Positive Ground State Solution Research Articles

Existence and concentration behavior of positive solutions for a quasilinear Schrödinger equation

In this paper, we study the existence and concentration of positive solutions for the quasilinear Schrödinger equation−h2Δu+V(x)u−h2Δ(u2)u=K(x)|u|p−2u,x∈RN, where h>0 is a small real parameter, N⩾3, 4<p<22⁎=4NN−2, and V(x) and K(x) are uniformly continuous functions with V(x) having at least one minimum and K(x) having at least one maximum, then a positive ground state solution uh(x) exists. Moreover, we establish the concentration property of uh(x) as h→0+.

On a nonlinear Schrödinger equation with indefinite potential

We investigate the semilinear Schrödinger equation −Δu+V(x)u=Q(x)f(u),inRN, where N≥3, V(x)∈LlocN/2(RN) and Q(x)∈L∞(RN), V(x) and Q(x) respectively tend to some positive limits V∞ and Q∞ as |x|→∞, and f∈C(R) is a superlinear and subcritical function. Assuming some weak one-sided asymptotic estimates for V(x) and Q(x), we prove that the above equation has a positive ground state solution and a least energy nodal solution via the minimization method constrained to Nehari type sets.

On ground state solutions for the Schrödinger–Poisson equations with critical growth

In this paper, we study the existence of ground state solutions for the following Schrödinger–Poisson equation{−Δu+V(x)u+λϕu=μ|u|q−1u+|u|4u,inR3,−Δϕ=u2,inR3, where μ is a positive parameter. Under some certain assumptions on V, we prove that for every λ>0 and q∈(2,5), such a class of Schrödinger–Poisson equation with critical growth has at least a positive ground state solution via variational methods. Some recent results from the literature are extended.

Existence and Concentration of Positive Ground State Solutions for Schrödinger-Poisson Systems

Abstract In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson system where ε &gt; 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that b(x) has a maximum. We prove that the system has a positive ground state solution for all λ ≠ 0 and sufficiently small ε &gt; 0. Moreover, for each λ ≠ 0 we show that uε converges to the positive ground state solution of the associated limit problem and concentrates to a maximum point of b(x) in certain sense as ε → 0. Furthermore, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.

Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics

We consider a semilinear elliptic problem−Δu+u=(Iα⁎|u|p)|u|p−2uinRN, where Iα is a Riesz potential and p>1. This family of equations includes the Choquard or nonlinear Schrödinger–Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groundstates.

Multiple solutions for a semilinear elliptic system in

In this paper, we study the following semilinear elliptic system urn:x-wiley:01704214:media:mma2764:mma2764-math-0001 where N &gt; 2, f(x,t) and g(x,t) are continuous functions and satisfy additional conditions. By using critical point theory of strongly indefinite functionals, we obtain a positive ground state solution and infinitely many geometrically distinct solutions when f(x,t) and g(x,t) are periodic in X and odd in t. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Existence and concentration of ground state solution to a critical p-Laplacian equation

In this paper, we consider the existence and concentration behavior of positive ground state solution to the following problem � −h p Δpu+V(x)|u| p−2 u = K(x)|u| q−2 u+ |u| p ∗ −2 u, x ∈ R N , u ∈W 1,p (R N ), u > 0, x ∈ R N , where h is a small positive parameter, 1 0, the equation has a positive ground state solution. Furthermore, we establish the concentration property of such solutions when h tends to zero.

Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity

In this paper we investigate the existence of positive ground state solution for the following class of elliptic equations−Δu+V(x)u=K(x)f(u)inRN, where N⩾3, V, K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. Here, we prove a Hardy-type inequality and use it together with variational method to get a ground state solution.

Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in $${\mathbb{R}^{3}}$$

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrodinger–Poisson system $$\left\{\begin{array}{ll}-\varepsilon^{2}\Delta u + a(x)u + \lambda\phi(x)u = b(x)f(u), & x \in \mathbb{R}^{3},\\-\varepsilon^{2}\Delta\phi = u^{2}, \ u \in H^{1}(\mathbb{R}^{3}), &x \in \mathbb{R}^{3},\end{array}\right.$$ where e > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that a(x) has at least one minimum and b(x) has at least one maximum. We first prove the existence of least energy solution (ue, φe) for λ ≠ 0 and e > 0 sufficiently small. Then we show that ue converges to the least energy solution of the associated limit problem and concentrates to some set. At the same time, some properties for the least energy solution are also considered. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.

Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth

In this paper we concern with the multiplicity and concentration of positive solutions for the semilinear Kirchhoff type equation{−(ε2a+bε∫R3|∇u|2)Δu+M(x)u=λf(u)+|u|4u,x∈R3,u∈H1(R3),u>0,x∈R3, where ε>0 is a small parameter, a, b are positive constants and λ>0 is a parameter, and f is a continuous superlinear and subcritical nonlinearity. Suppose that M(x) has at least one minimum. We first prove that the system has a positive ground state solution uε for λ>0 sufficiently large and ε>0 sufficiently small. Then we show that uε converges to the positive ground state solution of the associated limit problem and concentrates to a minimum point of M(x) in certain sense as ε→0. Moreover, some further properties of the ground state solutions are also studied. Finally, we investigate the relation between the number of positive solutions and the topology of the set of the global minima of the potentials by minimax theorems and the Ljusternik–Schnirelmann theory.

Positive ground state solutions for the critical Klein–Gordon–Maxwell system with potentials

This paper deals with the Klein–Gordon–Maxwell system when the nonlinearity exhibits critical growth. We prove the existence of positive ground state solutions for this system when a periodic potential V is introduced. The method combines the minimization of the corresponding Euler–Lagrange functional on the Nehari manifold with the Brézis and Nirenberg technique.

MULTIPLE SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS IN UNBOUNDED CYLINDER DOMAINS

In this paper, we show that if $Q(x)$ satisfies some suitable conditions, then the quasilinear elliptic Dirichlet problem $-\Delta_p u + |u|^{p-2} u = Q(x)|u|^{q-2}u$ in an unbounded cylinder domain $\Omega$ has at least two solutions in which one is a positive ground state solution and the other is a nodal solution.

Ground states for a system of Schrödinger equations with critical exponent

We study the following system of nonlinear Schrödinger equations:{−Δu+μu=|u|p−1u+λv,x∈RN,−Δv+νv=|v|2⁎−2v+λu,x∈RN, where N⩾3, 2⁎=2NN−2, 1<p<2⁎−1 and μ,ν,λ are positive parameters satisfying 0<λ<μν. We show that, there is some critical value μ0∈(0,1), such that this system has a positive ground state solution if 0<μ⩽μ0. In the case μ>μ0, there exists λμ,ν∈[(μ−μ0)ν,μν) such that, this system has no ground state solutions if λ<λμ,ν; while this system has a positive ground state solution if λ>λμ,ν. In particular, if p=2⁎−1, the system has no nontrivial solutions. Some further properties of the ground state solutions are also studied. This seems to be the first result for such a critical Schrödinger system.

MULTIPLE SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS IN UNBOUNDED CYLINDER DOMAINS

In this paper, we show that if $Q(x)$ satisfies some suitable conditions, then the quasilinear elliptic Dirichlet problem $-\Delta_p u+|u|^{p-2} u=Q(x)|u|^{q-2}u$ in an unbounded cylinder domain $\Omega$ has at least two solutions in which one is a positive ground state solution and the other is a nodal solution.

A note on a superlinear and periodic elliptic system in the whole space

This paper is concerned with the following periodic Hamiltonianelliptic system$ -\Delta u+V(x)u=g(x,v)$ in $R^N,$$ -\Delta v+V(x)v=f(x,u)$ in $R^N,$ $ u(x)\to 0$ and $v(x)\to 0$ as $|x|\to\infty,$ where the potential $V$ is periodic and has a positive bound frombelow, $f(x,t)$ and $g(x,t)$ are periodic in $x$ and superlinear butsubcritical in $t$ at infinity. By using generalized Nehari manifoldmethod, existence of a positive ground state solution as wellas multiple solutions for odd $f$ and $g$ are obtained.

Locating peaks of a Schrödinger equation with sign-changing nonlinearity

In this paper, we study the concentration phenomenon of a positive ground state solution of a nonlinear Schrödinger equation on R N . The coefficient of the nonlinearity of the equation changes sign. We prove that the solution has a maximum point at x 0 ∈ Ω + = { x ∈ R N : Q ( x ) > 0 } where the energy attains its minimum.

Solitary waves for a class of quasilinear Schrödinger equations in dimension two

In this paper we prove the existence and concentration behavior of positive ground state solutions for quasilinear Schrodinger equations of the form −e2Δu + V(z)u − e2 [Δ(u2)]u = h(u) in the whole two-dimension space where e is a small positive parameter and V is a continuous potential uniformly positive. The main feature of this paper is that the nonlinear term h(u) is allowed to enjoy the critical exponential growth with respect to the Trudinger–Moser inequality and also the presence of the second order nonhomogeneous term [Δ(u2)]u which prevents us to work in a classical Sobolev space. Using a version of the Trudinger–Moser inequality, a penalization technique and mountain-pass arguments in a nonstandard Orlicz space we establish the existence of solutions that concentrate near a local minimum of V.

SEMI-CLASSICAL STATES FOR QUASILINEAR SCHRÖDINGER EQUATIONS ARISING IN PLASMA PHYSICS

In this paper, the existence and qualitative properties of positive ground state solutions for the following class of Schrödinger equations -ε2Δu + V(x)u - ε2[Δ(u2)]u = f(u) in the whole two-dimensional space are established. We develop a variational method based on a penalization technique and Trudinger–Moser inequality, in a nonstandard Orlicz space context, to build up a one parameter family of classical ground state solutions which concentrates, as the parameter approaches zero, around some point at which the solutions will be localized. The main feature of this paper is that the nonlinearity f is allowed to enjoy the critical exponential growth and also the presence of the second order nonhomogeneous term -ε2[Δ(u2)]u which prevents us from working in a classical Sobolev space. Our analysis shows the importance of the role played by the parameter ε for which is motivated by mathematical models in physics. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.

Multiple Solutions for Asymptotically Linear Elliptic Systems

This paper is concerned with the following periodic Hamiltonian elliptic system $$\left\{ {\begin{array}{lll} { - \Delta u + V(x)u = g(x,v)} & {{\text{in}}} & {\mathbb{R}^N }, { - \Delta u + V(x)v = f(x,v)} & {{\text{in}}} & {\mathbb{R}^N }, {u(x) \to 0\,\,\text{and}\,\,v(x) \to 0} & {{\text{as}}} & {|x| \to \infty }, \end{array} } \right.$$ where the potential V is periodic and has a positive bound from below, f(x, t) and g(x, t) are periodic in x, asymptotically linear in t as \(|t| \rightarrow \infty\). By using critical point theory of strongly indefinite functionals, existence of a positive ground state solution as well as infinitely many geometrically distinct solutions for odd f and g are obtained.

A note on solutions for asymptotically linear elliptic systems

In this paper, we are concerned with the elliptic system of $$ \left\{ {\begin{array}{*{20}c} { - \Delta u + V(x)u = g(x,v), x \in R^N ,} \\ { - \Delta v + V(x)v = f(x,u), x \in R^N ,} \\ \end{array} } \right. $$ where V (x) is a continuous potential well, ƒ, g are continuous and asymptotically linear as t → ∞. The existence of a positive solution and ground state solution are established via variational methods.