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.     

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.     

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.     

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).     

Gradient estimates for solutions to quasilinear elliptic equations with critical sobolev growth and hardy potential

This note is a continuation of the work[17].We study the following quasilinear elliptic equations(■)where 1 p N,0 ≤μ ((N-p)/p)~p and Q ∈ L~∞(R~N).Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.     

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.     

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].     

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.     

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes

In the present paper we perform the homogenization of the semilinear elliptic problem

$\{\begin{array}{cc}{u}^{\epsilon }\ge 0\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ ?divA(x)D{u}^{\epsilon }=F(x,u^{\epsilon })\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ u^{\epsilon }=0\hfill & \text{on}?{\mathrm{\Omega }}^{\epsilon }.\hfill \end{array}$

In this problem $F(x,s)$ is a Carathéodory function such that $0\le F(x,s)\le h(x)/\mathrm{\Gamma }(s)$ a.e. $x\in \mathrm{\Omega }$ for every $s>0$, with h in some $L^r(\mathrm{\Omega })$ and Γ a $C^1([0,+\infty [)$ function such that $\mathrm{\Gamma }(0)=0$ and ${\mathrm{\Gamma }}^{\prime }(s)>0$ for every $s>0$. On the other hand the open sets ${\mathrm{\Omega }}^{\epsilon }$ are obtained by removing many small holes from a fixed open set Ω in such a way that a “strange term” $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on x.We already treated this problem in the case of a “mild singularity”, namely in the case where the function $F(x,s)$ satisfies $0\le F(x,s)\le h(x)(\frac{1}{s}+1)$. In this case the solution $u^{\epsilon }$ to the problem belongs to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ and its definition is a “natural” and rather usual one.In the general case where $F(x,s)$ exhibits a “strong singularity” at $u=0$, which is the purpose of the present paper, the solution $u^{\epsilon }$ to the problem only belongs to $H_{\text{loc}}^1({\mathrm{\Omega }}^{\epsilon })$ but in general does not belong to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ anymore, even if $u^{\epsilon }$ vanishes on $?{\mathrm{\Omega }}^{\epsilon }$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the “strange term” $\mu u^0$ still appears in the left-hand side while the source term $F(x,u^0)$ is not modified in the right-hand side.     

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 $f\left(u\right)$.     

On a p-Laplace equation with multiple critical nonlinearities

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{u}^{p?1}={|x|}^{?s}{u}^{p^{?}(s)?1}+u^{p^{?}?1}$ admits a positive weak solution in ${\mathbb{R}}^n$ of class $D_1^p({\mathbb{R}}^n)\cap C^1({\mathbb{R}}^n?\{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1={[(n?p)/p]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in D_1^p({\mathbb{R}}^n)$ is a weak solution in ${\mathbb{R}}^n$ of $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{|u|}^{p?2}u={|x|}^{?s}{|u|}^{p^{?}(s)?2}u+{|u|}^{q?2}u$, then $u\equiv 0$ when either $1<q<p^{?}$, or $q>p^{?}$ and u is also of class $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^n?\{0\})$.