scholarly journals Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials

2019 ◽  
Vol 9 (1) ◽  
pp. 496-515 ◽  
Author(s):  
Sitong Chen ◽  
Xianhua Tang
Keyword(s):  
Ground State ◽  
Radial Symmetry ◽  
Ground State Solution ◽  
Bounded Sequence ◽  
Energy Solution ◽  
Related Literature ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger ◽  
Least Energy Solution ◽  
Least Energy

Abstract This paper is dedicated to studying the nonlinear Schrödinger equations of the form $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(u), & x\in \mathbb{R}^N; \\ u\in H^1(\mathbb{R}^N), \end{array} \right. \end{array}$$ where V ∈ 𝓒1(ℝN, [0, ∞)) satisfies some weak assumptions, and f ∈ 𝓒(ℝ, ℝ) satisfies the general Berestycki-Lions assumptions. By introducing some new tricks, we prove that the above problem admits a ground state solution of Pohožaev type and a least energy solution. These results generalize and improve some ones in [L. Jeanjean, K. Tanka, Indiana Univ. Math. J. 54 (2005), 443-464], [L. Jeanjean, K. Tanka, Proc. Amer. Math. Soc. 131 (2003) 2399-2408], [H. Berestycki, P.L. Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related literature. In particular, our assumptions are “almost” necessary when V(x) ≡ V∞ > 0, moreover, our approach could be useful for the study of other problems where radial symmetry of bounded sequence either fails or is not readily available, or where the ground state solutions of the problem at infinity are not sign definite.

Ground State Solutions for the Nonlinear Schrödinger–Bopp–Podolsky System with Critical Sobolev Exponent

2020 ◽  
Vol 20 (3) ◽  
pp. 511-538 ◽  
Author(s):  
Lin Li ◽  
Patrizia Pucci ◽  
Xianhua Tang
Keyword(s):  
Ground State ◽  
Critical Sobolev Exponent ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger ◽  
Sobolev Exponent

AbstractIn this paper, we study the existence of ground state solutions for the nonlinear Schrödinger–Bopp–Podolsky system with critical Sobolev exponent\left\{\begin{aligned} \displaystyle{}{-}\Delta u+V(x)u+q^{2}\phi u&% \displaystyle=\mu|u|^{p-1}u+|u|^{4}u&&\displaystyle\phantom{}\mbox{in }\mathbb% {R}^{3},\\ \displaystyle{-}\Delta\phi+a^{2}\Delta^{2}\phi&\displaystyle=4\pi u^{2}&&% \displaystyle\phantom{}\mbox{in }\mathbb{R}^{3},\end{aligned}\right.where {\mu>0} is a parameter and {2<p<5}. Under certain assumptions on V, we prove the existence of a nontrivial ground state solution, using the method of the Pohozaev–Nehari manifold, the arguments of Brézis–Nirenberg, the monotonicity trick and a global compactness lemma.

scholarly journals Ground state solution of semilinear Schrödinger system with local super-quadratic conditions

2021 ◽  
pp. 1-18
Author(s):  
Jing Chen ◽  
Yiqing Li
Keyword(s):  
Variational Method ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
The Other ◽  
Ground State Solutions ◽  
Approximation Argument ◽  
Least Energy Solutions ◽  
Least Energy ◽  
Schrödinger System

In this paper, we dedicate to studying the following semilinear Schrödinger system equation*-Δu+V1(x)u=Fu(x,u,v)amp;mboxin~RN,r-Δv+V2(x)v=Fv(x,u,v)amp;mboxin~RN,ru,v∈H1(RN),endequation* where the potential Vi are periodic in x,i=1,2, the nonlinearity F is allowed super-quadratic at some x ∈ R N and asymptotically quadratic at the other x ∈ R N . Under a local super-quadratic condition of F, an approximation argument and variational method are used to prove the existence of Nehari–Pankov type ground state solutions and the least energy solutions.

scholarly journals Ground state solutions for Kirchhoff-type equations with general nonlinearity in low dimension

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Chen ◽  
Yiqing Li
Keyword(s):  
Ground State ◽  
Ground State Solution ◽  
Nonlocal Term ◽  
Sobolev’S Inequality ◽  
Kirchhoff Type ◽  
Least Energy Solution ◽  
Kirchhoff Type Problem ◽  
Low Dimension ◽  
New Energy ◽  
Least Energy

AbstractThis paper is dedicated to studying the following Kirchhoff-type problem: $$ \textstyle\begin{cases} -m ( \Vert \nabla u \Vert ^{2}_{L^{2}(\mathbb{R} ^{N})} )\Delta u+V(x)u=f(u), & x\in \mathbb{R} ^{N}; \\ u\in H^{1}(\mathbb{R} ^{N}), \end{cases} $$ { − m ( ∥ ∇ u ∥ L 2 ( R N ) 2 ) Δ u + V ( x ) u = f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where $N=1,2$ N = 1 , 2 , $m:[0,\infty )\rightarrow (0,\infty )$ m : [ 0 , ∞ ) → ( 0 , ∞ ) is a continuous function, $V:\mathbb{R} ^{N}\rightarrow \mathbb{R} $ V : R N → R is differentiable, and $f\in \mathcal{C}(\mathbb{R} ,\mathbb{R} )$ f ∈ C ( R , R ) . We obtain the existence of a ground state solution of Nehari–Pohozaev type and the least energy solution under some assumptions on V, m, and f. Especially, the existence of nonlocal term $m(\|\nabla u\|^{2}_{L^{2}(\mathbb{R} ^{N})})$ m ( ∥ ∇ u ∥ L 2 ( R N ) 2 ) and the lack of Hardy’s inequality and Sobolev’s inequality in low dimension make the problem more complicated. To overcome the above-mentioned difficulties, some new energy inequalities and subtle analyses are introduced.

scholarly journals Existence of Positive Ground State Solution for the Nonlinear Schrödinger-Poisson System with Potentials

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Lu Xiao ◽  
Haiqin Qian ◽  
Mengmeng Qu ◽  
Jun Wang
Keyword(s):  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Poisson System ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger

In the present paper we study the existence of positive ground state solutions for the nonautonomous Schrödinger-Poisson system with competing potentials. Under some assumptions for the potentials we prove the existence of positive ground state solution.

scholarly journals Ground state solutions for Hamiltonian elliptic systems with super or asymptotically quadratic nonlinearity

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yubo He ◽  
Dongdong Qin ◽  
Dongdong Chen
Keyword(s):  
Ground State ◽  
Spectral Gap ◽  
Elliptic Systems ◽  
Technical Condition ◽  
Sigma Delta ◽  
Ground State Solutions ◽  
Least Energy Solution ◽  
Hamiltonian Elliptic System ◽  
Least Energy ◽  
Quadratic Case

Abstract This article concerns the Hamiltonian elliptic system: $$ \textstyle\begin{cases} -\Delta \varphi +V(x)\varphi =G_{\psi }(x,\varphi ,\psi ) & \mbox{in } \mathbb {R}^{N}, \\ -\Delta \psi +V(x)\psi =G_{\varphi }(x,\varphi ,\psi ) & \mbox{in } \mathbb {R}^{N}, \\ \varphi , \psi \in H^{1}(\mathbb {R}^{N}). \end{cases} $$ { − Δ φ + V ( x ) φ = G ψ ( x , φ , ψ ) in  R N , − Δ ψ + V ( x ) ψ = G φ ( x , φ , ψ ) in  R N , φ , ψ ∈ H 1 ( R N ) . Assuming that the potential V is periodic and 0 lies in a spectral gap of $\sigma (-\Delta +V)$ σ ( − Δ + V ) , least energy solution of the system is obtained for the super-quadratic case with a new technical condition, and the existence of ground state solutions of Nehari–Pankov type is established for the asymptotically quadratic case. The results obtained in the paper generalize and improve related ones in the literature.

scholarly journals Multiplicity results for some nonlinear elliptic problems

2004 ◽  
Vol 76 (2) ◽  
pp. 247-268
Author(s):  
Kuan-Ju Chen
Keyword(s):  
Ground State ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Ground State Solution ◽  
State Solution ◽  
Energy Solution ◽  
Nonlinear Elliptic Problems ◽  
Multiplicity Results ◽  
Nonlinear Elliptic ◽  
Half Strip

AbstractIn this paper, first, we study the existence of the positive solutions of the nonlinear elliptic equations in unbounded domains. The existence is affected by the properties of the geometry and the topology of the domain. We assert that if there exists a (PS)c-sequence with c belonging to a suitable interval depending by the equation, then a ground state solution and a positive higher energy solution exist, too. Next, we study the upper half strip with a hole. In this case, the ground state solution does not exist, however there exists at least a positive higher energy solution.

scholarly journals On the location of a peak point of a least energy solution for Hénon equation

2011 ◽  
Vol 30 (4) ◽  
pp. 1055-1081 ◽  
Author(s):  
Jaeyoung Byeon ◽  
◽  
Sungwon Cho ◽  
Junsang Park ◽  
◽  
...  
Keyword(s):  
Energy Solution ◽  
Peak Point ◽  
Least Energy Solution ◽  
Hénon Equation ◽  
Least Energy
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