Nontrivial Ground State Solution Research Articles

Existence and multiplicity of non-trivial solutions for a class of modified Schrödinger–Poisson systems

In the present paper, we study the following modified Schrödinger–Poisson system (P){−△u+V(x)u+ϕu−12u△u2=∣u∣p−1u+h(x),inR3,−△ϕ=u2,inR3, where 4≤p+1<12, h∈L2(R3) and h≠0. This model describes the interaction of a charge particle with an electro-magnetic field. Under some hypotheses on V(x), we prove the existence of a non-trivial ground state solution and two non-trivial ground state solutions for the system (P).

On the ground state solution for a critical fractional Laplacian equation

In this paper, we study the existence of ground state solutions for the following critical fractional Laplacian equation {(−Δ)α2u=|u|2α∗−2u+μuinΩ,u=0on∂Ω, where 0<α<2,Ω⊂RN,N>(1+2)α is a bounded domain. We will prove that the problem possesses a nontrivial ground state solution for any μ>0.

Ground state solutions for singular p-Laplacian equation in [formula omitted

In this work, we study the ground state solution for the class of singular quasilinear elliptic problem of the form − div ( | x | − a p | ∇ u | p − 2 ∇ u ) + h ( | x | ) | u | m − 2 u = H ( | x | ) | u | q − 2 u , x ∈ R N , where 1 < p < N , a < N − p p and h ( r ) , H ( r ) ⩾ 0 . We prove the compactly embedding results of the weighted Z = D a , r 1 , p ( R N ) ∩ L h , r m ( R N ) space of radially symmetric functions and then obtain the existence of nontrivial ground state solution.