The existence and multiplicity of solutions of a fractional Schrdinger-Poisson system with critical growth

In this paper, we study the existence and multiplicity of solutions for the following fractional Schr¨odinger-Poisson system:ε~(2s)(-?)~su + V(x)u + ?u = |u|~2_s~*-2 u + f(u) in R~3,ε~(2s)(-?)~s? = u~2 in R~3,(0.1)where 3/4 s 1, 2_s~*:=6/(3-2s)is the fractional critical exponent for 3-dimension, the potential V : R~3→ R is continuous and has global minima, and f is continuous and supercubic but subcritical at infinity. We prove the existence and multiplicity of solutions for System(0.1) via variational methods.     

Existence of Semiclassical States for a Quasilinear Schr?dinger Equation on R~N with Exponential Critical Growth

We study a quasilinear Schr?dinger equation{-ε~NΔNu+V(x)|u|~(N-2)u=Q(x)f(u) in R~N,0u∈W~(1,N)(R~N),u(x)|x|→∞→0,where V,Q are two continuous real functions on R~N and ε0 is a real parameter.Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger–Moser inequality,we are able to establish the existence and concentration of the semiclassical solutions by variational methods.     

The existence and concentration of weak solutions to a class of p-Laplacian type problems in unbounded domains

In this paper,we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem-εp△pu+V(z)|u|p-2u-f(u)=0 in Ω,u=0 on ■Ω,u0 in Ω,Np2,where Ω is a domain in RN,possibly unbounded,with empty or smooth boundary,εis a small positive parameter,f∈C1(R+,R)is of subcritical and V:RN→R is a locally Hlder continuous function which is bounded from below,away from zero,such that infΛVmin ■ΛV for some open bounded subset Λ of Ω.We prove that there is anε00 such that for anyε∈(0,ε0],the above mentioned problem possesses a weak solution uεwith exponential decay.Moreover,uεconcentrates around a minimum point of the potential V inΛ.Our result generalizes a similar result by del Pino and Felmer(1996)for semilinear elliptic equations to the p-Laplacian type problem.     

Infinitely Many Solutions for Schr?dinger–Choquard–Kirchhoff Equations Involving the Fractional p-Laplacian

In this article,we show the existence of infinitely many solutions for the fractional pLaplacian equations of Schr?dinger-Kirchhoff type equation ■ ,where(-△)_p~s is the fractional p-Laplacian operator,[u]_(s,p) is the Gagliardo p-seminorm,0 s 1 q p N/s,α∈(0,N),M and V are continuous and positive functions,and k(x) is a non-negative function in an appropriate Lebesgue space.Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma,we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters λ and β.     

MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY

In this article,we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity-△u+(λa(x)+1)u=(1/|x|α*F(u))f(u) in R~N,where N≥3,0 αmin{N,4},λ is a positive parameter and the nonnegative potential function a(x) is continuous.Using variational methods,we prove that if the potential well int(a~(-1)(0)) consists of k disjoint components,then there exist at least 2~k-1 multi-bump solutions.The asymptotic behavior of these solutions is also analyzed as λ→+∞.     

Existence and Concentration of Bound States of a Class of Nonlinear Schrdinger Equations in R~2 with Potential Tending to Zero at Infinity

In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schrdinger equation -ε2 Δuε + V(x)uε = K(x)|uε|p-2 uεeα0 |uε|γ,uε0, uε∈H 1(R2),where 2 p ∞, α0 0, 0 γ 2. When the potential function V (x) decays at infinity like (1 + |x|)-α with 0 α≤ 2 and K(x) 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H1 (R2 )-solution uε exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schrdinger equation-Δu + V (ξ)u = K(ξ)|u| p-2 ue α0 |u|γ has local minimum points. Furthermore, the concentration property of uε is also established as ε tends to zero.     

EXISTENCE,MULTIPLICITY AND STABILITY RESULTS FOR POSITIVE SOLUTIONS OF NONLINEAR p-LAPLACIAN EQUATIONS

This paper studies the existence of positive solutions of the Dirichlet problem for the nonlinear equation involving p-Laplacian operator:-△pu=λf(u) on a bounded smooth domain Ω in Rn. The authors extend part of the Crandall-Rabinowitz bifurcation theory to this problem. Typical examples are checked in detail and multiplicity of the solutions are illustrated. Then the stability for the associated parabolic equation is considered and a Fujita-type result is presented.     

GROUND STATES FOR FRACTIONAL SCHR?DINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS AND CRITICAL GROWTH

In this article, we study the following fractional Schr?dinger equation with electromagnetic fields and critical growth (-?)_A~su + V(x)u = |u|~(2_s~*-2) u + λf(x, |u|~2)u, x ∈ R~N,where(-?)_A~s is the fractional magnetic operator with 0 s 1, N 2s, λ 0, 2_s~*=2N/(N-2s),f is a continuous function, V ∈ C(R~N, R) and A ∈ C(R~N, R~N) are the electric and magnetic potentials, respectively. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by Nehari method.     

Existence and Multiplicity Results for a Degenerate Elliptic Equation

The goal of this paper is to study the multiplicity result of positive solutions of a class of degenerate elliptic equations. On the basis of the mountain pass theorems and the sub- and supersolutions argument for p-Laplacian operators, under suitable conditions on the nonlinearity f（x, s）, we show the following problem：-△pu=λu^α-a（x）u^q in Ω,u│δΩ=0 possesses at least two positive solutions for large λ, where Ω is a bounded open subset of R^N, N ≥ 2, with C^2 boundary, λ is a positive parameter, Ap is the p-Laplacian operator with p 〉 1, α, q are given constants such that p - 1 〈α 〈 q, and a（x） is a continuous positive function in Ω^-.     

Existence of positive solutions to some nonlinear equations on locally finite graphs

Let G =(V,E) be a locally finite graph,whose measure μ(x) has positive lower bound,and A be the usual graph Laplacian.Applying the mountain-pass theorem due to Ambrosetti and Rabinowitz(1973),we establish existence results for some nonlinear equations,namely △u+hu=f(x,u),x∈V.In particular,we prove that if h and f satisfy certain assumptions,then the above-mentioned equation has strictly positive solutions.Also,we consider existence of positive solutions of the perturbed equation △u+hu=f(x,u)+∈g.Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.     

MULTIPLICITY RESULTS FOR FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE

In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {△2u- a+b∫R3|▽u|2dx △u+V(x)u=f(x, u), x∈ R3,u∈H2(R3),wherea, b 0 are constants and the primitive of the nonlinearityfis of superlinear growth near infinity inuand is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the AmbrosettiRabinowitz type condition.     

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.     

POSITIVE SOLUTIONS FOR WEAKLY COUPLED NONLINEAR ELLIPTIC SYSTEMS

In this article,we consider the existence of positive solutions for weakly cou-pled nonlinear elliptic systems {-△u+u (1+a(x))|u| p-1 u+μ|u| α-2 u|v|β+λv in R~N,-△v+v=(1+bx))|v|p-1v+μ|u|α|v|β-2v+λu in R N.(0.1) To find nontrivial solutions,we first investigate autonomous systems.In this case,results of bifurcation from semi-trivial solutions are obtained by the implicit function theorem.Next,the existence of positive solutions of problem(0.1) is obtained by variational methods.     

POSITIVE SOLUTIONS TO THE p-LAPLACIAN WITH SINGULAR WEIGHTS＊

By the Mountain Pass Theorem,we study existence and multiplicity of positive solutions of p-laplacian equation of the form -△pu =λf(x,u),the nonlinearity f(x,u) grows as uσ at infinity with a singular ...     

SELF-SIMILAR SINGULAR SOLUTION OF A P-LAPLACIAN EVOLUTION EQUATION WITH GRADIENT ABSORPTION TERM＊

In this paper we deal with the self-similar singular solution of the p-Laplacian evolution equation ut = div(|△↓u|^p-2△↓u) -|△↓|^q for p > 2 and q > 1 in R^n× (0, ∞). We prove that when p > q + n/(n + 1) there exist self-similar singular solutions, while p≤q+n/(n+1) there is no any self-similar singular solution. In case of existence, the self-similar singular solutions are the self-similar very singular solutions,which have compact support. Moreover, the interface relation is obtained.     

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrödinger equations

The classical critical Trudinger-Moser inequality in R~2 under the constraint ∫_(R_2)(|▽u|~2+|u|~2)dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for any τ 0,it holds that ■ and 4π is sharp.However,if we consider the less restrictive constraint ∫_(R_2)(|▽u|~2+|u|~2)dx≤1,where V(x) is nonnegative and vanishes on an open set in R~2,it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x) makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality ■ when V is nonnegative and vanishes on an open set in R~2.As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u) in R~2,(0.1)where V(x)≥0 and vanishes on an open set of R~2 and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential) has been the standard assumption when one deals with the existence of solutions to the above Schr?dinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.     

一类Kirchhoff型系统解的存在性

In this paper,we are interested in the existence of positive solutions for the Kirchhoff type problems{-(a_1 + b_1M_1(∫_?|▽u|~pdx))△_(_pu) = λf(u,v),in ?,-(a_2 + b_2M_2(∫?|▽v|~qdx))△_(_qv) = λg(u,v),in ?,u = v = 0,on ??,where 1 p,q N,M i:R_0~+→ R~+(i = 1,2) are continuous and increasing functions.λ is a parameter,f,g ∈ C~1((0,∞) ×(0,∞)) × C([0,∞) × [0,∞)) are monotone functions such that f_s,f_t,g_s,g_t ≥ 0,and f(0,0) 0,g(0,0) 0(semipositone).Our proof is based on the sub-and super-solutions techniques.     

Multiple positive solutions for a fourth-order p-Laplacian Sturm-Liouville boundary value problems on time scales

In this paper, we investigate the existence of multiple positive solutions for the following fourthorder p-Laplacian Sturm-Liouville boundary value problems on time scales ﹛[φp(u△△(t))u △△= f(t, u(σ(t))), t ∈ [a, b],α0 u(a)- β0u△(a) = 0, γ0 u(σ(b)) + δ0 u△(σ(b)) = 0,α0(φp(u△△))(a)- β0(φp(u△△))△(a) = 0,γ0(φp(u△△))(σ(b)) + δ0(φp(u△△))△(σ(b)) = 0,where φp(s) is the p-Laplacian operator. Under growth conditions on the nonlinearity f some existence results of at least two and three positive solutions for the above problem are obtained by virtue of fixed point theorems on cone. In particular, the nonlinearity f may be both sublinear and superlinear.     

SIGN-CHANGING SOLUTIONS FOR SCHRDINGER EQUATIONS WITH VANISHING AND SIGN-CHANGING POTENTIALS

In this article, we study the existence of sign-changing solutions for the following Schrdinger equation-△u + λV(x)u = K(x)|u|p-2u x ∈ RN, u → 0 as |x| → +∞,where N ≥ 3, λ 0 is a parameter, 2 p 2N N-2, and the potentials V(x) and K(x) satisfy some suitable conditions. By using the method based on invariant sets of the descending flow,we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].     

BLOWING UP AND MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER EQUATION WITH CRITICAL NONLINEARITY

In this paper,we consider the following nonlinear elliptic problem:△~2u=|u|~(8/(n-4))u+μ|u|~(q-1)u,in Ω,△u = u = 0 on δΩ,where Ω is a bounded and smooth domain in R~n,n ∈ {5,6,7},μ is a parameter and q ∈]4/(n- 4),(12- n)/(n- 4)[.We study the solutions which concentrate around two points of Ω.We prove that the concentration speeds are the same order and the distances of the concentration points from each other and from the boundary are bounded.For Ω =(Ω_α)α a smooth ringshaped open set,we establish the existence of positive solutions which concentrate at two points of Ω.Finally,we show that for μ 0,large enough,the problem has at least many positive solutions as the LjusternikSchnirelman category of Ω.