Claudio Bonanno
In this paper we study the multiplicity of positive solutions for nonlinear elliptic equations on $\R^N$. The number of solutions is greater or equal than the number of disjoint intervals on which the nonlinear term is negative. Applications are given to multiplicity of standing waves for the nonlinear Schrödinger and Klein-Gordon equations.
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We prove some multiplicity results for a nonlinear equation of Schroedinger type with potential functions
We establish the multiplicity of positive solutions to a quasilinear Neumann problem in expanding balls and hemispheres with critical exponent in the boundary condition.
We consider the existence and multiplicity of solutions for a class of quasi-linear Schrödinger equations which include the modified nonlinear Schrödinger equations. A new perturbation approach is used to treat the sub-cubic nonlinearity.
we study on compact Riemannian manifolds with boundary, the problems of existence and multiplicity of solutions to a Neumann problem involving the p-Laplacian operator and critical Sobolev exponents.
In this paper, we study the multiplicity of positive solutions for the p-Laplacian systems with sign-changing weight functions. Using the decomposition of the Nehari manifold, we prove that an elliptic system has at least two positive solutions.
We prove the multiplicity of spike-layer type solutions to the Dirichlet problem for the equation $-Δ_p u = u^{q-1}$ in expanding annuli.
In this article we study modified Kirchhoff-Schrödinger equations involving critical Stein-Weiss type nonlinearity and $p$-Laplacian
The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory.
We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian.
The present paper studies the fractional $p$-Laplacian boundary value problems with jumping nonlinearities at zero or infinity and obtain the existence of multiple solutions and sign-changing solutions by constructing the suitable pseudo-gradient vector field of the corresponding energy functional.
We consider a class of parametric Schrödinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. By using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.
In this paper we prove the existence of infinitely many nontrivial solutions for the class of $(p,\, q)$ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_Ω(Ω)$ nonnegative solutions.
In this work, we study the existence and multiplicity of solutions for a class of problems involving the $ϕ$-Laplacian operator in a bounded domain, where the nonlinearity has a critical growth. The main tool used is the variational method combined with the genus theory for even functionals.
In this paper, we prove multiplicity of solutions for a class of quasilinear problems in $ \mathbb{R}^{N} $ involving variable exponents and nonlinearities of concave-convex type. The main tools used are variational methods, more precisely, Ekeland's variational principle and Nehari manifolds.
Multiplicity Results for the Nonlinear Weighted Laplacian Equation on the Sierpiński Gasket
研究 Sierpiński 垫片上加权拉普拉斯方程 Dirichlet 边值问题，通过直接法、山路定理及相关多重性定理，在非线性项满足适当增长条件下给出解的存在性与多重性结果。
We are interested in the problem of existence of soliton-like solutions for the nonlinear Klein-Gordon equation. In particular we study some necessary and sufficient conditions on the nonlinear term to obtain solitons of a given charge. We remark that the conditions we consider can be easily verified. Moreover we show that multiplicity of solitons of the same charge is guaranteed by the "shape" of the nonlinear term for equations on $\R^{N}$, hence without appealing to topological or geometrical properties of the domain.
Even without a variational background, a multiplicity result of positive solutions with ordered $L^{p}(Ω)$-norms is provided to the following boundary value problem \begin{equation*} \left \{ \begin{array}{ll} -a(\int_Ωu^{p}dx)Δu = f(u) & \mbox{in $Ω$,}\\ u=0 & \mbox{on $\partialΩ$,} \end{array}\right. \end{equation*} where $Ω$ is a bounded domain and $a$, $f$ are continuous real functions with $a$ vanishing in many positive points.
Uniqueness, non-degeneracy, and exact multiplicity of positive solutions for superlinear elliptic problems
本文利用打靶法与比较定理，研究变号权超线性方程正解的非退化性与精确多重性，完善了Feltrin和Zanolin的结果，并讨论数值算例与开放问题。
In this paper we provide a new technique to find solutions to the Klein-Gordon-Maxwell system. The method, based on an iterative argument, permits to improve previous results where the reduction method was used. We also show how this device permits to obtain a multiplicity result in the physically significant context known as "the positive potential case".
Multiplicity and Nonrelativistic limit of Bound States of Nonlinear Dirac Equations on Noncompact Metric Graphs with Localized Nonlinearities
研究非紧度量图上具有局部非线性项的狄拉克方程在L^2约束下归一化解的多重性，并证明在光速趋于无穷时解收敛到非线性薛定谔方程的解，涵盖质量次临界、临界和超临界情形。
We study a Dirichlet type problem for an equation involving the fractional Laplacian and a reaction term subject to either subcritical or critical growth conditions, depending on a positive parameter. Applying a critical point result of Bonanno, we prove existence of one or two positive solutions as soon as the parameter lies under a (explicitly determined) threshold. As an application, we find two positive solutions for a fractional Brezis-Nirenberg problem.
In this paper, we study existence, multiplicity and concentration of positive solutions for the following class of quasilinear problems \[ - Δ_Φu + V(εx)ϕ(\vert u\vert)u = f(u)\quad \mbox{in} \quad \mathbb{R}^{N} \,\,\, ( N\geq 2 ), \] where $Φ(t) = \int_{0}^{\vert t\vert}ϕ(s)sds$ is a N-function, $ Δ_Φ$ is the $Φ$-Laplacian operator, $ε$ is a positive parameter, $V : \mathbb{R}^{N} \rightarrow \mathbb{R} $ is a continuous function and $f : \mathbb{R} \rightarrow \mathbb{R} $ is a $C^{1}$-function.
Note on the multiplicity of solutions for nonlinear scalar field equations with a critical inverse-square potential
本文研究了非线性标量场方程在临界反平方势下的解的乘积问题，证明了无限多个径向和非径向解的存在性。
In the present paper we deal with a quasilinear problem involving a singular term and a parametric superlinear perturbation. We are interested in the existence, nonexistence and multiplicity of positive solutions as the parameter $λ>0$ varies. In our first result, the superlinear perturbation has an arbitrary growth and we obtain the existence of a solution for the problem by using the sub-supersolution method. For the second result, the superlinear perturbation has subcritical growth and we employ the Mountain Pass Theorem to show the existence of a second solution.
By using the penalization method and the Ljusternik-Schnirelmann theory, we investigate the multiplicity of positive solutions of the following fractional Schrödinger equation $$ \e^{2s}(-Δ)^{s} u + V(x)u = f(u) \mbox{ in } \mathbb{R}^{N} $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-Δ)^{s}$ is the fractional Laplacian, $V$ is a positive continuous potential with local minimum, and $f$ is a superlinear function with subcritical growth. We also obtain a multiplicity result when $f(u)=|u|^{q-2}u+λ|u|^{r-2}u$ with $2<q<2^{*}_{s}\leq r$ and $λ>0$, by combining a truncation arg…
Given $k\in\mathbb N$, we define a class of continuous piecewise functions $f$ having abrupt but controlled magnitude changes so that the problem $$Δu +f(u)=0,\quad x\in \mathbb R^N, N> 2, $$ has at least $k$ radially symmetric ground state solutions.
In this paper we focus our attention on the following nonlinear fractional Schrödinger equation with magnetic field \begin{equation*} \varepsilon^{2s}(-Δ)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N\geq 3$, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^N$ are continuous potentials and $f:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick-S…
We investigate positive steady states of an indefinite superlinear reaction-diffusion equation arising from population dynamics, coupled with a nonlinear boundary condition. Both the equation and the boundary condition depend upon a positive parameter $λ$, which is inversely proportional to the diffusion rate. We establish several multiplicity results when the diffusion rate is large and analyze the asymptotic profiles and the stability properties of these steady states as the diffusion rate grows to infinity. In particular, our results show that in some cases bifurcation from zero and from in…
In this paper, we prove a multiplicity result of solutions for the following stationary Schrödinger-Poisson-Slater equations \begin{equation}\label{eq-abstract} -Δu - λu + (\left | x \right |^{-1}\ast \left | u \right |^2) u - |u|^{p-2}u = 0 \ \mbox{ in } \ \mathbb{R}^{3}, \end{equation} where $λ\in \R$ is a parameter, and $p\in (2,6)$. The solutions we obtained have a prescribed $L^2$-norm. Our proofs are mainly inspired by a recent work of Bartsch and De Valeriola [7].
Multiplicity of positive solutions for mixed local-nonlocal singular critical problems
本文证明混合局部-非局部奇异临界椭圆问题至少存在两个正弱解，推广了奇异与临界问题的现有结果。
In this paper we develop a new theory for the existence, localization and multiplicity of positive solutions for a class of non-variational,quasilinear, elliptic systems. In order to do this, we provide a fairly general abstract framework for the existence of fixed points of nonlinear operators acting on cones that satisfy an inequality of Harnack type. Our methodology relies on fixed point index theory. We also provide a non-existence result and an example to illustrate the theory.
We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degre…
Existence and Multiplicity of Normalized Solutions to Schrödinger Equations with General Nonlinearities in Bounded Domains
本文利用变分方法研究有界区域中具一般非线性项的薛定谔方程的归一化解，得到两个正归一化解，证明星形有界区域解的多重性及球域中非径向归一化解的存在性。
In this paper we give a multiplicity result for the following Chern-Simons-Schrödinger equation \[ -Δu+2q u \int_{|x|}^{\infty}\frac{u^{2}(s)}{s}h_u(s)ds +q u\frac{h^{2}_u(|x|)}{|x|^{2}} = g(u), \quad\hbox{in }\mathbb{R}^2, \] where $\displaystyle h_u(s)=\int_0^s τu^2(τ) \ d τ$, under very general assumptions on the nonlinearity $g$. In particular, for every $n\in \mathbb N$, we prove the existence of (at least) $n$ distinct solutions, for every $q\in (0,q_{n})$, for a suitable $q_n$.
In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation \begin{equation*} (-Δ)^{\fracα{2}} u=λf(x)|u|^{q-2}u+|u|^{2^{*}_α-2}u, \quad\text{in}\,\,Ω, u=0,\text{on}\,\,\partialΩ, \end{equation*} where $Ω\subset \mathbb{R}^{N}(N\geq 2)$ is a bounded domain with smooth boundary, $0<α<2$, $(-Δ)^{\fracα{2}}$ stands for the fractional Laplacian operator, $f\in C(Ω\times\mathbb{R},\mathbb{R})$ may be sign changing and $λ$ is a positive parameter. We will prove that there exists $λ_{*}>0$ such that the problem has at least two positive solutions for each $λ…
The paper is concerned with the existence of positive weak solutions for a new class of $\left( p,q\right) $-Laplacian elliptic systems in a bounded domain by means of the method of sub-super solutions. Particularly, we do not need any sign conditions for $γ\left( 0\right), g\left( 0\right), f\left( 0\right) $ and $h\left(0\right) $. Moreover, a multiplicity result is obtained when $γ\left(0\right)=g\left( 0\right)=f\left( 0\right)=h\left( 0\right)=0.$ Finally, we give some examples to verify our main results.
Multiplicity results for mixed local-nonlocal equations with singular and critical exponential nonlinearity in R^2
本文研究R^2中带可变奇异指数与临界指数非线性项的混合局部-非局部方程，利用下解-上解技巧与变分方法证明了至少两个正弱解的存在性。
Sufficient conditions for multiplicity and uniqueness of positive solutions of the Hénon equation
本文给出Hénon方程在单位球上正解多重性及一维唯一性的显式充分条件，通过Morse指标、扰动及特征值比较等方法证明。
Multiplicity Results for Mixed Local Nonlocal Equations With Indefinite Concave-Convex Type Nonlinearity
本文研究含不定凹凸型非线性项的混合局部-非局部方程多解性，通过纤维映射分析与Nehari流形约束极小化得到次临界和临界情形下至少两个非平凡非负解，还得到大λ时的不存在性结果。
Existence and multiplicity of peaked bound states for nonlinear Schrödinger equations on metric graphs
本文在一般度量图上构造非线性薛定谔方程在奇数度（≥3）顶点集中的单峰与多峰正束缚态，证明其非极小元，为度数≠1顶点集中的首个结果，基于Ljapunov-Schmidt约化。
Multiplicity of critical orbits to nonlinear, strongly indefinite functionals with sign-changing nonlinear part
本文建立带离散群作用的位错空间上强不定变号非线性泛函的无穷多临界轨道定理，并将其用于非线性光学 Schrödinger 方程及 Maxwell 电磁波的奇异势方程。
It is well known that a single nonlinear fractional Schrödinger equation with a potential $V(x)$ and a small parameter $\varepsilon $ may have a positive solution that is concentrated at the nondegenerate minimum point of $V(x)$. In this paper, we can find two different positive solutions for two weakly coupled fractional Schrödinger systems with a small parameter $\varepsilon $ and two potentials $V_{1}(x)$ and $V_{2}(x)$ having the same minimum point are concentrated at the same point minimum point of $V_{1}(x)$ and $V_{2}\left(x\right) $. In fact that by using the energy estimates, Nehari m…
We study a Dirichlet boundary value problem associated to an anisotropic differential operator on a smooth bounded of $\Bbb R^N$. Our main result establishes the existence of at least two different non-negative solutions, provided a certain parameter lies in a certain range. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with adequate variational methods and a variant of Mountain Pass lemma.
In this paper we deal with the following fractional Kirchhoff equation \begin{equation*} \left(p+q(1-s) \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^{2}}{|x-y|^{N+2s}} \, dx\,dy \right)(-Δ)^{s}u = g(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $N\geq 2$, $p>0$, $q$ is a small positive parameter and $g: \mathbb{R}\rightarrow \mathbb{R}$ is an odd function satisfying Berestycki-Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that $q$ is sufficiently small.
In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem \begin{equation*} \left\{ \begin{array}[c]{ll} -Δu - Δ(u^2)u = |u|^{p-2}u & \mbox{ in } Ωu= 0 &\mbox{ on } \partialΩ, \end{array} \right. \end{equation*} where $Ω$ is a smooth and bounded domain in $\mathbb R^{N},N\geq3$. More specifically we prove that, for $p$ near the critical exponent $22^{*}=4N/(N-2)$, the number of positive solutions is estimated below by topological invariants of the domain $Ω$: the Ljusternick-Schnirelmann category and the Poincaré polynomial.
In this paper we establish the existence of two positive solutions for the obstacle problem $$ \displaystyle \int_{\Re}\left[u'(v-u)'+(1+λV(x))u(v-u)\right] \geq \displaystyle \int_{\Re} f(u)(v-u), \forall v\in \Ka $$ where $f$ is a continuous function verifying some technical conditions and $\Ka$ is the convex set given by $$ \Ka =\left\{v\in H^{1}(\Re); v \geq φ\right\}, $$ with $φ\in H^{1}(\Re)$ having nontrivial positive part with compact support in $\Re$. \vspace{0.2cm} \noindent \emph{2000 Mathematics Subject Classification} : 34B18, 35A15, 46E39. \noindent \emph{Key words}: Obstacle pro…
In this paper we deal with the following fractional Choquard equation \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{sp}(-Δ)^{s}_{p} u + V(x)|u|^{p-2}u = \varepsilon^{μ-N}\left(\frac{1}{|x|^μ}*F(u)\right)f(u) \mbox{ in } \mathbb{R}^{N},\\ u\in W^{s,p}(\R^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $p\in (1, \infty)$, $N>sp$, $(-Δ)^{s}_{p}$ is the fractional $p$-Laplacian, $V$ is a positive continuous potential, $0<μ<sp$, and $f$ is a continuous superlinear function with subcritical growth. …
In this article we are interested in the following non-linear Schrödinger equation with non-local regional diffusion $$ (-Δ)_{ρ_ε}^αu + u = f(u) \hbox{ in } \mathbb{R}^n, \quad u \in H^α(\mathbb{R}^n), \qquad\qquad(P_ε) $$ where $ε>0$, $0< α< 1$, $(-Δ)_{ρ_ε}^α$ is a variational version of the regional laplacian, whose range of scope is a ball with radius $ρ_ε(x)=ρ(εx)>0$, where $ρ$ is a continuous function. We give general conditions on $ρ$ and $f$ which assure the existence and multiplicity of solution for $(P_ε)$.
Uniqueness and multiplicity of positive radial solutions to the super-critical Brezis-Nirenberg problem in an annulus
本文研究环形区域中超临界Brezis-Nirenberg问题，建立三维情形正径向解的唯一性结果，证明内半径足够小且外半径在某范围时至少有三个正径向解，指数在临界Sobolev指数与Joseph-Lundgren指数间时对任意正整数$k$至少有$k$个正径向解。
In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem $$\left\{\begin{array}{rcl} \mathcal{L}_{\varepsilon}u = f(u) \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u \in H^1 (\mathbb{R}^3), \end{array} \right.$$ where $\varepsilon$ is a small positive parameter, $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function, $\mathcal{L}_{\varepsilon}$ is a nonlocal operator defined by $$ \mathcal{L}_{\varepsilon} u = M \left(\frac{1}{\varepsilon} \int_{\mathbb{R}^3} |\nabla u|^2 + \frac{1}{…
This paper studies the multiplicity of normalized solutions to the Schrödinger equation with mixed nonlinearities \begin{equation*} \begin{cases} -Δu=λu+h(εx)|u|^{q-2}u+η|u|^{p-2}u,\quad x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=a^2, \end{cases} \end{equation*} where $a, ε, η>0$, $q$ is $L^2$-subcritical, $p$ is $L^2$-supercritical, $λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $h$ is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $ε$ is small enoug…
In the present work we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation $(-Δ)_{p}^{s}u + |u|^{p-2}u + (\ln|\cdot|\ast |u|^{p})|u|^{p-2}u = f(u) \textrm{ \ in \ } \mathbb{R}^N $ , where $ N=sp $, $ s\in (0, 1) $, $ p>2 $, $ a>0 $, $ λ>0 $ and $f: \mathbb{R}\rightarrow \mathbb{R} $ a continuous nonlinearity with exponential critical and subcritical growth. We guarantee the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under critical and subcritical growth. Morever, when $ f $ has subcritical…
This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \begin{equation} \tag{$\mathcal E$} (-Δ)^s u = a(x) |u|^{2^*_s-2}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, \quad u \in \dot{H}^s(\mathbb{R}^{N}), \end{equation} where $s \in (0,1)$, $N>2s$, $2_s^*:=\frac{2N}{N-2s}$, $0< a\in L^\infty(\mathbb{R}^{N})$ and $f$ is a nonnegative nontrivial functional in the dual space of $\dot{H}^s$. We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that $a$ is a contin…
Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of Fadell and Rabinowitz. We also consider the critical case and prove existence of positive least energy solution when the parameter $β$ is sufficiently large.
In this paper, we study the multiplicity and concentration of the positive solutions to the following critical Kirchhoff type problem: \begin{equation*} -\left(\varepsilon^2 a+\varepsilon b\int_{\R^3}|\nabla u|^2\mathrm{d} x\right)Δu + V(x) u = f(u)+u^5\ \ {\rm in } \ \ \R^3, \end{equation*} where $\varepsilon$ is a small positive parameter, $a$, $b$ are positive constants, $V \in C(\mathbb{R}^3)$ is a positive potential, $f \in C^1(\R^+, \R)$ is a subcritical nonlinear term, $u^5$ is a pure critical nonlinearity. When $\varepsilon>0$ small, we establish the relationship between the number of …
We investigate the existence of infinitely many radially symmetric solutions to the following problem $$(-Δ_p)^s u=g(u) \ \ \textrm{ in } \ \ \mathbb{R}^N, \ \ u\in W^{s,p}(\mathbb{R}^N),$$ where $s\in (0,1)$, $2 \leq p < \infty$, $sp \leq N $, $2 \leq N \in \mathbb{N}$ and $(-Δ_p)^s$ is the fractional $p$-Laplacian operator. We treat both of cases $sp=N$ and $sp<N.$ The nonlinearity $g$ is a function of Berestycki-Lions type with critical exponential growth if $sp=N$ and critical polynomial growth if $sp<N$. We also prove the existence of a ground state solution for the same problem.
In this paper, we deal with the existence and multiplicity of solutions to the nonuniformly elliptic equation of the N-Lapalcian type with a potential and a nonlinear term of critical exponential growth and satisfying the Ambrosetti-Rabinowitz condition. In spite of a possible failure of the Palais-Smale compactness condition, in this article we apply minimax method to obtain the weak solution to such an equation. In particular, in the case of $N-$Laplacian, using the minimization and the Ekeland variational principle, we obtain multiplicity of weak solutions. Finally, we will prove the above …
A proof of a conjecture on the multiplicity of positive solutions of an indefinite superlinear problem
本文证明了一类超线性不定边值问题正解多重性的一个2000年猜想，给出了正面解答。
This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent \begin{equation*} \begin{cases} -Δu+(V_1(x)+λ_1)u=μ_1(|x|^{-4}*u^{2})u+β(|x|^{-4}*v^{2})u, \ \ &x\in R^N, -Δv+(V_2(x)+λ_2)v=μ_2(|x|^{-4}*v^{2})v+β(|x|^{-4}*u^{2})v, \ \ &x\in R^N, \end{cases} \end{equation*} where $N\geq 5$, $λ_1$, $λ_2\geq 0$ with $λ_1+λ_2\neq 0$, $V_1(x), V_{2}(x)\in L^{\frac{N}{2}}(R^N)$ are nonnegative functions and $μ_1$, $μ_2$, $β$ are positive constants. Such system arises from mathematical models in Bose-Einstein condensates theory and nonlinear optics. By variational methods…
In this paper we study multiplicity and qualitative behavior of solutions for semilinear elliptic problems with neumann boundary condition and asymptotically linear smooth nonlinearity. We provide sufficient conditions on the number of eigenvalues the derivative of the nonlinearity crosses to guarantee existence of at least five nontrivial solutions. The techniques we use are a combination of minimization, Leray-Schauder degree, Morse Theory and Reduction method a la Castro-Lazer.
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