Existence of ground state solution for the first order discrete Hamiltonian systems

AbstractIn this paper we are concerned with the quasilinear Schrödinger equation−Δu + V(x)u − Δ(uwhere N ≥ 3, 4 < p < 4N/(N − 2), and V(x) and q(x) go to some positive limits V

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Multiplicity results for some nonlinear elliptic problems

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700008934 ◽

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.

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Ground state solution for a class of magnetic equation with general convolution nonlinearity

AIMS Mathematics ◽

10.3934/math.2021528 ◽

2021 ◽

Vol 6 (8) ◽

pp. 9100-9108

Author(s):

Li Zhou ◽

◽

Chuanxi Zhu ◽

Keyword(s):

Ground State ◽

Ground State Solution ◽

State Solution

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Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

Asymptotic Analysis ◽

10.3233/asy-211737 ◽

2021 ◽

pp. 1-19

Author(s):

Jing Zhang ◽

Lin Li

Keyword(s):

Schrödinger Equation ◽

Ground State ◽

Schrodinger Equation ◽

Critical Sobolev Exponent ◽

Ground State Solution ◽

Nehari Manifold ◽

State Solution ◽

Hardy Potential ◽

Asymptotically Periodic ◽

Sobolev Exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

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