A Landesman–Lazer type result for periodic parabolic problems on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:math> at resonance

Ćwiszewski, Aleksander; Łukasiak, Renata

doi:10.1016/j.na.2015.06.001

Cited by 2 publications

(2 citation statements)

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“…Both cases can be put in similar settings, however, are different geometrically. The resonant case was studied in [8]. Here we consider the non-resonant case.…”

Section: Introductionmentioning

confidence: 99%

Periodic solutions for nonresonant parabolic equations on $${\mathbb {R}}^N$$ with Kato–Rellich type potentials

Ćwiszewski

Łukasiak

2021

J. Fixed Point Theory Appl.

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A criterion for the existence of T-periodic solutions of nonautonomous parabolic equation $$u_t = \Delta u + V(x)u + f(t,x,u)$$ u t = Δ u + V ( x ) u + f ( t , x , u ) , $$x\in {\mathbb {R}}^N$$ x ∈ R N , $$t>0$$ t > 0 , where V is Kato–Rellich type potential and f diminishes at infinity, will be provided. It is proved that, under the nonresonance assumption, i.e. $${\mathrm {Ker}} (\Delta + V)=\{0\}$$ Ker ( Δ + V ) = { 0 } , the equation admits a T-periodic solution. Moreover, in case there is a trivial branch of solutions, i.e. $$f(t,x,0)=0$$ f ( t , x , 0 ) = 0 , there exists a nontrivial solution provided the total multiplicities of positive eigenvalues of $$\Delta +V$$ Δ + V and $$\Delta + V + f_0$$ Δ + V + f 0 , where $$f_0$$ f 0 is the partial derivative $$f_u(\cdot ,\cdot ,0)$$ f u ( · , · , 0 ) of f, are different mod 2.

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“…Both cases can be put in similar settings, however, are different geometrically. The resonant case was studied in [8]. Here we consider the non-resonant case.…”

Section: Introductionmentioning

confidence: 99%

Periodic solutions for nonresonant parabolic equations on $${\mathbb {R}}^N$$ with Kato–Rellich type potentials

Ćwiszewski

Łukasiak

2021

J. Fixed Point Theory Appl.

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“…An improvement of Henry's general result was given in [10] where initial values could converge in X (not necessarily in X α ). Averaging appeared useful in studying periodic solutionssee [15], [5], [6], [7], [8], [16], [17] as well as [9].…”

Section: Introductionmentioning

confidence: 99%