\left(\frac{N-2}{2}\right)}^2$ and $2^1\left(s\right)≔\frac{2(N-s)}{N-2}$ be the critical Sobolev–Hardy exponents. Via variational methods and the analytic technique, we prove the existence of a nontrivial solution to the singular semilinear problem $-\Delta u-\mu \frac{u}{{\left|x\right|}^2}+u=\frac{{\left|u\right|}^{2^1\left(s\right)-2}}{{\left|x\right|}^s}u+f\left(u\right),u\in H_r^1\left({\mathbb{R}}^N\right)$, for $N\ge 4,0\le \mu \le \bar{\mu }-1$ and suitable functions $f\left(u\right)$.