Abstract
Based on Nehari manifold, Schwarz symmetric methods and critical point theory, we prove the existence of positive radial ground states for a class of Schrodinger-Poisson systems in , which doesn’t require any symmetry assumptions on all potentials. In particular, the positive potential is interesting in physical applications.
Highlights
In this paper, we consider the following nonlinear Schrodinger-Poisson systems−∆u +V ( x)u − λρ ( x) Φu + Q ( x) u p= −2 u 0,= −∆Φ ρ (x)u2, x∈ 3, x∈ 3, (1.1)where λ > 0, 2 < p < 4 ; V ( x), ρ ( x) and Q ( x) are positive potentials defined in 3 .In recent years, such systems have been paid great attention by many authors concerning existence, nonexistence, multiplicity and qualitative behavior
The systems are to describe the interaction of nonlinear Schrodinger field with an electromagnetic field
When λ = −1, V= ( x) ρ= ( x) 1, Q ( x) = −1, the existence of nontrivial solution for the problem (1.1) was proved as p ∈ (4, 6) in [1], and non-existence result for p ∈ (0, 2] or p ∈ (6, +∞) was proved in [2]
Summary
Where λ > 0 , 2 < p < 4 ; V ( x) , ρ ( x) and Q ( x) are positive potentials defined in 3. (2015) Ground States for a Class of Nonlinear Schrodinger-Poisson Systems with Positive Potential. Sanchel and Soler [6] considered the following Schrodinger-Poisson-Slater systems. The solution is obtained by using the minimization argument and ω as a Lagrange multiplier It is not known if the solution for the problem (1.2) is radial. Without requiring any symmetry assumptions on V ( x) , ρ ( x) and Q ( x) , we obtain the existence of positive radial ground state solution for the problem (1.1). The positive potential Q ( x) implies that we are dealing with systems of particles having positive mass.
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