Behavior Of Least Energy Solutions Research Articles

A critical phenomenon for the least energy solutions to nonlinear scalar field equations

We study the existence of least energy solutions to the nonlinear scalar field equation: (1) − Δ u + λ u + V ( x ) u = Q ( x ) | u | p u , u ∈ H 1 ( R N ) , Where V ( x ) , Q ( x ) ∈ L ∞ ( R N ) are real functions satisfying suitable assumptions. By considering the Nehari type constraint N λ := { u ∈ H 1 ( R N ) ∖ { 0 } : ∫ R N ( | ∇ u | 2 + λ | u | 2 + V ( x ) | u | 2 ) d x = ∫ R N Q ( x ) | u | p + 2 d x } , it is shown that (1) exists at least a nontrivial least energy solutions if \\lambda _{*}:=-\\inf \\sigma (-\\Delta +V) $ ]]> λ > λ ∗ := − inf σ ( − Δ + V ) . In this argument, we point out λ ∗ is a critical value for the existence of least energy solutions restricted to N λ . That is, there exists global least energy solutions in N λ if \\lambda _{*} $ ]]> λ > λ ∗ , but not if λ < λ ∗ . Moreover, the asymptotic behavior of least energy solutions as λ → + ∞ is analyzed.

Existence and limit behavior of least energy solutions to constrained Schrödinger–Bopp–Podolsky systems in $${\mathbb {R}}^3$$

Consider the following Schrödinger–Bopp–Podolsky system in $${\mathbb {R}}^3$$ under an $$L^2$$ -norm constraint, $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + \omega u + \phi u = u|u|^{p-2},\\ -\Delta \phi + a^2\Delta ^2\phi =4\pi u^2,\\ \Vert u\Vert _{L^2}=\rho , \end{array}\right. } \end{aligned}$$ where $$a,\rho >0$$ are fixed, with our unknowns being $$u,\phi :{\mathbb {R}}^3\rightarrow {\mathbb {R}}$$ and $$\omega \in {\mathbb {R}}$$ . We prove that if $$2<p<3$$ (resp., $$3<p<10/3$$ ) and $$\rho >0$$ is sufficiently small (resp., sufficiently large), then this system admits a least energy solution. Moreover, we prove that if $$2<p<14/5$$ and $$\rho >0$$ is sufficiently small, then least energy solutions are radially symmetric up to translation, and as $$a\rightarrow 0$$ , they converge to a least energy solution of the Schrödinger–Poisson–Slater system under the same $$L^2$$ -norm constraint.

Asymptotic behavior of least energy solutions to the Finsler Lane-Emden problem with large exponents

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On the behavior of least energy solutions of a fractional ( p , q ( p ) )-Laplacian problem as p goes to infinity

We study the behavior as p → ∞ of u p , a positive least energy solution of the problem [ ( − Δ p ) α + ( − Δ q ( p ) ) β ] u = μ p | u ( x u ) | p − 2 u ( x u ) δ x u in Ω u = 0 in R N ∖ Ω | u ( x u ) | = ‖ u ‖ ∞ , where Ω ⊂ R N is a bounded, smooth domain, δ x u is the Dirac delta distribution supported at x u , lim p → ∞ q ( p ) p = Q ∈ ( 0 , 1 ) if 0 &lt; β &lt; α &lt; 1 ( 1 , ∞ ) if 0 &lt; α &lt; β &lt; 1 and lim p → ∞ μ p p &gt; R − α , with R denoting the inradius of Ω.

Asymptotic behavior of least energy solutions to the Lane-Emden system near the critical hyperbola

We consider the Lane-Emden system{−Δu=vpin Ω,−Δv=uqin Ω,u,v>0in Ω,u=v=0on ∂Ω where Ω is a smooth bounded domain in Rn for n≥3 and 0<p<q<∞. The asymptotic behavior of least energy solutions near the critical hyperbola was studied by Guerra [17] when p≥1 and the domain is convex. In this paper, we cover all the remaining cases p<1 and extend the results to arbitrary smooth bounded domains.

Least energy solutions for indefinite biharmonic problems via modified Nehari–Pankov manifold

In this paper, by using a modified Nehari–Pankov manifold, we prove the existence and the asymptotic behavior of least energy solutions for the following indefinite biharmonic equation: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is a parameter, [Formula: see text] is a nonnegative potential function with nonempty zero set [Formula: see text], [Formula: see text] is a positive function such that the operator [Formula: see text] is indefinite and non-degenerate for [Formula: see text] large. We show that both in subcritical and critical cases, equation [Formula: see text] admits a least energy solution which for [Formula: see text] large localized near the zero set [Formula: see text].

Solutions for the Problems Involving Fractional Laplacian and Indefinite Potentials

Abstract In this paper, we consider a class of Schrödinger equations involving fractional Laplacian and indefinite potentials. By modifying the definition of the Nehari–Pankov manifold, we prove the existence and asymptotic behavior of least energy solutions. As the fractional Laplacian is nonlocal, when the bottom of the potentials contains more than one isolated components, the least energy solutions may localize near all the isolated components simultaneously. This phenomenon is different from the Laplacian.

Asymptotic Behavior of Least Energy Solutions for a Critical Elliptic System

Asymptotic Behavior of Least Energy Solutions for a Critical Elliptic System

Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian

In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen as a nonlocal analog of the results of Han (1991) [24] and Rey (1990) [35].

Asymptotic behavior of least energy solutions for a 2D nonlinear Neumann problem with large exponent

In this paper, we consider the following elliptic problem with the nonlinear Neumann boundary condition:(Ep){−Δu+u=0onΩ,u>0onΩ,∂u∂ν=upon∂Ω, where Ω is a smooth bounded domain in R2, ν is the outer unit normal vector to ∂Ω, and p>1 is any positive number.We study the asymptotic behavior of least energy solutions to (Ep) when the nonlinear exponent p gets large. Following the arguments of X. Ren and J.C. Wei [13,14], we show that the least energy solutions remain bounded uniformly in p, and it develops one peak on the boundary, the location of which is controlled by the Green function associated to the linear problem.

Asymptotic behavior of least energy solutions for a singularly perturbed problem with nonlinear boundary condition

We consider the problem of finding a positive harmonic function \(u_\varepsilon \) in a bounded domain \(\Omega \subset \mathbb R ^N (N\ge 3)\) satisfying a nonlinear boundary condition of the form \(\varepsilon \partial _{\nu } u +u =|u|^{p-2}u,\,x\in \partial \Omega \), where \(\varepsilon \) is a positive parameter and \(2<p<2_*:=2(N-1)/(N-2)\). To be more precise, by using min-max methods, we study the existence of least energy solution \(u_\varepsilon \) of the problem depending on the parameter \(\varepsilon \). We provide a detailed description of the shape of \(u_\varepsilon \) and prove that the maximum of \(u_\varepsilon \) is achieved at a point \(z_{\varepsilon }\), which lies on the boundary \(\partial \Omega \) and concentrates at the mean curvature maximum point of the boundary \(\partial \Omega \). This problem is related to the existence of extremals for a Sobolev inequality involving the trace embedding and the asymptotic behavior of the best constants in expanding domains.

Asymptotic behavior of least energy solutions for a biharmonic problem with nearly critical growth

Asymptotic behavior of least energy solutions for a biharmonic problem with nearly critical growth

Single-point condensation phenomena for a four-dimensional biharmonic Ren–Wei problem

We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (Ep): \((E_p) \left \{ \begin{array}{ll} \Delta^2 u = u^p & \hbox{in} \; \Omega, \\ u > 0 & \hbox{in} \; \Omega, \\ u |_{\partial\Omega} = \Delta u |_{\partial\Omega} = 0 \end{array} \right.\) as p gets large, where Ω is a smooth bounded domain in R4. In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc. AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky pattern, under the assumption that the domain is convex.

Solutions of an elliptic system with a nearly critical exponent

Consider the problem−Δuɛ=vɛp,vɛ>0inΩ,−Δvɛ=uɛqɛ,uɛ>0inΩ,uɛ=vɛ=0on∂Ω, where Ω is a bounded convex domain in RN, N>2, with smooth boundary ∂Ω. Here p,qɛ>0, andɛ:=Np+1+Nqɛ+1−(N−2). This problem has positive solutions for ɛ>0 (with pqɛ>1) and no non-trivial solution for ɛ⩽0. We study the asymptotic behavior of least energy solutions as ɛ→0+. These solutions are shown to blow-up at exactly one point, and the location of this point is characterized. In addition, the shape and exact rates for blowing up are given.

Asymptotic Behavior of Least Energy Solutions of a Biharmonic Equation in Dimension Four

In this paper we consider a biharmonic equation on a bounded domain in R 4 with large exponent in the nonlinear term. We study asymptotic behavior of positive solutions obtained by minimizing suitable functionals. Among other results, we prove that c p , the minimum of energy functional with the nonlinear exponent equal to p, is like ρ 4 e/p as p → +∞, where ρ 4 = 32ω 4 and ω 4 is the area of the unit sphere S 3 in R 4 . Using this result, we compute the limit of the L ∞ -norm of least energy solutions as p → +∞. We also show that such solutions blow up at exactly one point which is a critical point of the Robin function.

Asymptotic behavior of least energy solutions to a four-dimensional biharmonic semilinear problem

In this paper, we study the following fourth order elliptic problem $(E_p)$: \begin{eqnarray*} (E_p) \left \{ \begin{array}{l} \Delta^2 u = u^p \quad \mbox{in} \Omega, u > 0 \quad \mbox{in} \Omega, u |_{\partial\Omega} = \Delta u |_{\partial\Omega} = 0 \end{array} \right. \end{eqnarray*} where $\Omega$ is a smooth bounded domain in $\mathbf{R}^4$, $\Delta^2 = \Delta\Delta$ is a biharmonic operator and $p >1$ is any positive number. We investigate the asymptotic behavior as $p \to \infty$ of the least energy solutions to $(E_p)$. Combining the arguments of Ren-Wei [8] and Wei [10], we show that the least energy solutions remain bounded uniformly in $p$, and on convex bounded domains, they have one or two ``peaks'' away form the boundary. If it happens that the only one peak point appears, we further prove that the peak point must be a critical point of the Robin function of $\Delta^2$ under the Navier boundary condition.

Spikes in two coupled nonlinear Schrödinger equations

Here we study the interaction and the configuration of spikes in a double condensate by analyzing least energy solutions of two coupled nonlinear Schrödinger equations which model Bose–Einstein condensates of two different hyperfine spin states. When the interspecies scattering length is positive and large enough, spikes of a double condensate repel each other and behave like two separate spikes. In contrast, spikes of a double condensate attract each other and behave like a single spike if the interspecies scattering length is negative and large enough. Our mathematical arguments can prove such physical phenomena. We first use Nehari's manifold to construct least energy solutions, and then use some techniques of singular perturbation problems to derive the asymptotic behavior of least energy solutions. It is shown that the interaction term determines the locations of the two spikes and the asymptotic shape of least energy solutions.

Asymptotic behavior of least energy solutions to a semilinear Dirichlet problem near the critical exponent

Asymptotic behavior of least energy solutions to a semilinear Dirichlet problem near the critical exponent