Ground states and multiple solutions for Choquard-Pekar equations with indefinite potential and general nonlinearity

2021 ◽  
Vol 500 (2) ◽  
pp. 125143
Author(s):  
Dongdong Qin ◽  
Lizhen Lai ◽  
Shuai Yuan ◽  
Qingfang Wu
Keyword(s):  
Multiple Solutions ◽  
Ground States ◽  
Indefinite Potential ◽  
General Nonlinearity

Multiple Solutions for the Schrödinger-Poisson Equation with a General Nonlinearity

2021 ◽  
Vol 41 (3) ◽  
pp. 703-711
Author(s):  
Yongsheng Jiang ◽  
Na Wei ◽  
Yonghong Wu
Keyword(s):  
Poisson Equation ◽  
Multiple Solutions ◽  
General Nonlinearity

scholarly journals Multiple solutions for Robin (p, q)-equations plus an indefinite potential and a reaction concave near the origin

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Andrea Scapellato
Keyword(s):  
Multiple Solutions ◽  
Principal Eigenvalue ◽  
Robin Problem ◽  
Smooth Solutions ◽  
Potential Term ◽  
Indefinite Potential

AbstractWe consider a Robin problem driven by the (p, q)-Laplacian plus an indefinite potential term. The reaction is either resonant with respect to the principal eigenvalue or $$(p-1)$$ ( p - 1 ) -superlinear but without satisfying the Ambrosetti-Rabinowitz condition. For both cases we show that the problem has at least five nontrivial smooth solutions ordered and with sign information. When $$q=2$$ q = 2 (a (p, 2)-equation), we show that we can slightly improve the conclusions of the two multiplicity theorems.

scholarly journals Multiple solutions for eigenvalue problems involving an indefinite potential and with (p1(x), p2(x)) balanced growth

2019 ◽  
Vol 27 (1) ◽  
pp. 289-307 ◽  
Author(s):  
Vasile-Florin Uţă
Keyword(s):  
Boundary Condition ◽  
Continuous Spectrum ◽  
Dirichlet Boundary Condition ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
The Other ◽  
Balanced Growth ◽  
Indefinite Potential

Abstract In this paper we are concerned with the study of the spectrum for a class of eigenvalue problems driven by two non-homogeneous differential operators with different variable growth and an indefinite potential in the following form $$\eqalign{ & - {\rm{div}}\left[ {{\cal H}(x,|\nabla u|)\nabla u + \Im (x,|\nabla u|)\nabla u} \right] + V(x)|u{|^{m(x) - 2}}u = \cr & = \lambda \left( {|u{|^{{q_1}(x) - 2}} + |u{|^{{q_2}(x) - 2}}} \right)u\;{\rm{in}}\;\Omega , \cr}$$ which is subjected to Dirichlet boundary condition. The proofs rely on variational arguments and they consist in finding two Rayleigh-type quotients, which lead us to an unbounded continuous spectrum on one side, and the nonexistence of the eigenvalues on the other.

scholarly journals Ground states and multiple solutions for Hamiltonian elliptic system with gradient term

2020 ◽  
Vol 10 (1) ◽  
pp. 331-352
Author(s):  
Wen Zhang ◽  
Jian Zhang ◽  
Heilong Mi
Keyword(s):  
Variational Method ◽  
Elliptic System ◽  
Ground State ◽  
Multiple Solutions ◽  
Ground States ◽  
Existence Results ◽  
Perturbation Approach ◽  
Gradient Term ◽  
Global Information ◽  
Hamiltonian Elliptic System

Abstract This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term $$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N}.\\ \end{array} \right. \end{array}$$ Compared with some existing issues, the most interesting feature of this paper is that we assume that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and infinitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.

scholarly journals Multiple solutions for resonant problems of the Robin p-Laplacian plus an indefinite potential

2017 ◽  
Vol 56 (3) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Multiple Solutions ◽  
Indefinite Potential
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