Multiple solutions for Schrödinger–Kirchhoff equations with indefinite potential

2021 ◽  
pp. 107672
Author(s):  
Shuai Jiang ◽  
Shibo Liu
Keyword(s):  
Multiple Solutions ◽  
Kirchhoff Equations ◽  
Indefinite Potential

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 Multiple Solutions for Kirchhoff Equations under the Partially Sublinear Case

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Wenjun Feng ◽  
Xiaojing Feng
Keyword(s):  
Multiple Solutions ◽  
Infinitely Many Solutions ◽  
Kirchhoff Equations ◽  
Kirchhoff Type ◽  
Clark’S Theorem

We prove the infinitely many solutions to a class of sublinear Kirchhoff type equations by using an extension of Clark’s theorem established by Zhaoli Liu and Zhi-Qiang Wang.

Multiple solutions for -Kirchhoff equations in

2013 ◽  
Vol 86 ◽  
pp. 146-156 ◽  
Author(s):  
Caisheng Chen ◽  
Hongxue Song ◽  
Zhonghu Xiu
Keyword(s):  
Multiple Solutions ◽  
Kirchhoff Equations

scholarly journals Multiple solutions for weighted Kirchhoff equations involving critical Hardy-Sobolev exponent

2020 ◽  
Vol 10 (1) ◽  
pp. 673-683
Author(s):  
Zupei Shen ◽  
Jianshe Yu
Keyword(s):  
Multiple Solutions ◽  
Kirchhoff Equations ◽  
Sobolev Exponent

Abstract In this article, we consider a class of Kirchhoff equations with critical Hardy-Sobolev exponent and indefinite nonlinearity, which has not been studied in the literature. We prove very nicely that this equation has at least two solutions in ℝ3. And some known results in the literature are improved.

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