Multiple solutions for superlinear Dirichlet problems with an indefinite potential

2011 ◽  
Vol 192 (2) ◽  
pp. 297-315 ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Multiple Solutions ◽  
Dirichlet Problems ◽  
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.

Multiple Solutions for Resonant Hemivariational Inequalities via Minimax Methods

2009 ◽  
Vol 9 (3) ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Donal O’ Regan ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Multiple Solutions ◽  
Principal Eigenvalue ◽  
Point Theory ◽  
Hemivariational Inequalities ◽  
Dirichlet Problems ◽  
Nonsmooth Critical Point Theory ◽  
Minimax Methods ◽  
Nonsmooth Critical Point ◽  
Multiplicity Theorem ◽  
The Right

AbstractIn this paper we consider nonlinear Dirichlet problems driven by the p-Laplacian differential operator with a nonsmooth potential (hemivariational inequalities). We assume that the problem is resonant at infinity with respect to λ1 > 0 (the principal eigenvalue of the Dirichlet p-Lapalcian) from the right. Using minimax methods based on the nonsmooth critical point theory we prove an existence and a multiplicity theorem.

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.

Multiple solutions for parametric double phase Dirichlet problems

2020 ◽  
pp. 2050006 ◽  
Author(s):  
N. S. Papageorgiou ◽  
C. Vetro ◽  
F. Vetro
Keyword(s):  
Dirichlet Problem ◽  
Multiple Solutions ◽  
Critical Parameter ◽  
Constant Sign ◽  
Nontrivial Solutions ◽  
Dirichlet Problems ◽  
Double Phase ◽  
Variational Tools

We consider a parametric double phase Dirichlet problem. Using variational tools together with suitable truncation and comparison techniques, we show that for all parametric values [Formula: see text] the problem has at least three nontrivial solutions, two of which have constant sign. Also, we identify the critical parameter [Formula: see text] precisely in terms of the spectrum of the [Formula: see text]-Laplacian.

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