Steep Potential Well Research Articles

Existence and concentration of ground state solutions for critical Schrödinger equation with steep potential well

In this paper, we study the following Schrödinger equation −Δu+(1+μa(x))u=f(u)+|u|2∗−2u,inRN,where N≥3 and μ>0. We obtain a ground state solution for the above equation. Moreover, the concentration behavior of the ground state solution is also described as μ→∞.

Existence of solution for a partial differential inclusion in $$\pmb {{\mathbb {R}}}^{\varvec{N}}$$ R N with steep potential well

In this work, we study the existence of nontrivial solutions for the following class of partial differential inclusion problem: $$\begin{aligned} -\Delta u+\left( 1+\lambda V(x)\right) u\in \partial _t F(x,u) \quad \text {in} \quad \mathbb {R}^N, \qquad \quad (P_\lambda ) \end{aligned}$$ where $$N\ge 1$$ , $$\lambda >0$$ , V is a continuous function verifying some conditions and $$\partial _t F(x,u)$$ is the generalized gradient of F(x, t) with respect to t. Assuming that F(x, t) is a mensurable function for each $$t\in \mathbb {R}$$ and locally Lipschitzian for each $$x\in \mathbb {R}^N$$ , we have applied variational methods for locally Lipschitz functionals to get a solution for $$(P_\lambda )$$ when $$\lambda $$ is large enough.

Existence and multiplicity of solutions for the fractional Schrödinger equation with steep potential well and perturbation

In the present paper, we investigate the solutions of a fractional Schrodinger equation with sublinear perturbation and steep potential well. First, we obtain the existence of at least one solution by the minimisation result due to Mawhin and Willem. Second, by virtue of the symmetric mountain pass theorem, the existence criteria of infinitely many solutions are established. Lastly, an example is given to demonstrate the main results.

Positive solutions of a superlinear kirchhoff type equation in [formula omitted] (N ≥ 4)

In this paper, we study a class of superlinear Kirchhoff type equations with steep potential well:{−(a∫RN|∇u|2dx+b)Δu+λV(x)u=f(x,u)inRN,u∈H1(RN),where N ≥ 4, a, b > 0, and the parameter λ > 0. By proposing new techniques and introducing new superlinear hypotheses on f, we establish the existence of two positive solutions, one of which will blow up as the nonlocal term vanishes. Some existing results are improved and generalized in the literature. Moreover, the nonexistence of nontrivial solution is also obtained.

Ground state solutions for Klein–Gordon–Maxwell system with steep potential well

In this paper, we study the following Klein–Gordon–Maxwell system −Δu+(λa(x)+1)u−(2ω+ϕ)ϕu=f(x,u),inR3,−Δϕ=−(ω+ϕ)u2,inR3.Using variational methods, we obtain the existence of ground state solutions under some appropriate assumptions on a and f.

Existence and concentrating behavior of solutions for Kirchhoff type equations with steep potential well

In this paper, we consider the following Kirchhoff type equations(0.1){−(a+b∫R3|∇u|2)Δu+λV(x)u=q(x)f(u) in R3,u∈H1(R3), where a, b, λ>0, V∈C(R3,R) is a potential well, q(x) is a positive bounded function, f(s) is either asymptotically linear or asymptotically 3-linear in s at infinity. Under some other suitable conditions on V, q and f, the existence, nonexistence and concentrating behavior of solutions to problem (0.1) are obtained by using variational methods. We mainly extend the results in J. Sun and T. Wu (2014) [26], which dealt with Kirchhoff type equations with positive potential well, to Kirchhoff type equations with sign-changing potential well.

Existence and concentration result for a class of fractional Kirchhoff equations with Hartree-type nonlinearities and steep potential well

In this paper, we study the following fractional Kirchhoff equations{(a+b∫RN|(−△)α2u|2dx)(−△)αu+λV(x)u=(|x|−μ⁎G(u))g(u),u∈Hα(RN),N≥3, where a,b>0 are constants, and (−△)α is the fractional Laplacian operator with α∈(0,1),2<2α,μ⁎=2N−μN−2α≤2α⁎=2NN−2α, 0<μ<2α, λ>0, is real parameter. 2α⁎ is the critical Sobolev exponent. g satisfies the Berestycki–Lions-type condition (see [2]). By using Pohožaev identity and concentration-compact theory, we show that the above problem has at least one nontrivial solution. Furthermore, the phenomenon of concentration of solutions is also explored. Our result supplements the results of Lü (see [8]) concerning the Hartree-type nonlinearity g(u)=|u|p−1u with p∈(2,6−α).

The existence of nontrivial solution for a class of sublinear biharmonic equations with steep potential well

In this paper, we study the following biharmonic equation: \t\t\t{Δ2u−Δu+λV(x)u=α(x)f(u)+μK(x)|u|q−2uin RN,u∈H2(RN),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} \\Delta^{2}u - \\Delta u + \\lambda V( x)u = \\alpha( x) f(u) + \\mu K(x) \\vert u \\vert ^{q-2}u \\quad\\text{in } \\mathbb{R}^{N},\\\\ u\\in H^{2}(\\mathbb{R}^{N}), \\end{cases} $$\\end{document} where Delta^{2}u=Delta(Delta u), N>4, lambda>0, 1< q<2 and muin[0,mu_{0}]. By using Ekeland’s variational principle and Gigliardo–Nirenberg’s inequality, we prove the existence of nontrivial solution for the above problem.

Energy solutions and concentration problem of fractional Schr\xf6dinger equation

In this paper, we consider a fractional Schrödinger equation with steep potential well and sublinear perturbation. By virtue of variational methods, the existence criteria of infinitely many nontrivial high or small energy solutions are established. In addition, the phenomenon of the concentration of solutions is also explored. We also give some examples to demonstrate the main results.

Existence and concentration of ground state sign-changing solutions for Kirchhoff type equations with steep potential well and nonlinearity

We study the following class of elliptic equations: \begin{equation*} -\bigg(a+b\int_{{\mathbb R}^3}|\nabla u|^2dx\bigg)\Delta u+\lambda V(x)u=f(u), \quad x\in{\mathbb R}^3, \end{equation*} where $\lambda,a,b&gt; 0$, $V\in \mathcal{C}({\mathbb R}^3,{\mathbb R})$ and $V^{-1}(0)$ has nonempty interior. First, we obtain one ground state sign-changing solution $u_{b,\lambda}$ applying the non-Nehari manifold method. We show that the energy of $u_{b,\lambda}$ is strictly larger than twice that of the ground state solutions of Nehari-type. Next we establish the convergence property of $u_{b,\lambda}$ as $b\searrow0$. Finally, the concentration of $u_{b,\lambda}$ is explored on the set $V^{-1}(0)$ as $\lambda\rightarrow\infty$.

Ground state of solutions for a class of fractional Schrödinger equations with critical Sobolev exponent and steep potential well

In this paper, we study the following fractional Schrödinger equations: urn:x-wiley:mma:media:mma4527:mma4527-math-0001 where (−△)α is the fractional Laplacian operator with , 0≤s≤2α, λ&gt;0, κ and β are real parameter. is the critical Sobolev exponent. We prove a fractional Sobolev‐Hardy inequality and use it together with concentration compact theory to get a ground state solution. Moreover, concentration behaviors of nontrivial solutions are obtained when the coefficient of the potential function tends to infinity.

Existence of nontrivial solution for Schrödinger–Poisson systems with indefinite steep potential well

In this paper, we study a class of nonlinear Schrodinger–Poisson systems with indefinite steep potential well: $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+V_{\lambda }(x)u+K(x)\phi u=|u|^{p-2}u &{} \text { in }\mathbb {R}^{3},\\ -\Delta \phi =K\left( x\right) u^{2} &{} \ \text {in }\mathbb {R}^{3}, \end{array} \right. \end{aligned}$$ where $$3<p< 4$$ , $$V_{\lambda }(x)=\lambda a(x)+b(x)$$ with $$\lambda >0$$ and $$ K(x)\ge 0$$ for all $$x\in \mathbb {R}^{3}$$ . We require that $$a\in C( \mathbb {R}^{3}) $$ is nonnegative and has a potential well $$\Omega _{a}$$ , namely $$a(x)\equiv 0$$ for $$x\in \Omega _{a}$$ and $$a(x)>0$$ for $$x\in \mathbb {R}^{3}\setminus \overline{\Omega _{a}}$$ . Unlike most other papers on this problem, we allow that $$b\in C(\mathbb {R}^{3}) $$ is unbounded below and sign-changing. By introducing some new hypotheses on the potentials and applying the method of penalized functions, we obtain the existence of nontrivial solutions for $$\lambda $$ sufficiently large. Furthermore, the concentration behavior of the nontrivial solution is also described as $$\lambda \rightarrow \infty $$ .

Multiplicity and concentration of solutions for fractional Schrödinger equation with sublinear perturbation and steep potential well

In this paper, we investigate the following fractional Schrödinger equation with sublinear perturbation and steep potential well {(−△)su+λV(x)u=f(x,u)+α(x)|u|ν−2uinRN,u∈Hs(RN), where 0<s<1,2s<N,λ>0,1<ν<2, f∈C(RN×R) is of subcritical growth. By using variational methods, we prove that such a class of equations possess at least two nontrivial solutions. Moreover, the phenomenon of concentration of solutions is explored as well.

Spectral characteristics of steady-state Lévy flights in confinement potential profiles

The steady-state correlation characteristics of superdiffusion in the form of Lévy flights in one-dimensional confinement potential profiles are investigated both theoretically and numerically. Specifically, for Cauchy stable noise we calculate the steady-state probability density function for an infinitely deep rectangular potential well and for a symmetric steep potential well of the type . For these potential profiles and arbitrary Lévy index α, we obtain the asymptotic expression of the spectral power density.

Existence and asymptotic behavior of solutions for nonlinear Schrödinger-Poisson systems with steep potential well.

In this paper, we are concerned with a class of Schrödinger-Poisson systems with the asymptotically linear or asymptotically 3-linear nonlinearity. Under some suitable assumptions on V, K, a, and f, we prove the existence, nonexistence, and asymptotic behavior of solutions via variational methods. In particular, the potential V is allowed to be sign-changing for the asymptotically linear case.

Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well

In this paper we study the non-linear fractional Schrodinger equation with steep potential well \begin{eqnarray} (-\Delta)^{\alpha}u + \lambda V(x)u = f(x,u) in R^{n}, u\in H^{\alpha}(R^n), \end{eqnarray} where $(-\Delta)^\alpha$ ($\alpha \in (0,1)$) denotes the fractional Laplacian, $\lambda$ is a parameter, $V\in C(\mathbb{R}^n)$ and $V^{-1}(0)$ has nonempty interior. Under some suitable conditions, the existence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is also explored.

Existence and concentration of solutions for Schrödinger–Poisson system with steep potential well

This paper is concerned with the nonlinear Schrödinger–Poisson system urn:x-wiley:mma:media:mma3712:mma3712-math-0001 where λ &gt; 0 is a parameter. We require that V≥0 and has a bounded potential well V−1(0). Combining this with other suitable assumptions on K and f, the existence of nontrivial solutions is obtained by using variational methods. Moreover, the concentration of solutions is also explored on the set V−1(0) as λ→∞. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Elastic properties and atomic bonding character in metallic glasses

The elastic properties of glasses from different metallic systems were studied in the light of the atomic packing density and bonding character. We found that the electronegativity mismatch (Δe−) between the host- and the major solute-elements provides a plausible explanation to the large variation observed for Poisson's ratio (ν) among metallic glasses (MGs) (from 0.28 for Fe-based to 0.43 for Pd-based MGs), notwithstanding a similar atomic packing efficiency (Cg). Besides, it is found that ductile MGs correspond to Δe− smaller than 0.5 and to a relatively steep atomic potential well. Ductility is, thus, favored in MGs exhibiting a weak bond directionality on average and opposing a strong resistance to volume change.

Existence of ground state solutions for a super-biquadratic Kirchhoff-type equation with steep potential well

In this paper, we concern with the following Kirchhoff-type problemwhere are constants, is a parameter, is a nonnegative continuous potential, and is a subcritical growth nonlinearity. Under appropriate assumptions on and , we prove the existence of ground state solutions using variational methods. Furthermore, we also study the asymptotic behavior of ground state solutions.

Multi-bump solutions of Schrödinger–Poisson equations with steep potential well

In this paper, we study a system of Schrodinger–Poisson equation $$\left\{\begin{array}{ll}-\Delta u+(\lambda a(x)+a_0(x))u+K(x)\phi u=|u|^{p-2}u, & \quad x \in\mathbb{R}^3, \\- \Delta \phi=K(x)u^2,& \quad x \in \mathbb{R}^3 \end{array} \right.$$ where $${p \in (4,6)}$$ and $${\lambda}$$ is a parameter. We require that $${a(x) \geq 0}$$ and has a bounded potential well $${\Omega = a^{-1}(0)}$$ . Combining this with other suitable assumptions on Ω, a 0 and K, we obtain the existence of multi-bump-type solution $${u_\lambda}$$ when $${\lambda}$$ is large via variational methods.