doi: 10.3934/dcds.2021158
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Lower bounds for Orlicz eigenvalues

Instituto de Cálculo, FCEyN - Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Av. Cantilo s/n., Buenos Aires, Argentina

Received  April 2021 Revised  September 2021 Early access November 2021

In this article we consider the following weighted nonlinear eigenvalue problem for the
$ g- $
Laplacian
$ -{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb R^n, n\geq 1 $
with Dirichlet boundary conditions. Here
$ w $
is a suitable weight and
$ g = G' $
and
$ h = H' $
are appropriated Young functions satisfying the so called
$ \Delta' $
condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of
$ G $
,
$ H $
,
$ w $
and the normalization
$ \mu $
of the corresponding eigenfunctions.
We introduce some new strategies to obtain results that generalize several inequalities from the literature of
$ p- $
Laplacian type eigenvalues.
Citation: Ariel Salort. Lower bounds for Orlicz eigenvalues. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021158
References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces, $2^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

A. Anane, Simplicité et isolation de la premiere valeur propre du $p$-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 725-728.   Google Scholar

[3]

S. Bahrouni and A. Salort, Neumann and Robin type boundary conditions in fractional Orlicz-Sobolev spaces, ESAIM Control Optim. Calc. Var., 27 (2021), 23pp. doi: 10.1051/cocv/2020064.  Google Scholar

[4]

A. Cianchi, Hardy inequalities in Orlicz spaces, Trans. Amer. Math. Soc., 351 (1999), 2459-2478.  doi: 10.1090/S0002-9947-99-01985-6.  Google Scholar

[5]

M. Cuesta, Eigenvalue problems for the-Laplacian with indefinite weights, Electron. J. Differential Equations, 2001 (2001), 9pp.  Google Scholar

[6]

J. V. Da SilvaA. SalortA. Silva and J. Spedaletti, A constrained shape optimization problem in Orlicz-Sobolev spaces, J. Differential Equations, 267 (2019), 5493-5520.  doi: 10.1016/j.jde.2019.05.038.  Google Scholar

[7]

P. De Nápoli and J. P. Pinasco, Lyapunov-type inequalities for partial differential equations, J. Funct. Anal., 270 (2016), 1995-2018.  doi: 10.1016/j.jfa.2016.01.006.  Google Scholar

[8]

P. De Nápoli and J. P. Pinasco, A Lyapunov inequality for monotone quasilinear operators, Differential Integral Equations, 18 (2005), 1193-1200.   Google Scholar

[9]

L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[10]

A. Elbert, A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai, 30 (1979), 153-180.   Google Scholar

[11]

J. Fernández Bonder and A. Salort, Fractional order Orlicz-Sobolev spaces, J. Funct. Anal., 277 (2019), 333-367.  doi: 10.1016/j.jfa.2019.04.003.  Google Scholar

[12]

M. García-HuidobroV. LeR. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 207-225.  doi: 10.1007/s000300050073.  Google Scholar

[13]

J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205.  doi: 10.1090/S0002-9947-1974-0342854-2.  Google Scholar

[14]

J.-P. Gossez and R. Manásevich, On a nonlinear eigenvalue problem in Orlicz-Sobolev spaces, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 891-909.  doi: 10.1017/S030821050000192X.  Google Scholar

[15]

M. JleliM. Kirane and B. Samet, Lyapunov-type inequalities for fractional partial differential equations, Appl. Math. Lett., 66 (2017), 30-39.  doi: 10.1016/j.aml.2016.10.013.  Google Scholar

[16]

M. A. Krasnosel'skiĭ and J. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff Ltd., Groningen, 1961.  Google Scholar

[17]

A. Kufner, O. John and S. Fučík, Function Spaces, Springer Science & Business Media, 1977. Google Scholar

[18]

C. LeeC. YehC. Hong and R. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett., 17 (2004), 847-853.  doi: 10.1016/j.aml.2004.06.016.  Google Scholar

[19]

G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uralltseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361.  doi: 10.1080/03605309108820761.  Google Scholar

[20]

A. Liapounoff, Probléme général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 9 (1907), 203-474.   Google Scholar

[21]

M. Otani, A remark on certain nonlinear elliptic equations, Proc. Fac. Sci. Tokai Univ., 19 (1984), 23-28.   Google Scholar

[22]

B. Pachpatte, Lyapunov type integral inequalities for certain differential equations, Georgian Math. J., 4 (1997), 139-148.  doi: 10.1515/GMJ.1997.139.  Google Scholar

[23]

J. P. Pinasco, Lower bounds for eigenvalues of the one-dimensional $p$-Laplacian, Abstr. Appl. Anal., 2004 (2004), 147-153.  doi: 10.1155/S108533750431002X.  Google Scholar

[24]

A. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator, J. Differential Equations, 268 (2020), 5413-5439.  doi: 10.1016/j.jde.2019.11.027.  Google Scholar

[25]

A. Salort and H. Vivas, Fractional eigenvalues in Orlicz spaces with no $\Delta_2$ condition, preprint, 2020, arXiv: 2005.01847. Google Scholar

[26]

J. Sánchez and V. Vergara, A Lyapunov-type inequality for a $\phi$-Laplacian operator, Nonlinear Anal.: Theory, Methods and Applications, 74 (2011), 7071-7077.  doi: 10.1016/j.na.2011.07.027.  Google Scholar

[27]

I. Sim and Y. Lee, Lyapunov inequalities for one-dimensional-Laplacian problems with a singular weight function, J. Inequal. Appl., 2010 (2010), 1-9.  doi: 10.1155/2010/865096.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces, $2^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

A. Anane, Simplicité et isolation de la premiere valeur propre du $p$-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 725-728.   Google Scholar

[3]

S. Bahrouni and A. Salort, Neumann and Robin type boundary conditions in fractional Orlicz-Sobolev spaces, ESAIM Control Optim. Calc. Var., 27 (2021), 23pp. doi: 10.1051/cocv/2020064.  Google Scholar

[4]

A. Cianchi, Hardy inequalities in Orlicz spaces, Trans. Amer. Math. Soc., 351 (1999), 2459-2478.  doi: 10.1090/S0002-9947-99-01985-6.  Google Scholar

[5]

M. Cuesta, Eigenvalue problems for the-Laplacian with indefinite weights, Electron. J. Differential Equations, 2001 (2001), 9pp.  Google Scholar

[6]

J. V. Da SilvaA. SalortA. Silva and J. Spedaletti, A constrained shape optimization problem in Orlicz-Sobolev spaces, J. Differential Equations, 267 (2019), 5493-5520.  doi: 10.1016/j.jde.2019.05.038.  Google Scholar

[7]

P. De Nápoli and J. P. Pinasco, Lyapunov-type inequalities for partial differential equations, J. Funct. Anal., 270 (2016), 1995-2018.  doi: 10.1016/j.jfa.2016.01.006.  Google Scholar

[8]

P. De Nápoli and J. P. Pinasco, A Lyapunov inequality for monotone quasilinear operators, Differential Integral Equations, 18 (2005), 1193-1200.   Google Scholar

[9]

L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[10]

A. Elbert, A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai, 30 (1979), 153-180.   Google Scholar

[11]

J. Fernández Bonder and A. Salort, Fractional order Orlicz-Sobolev spaces, J. Funct. Anal., 277 (2019), 333-367.  doi: 10.1016/j.jfa.2019.04.003.  Google Scholar

[12]

M. García-HuidobroV. LeR. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 207-225.  doi: 10.1007/s000300050073.  Google Scholar

[13]

J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205.  doi: 10.1090/S0002-9947-1974-0342854-2.  Google Scholar

[14]

J.-P. Gossez and R. Manásevich, On a nonlinear eigenvalue problem in Orlicz-Sobolev spaces, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 891-909.  doi: 10.1017/S030821050000192X.  Google Scholar

[15]

M. JleliM. Kirane and B. Samet, Lyapunov-type inequalities for fractional partial differential equations, Appl. Math. Lett., 66 (2017), 30-39.  doi: 10.1016/j.aml.2016.10.013.  Google Scholar

[16]

M. A. Krasnosel'skiĭ and J. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff Ltd., Groningen, 1961.  Google Scholar

[17]

A. Kufner, O. John and S. Fučík, Function Spaces, Springer Science & Business Media, 1977. Google Scholar

[18]

C. LeeC. YehC. Hong and R. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett., 17 (2004), 847-853.  doi: 10.1016/j.aml.2004.06.016.  Google Scholar

[19]

G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uralltseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361.  doi: 10.1080/03605309108820761.  Google Scholar

[20]

A. Liapounoff, Probléme général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 9 (1907), 203-474.   Google Scholar

[21]

M. Otani, A remark on certain nonlinear elliptic equations, Proc. Fac. Sci. Tokai Univ., 19 (1984), 23-28.   Google Scholar

[22]

B. Pachpatte, Lyapunov type integral inequalities for certain differential equations, Georgian Math. J., 4 (1997), 139-148.  doi: 10.1515/GMJ.1997.139.  Google Scholar

[23]

J. P. Pinasco, Lower bounds for eigenvalues of the one-dimensional $p$-Laplacian, Abstr. Appl. Anal., 2004 (2004), 147-153.  doi: 10.1155/S108533750431002X.  Google Scholar

[24]

A. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator, J. Differential Equations, 268 (2020), 5413-5439.  doi: 10.1016/j.jde.2019.11.027.  Google Scholar

[25]

A. Salort and H. Vivas, Fractional eigenvalues in Orlicz spaces with no $\Delta_2$ condition, preprint, 2020, arXiv: 2005.01847. Google Scholar

[26]

J. Sánchez and V. Vergara, A Lyapunov-type inequality for a $\phi$-Laplacian operator, Nonlinear Anal.: Theory, Methods and Applications, 74 (2011), 7071-7077.  doi: 10.1016/j.na.2011.07.027.  Google Scholar

[27]

I. Sim and Y. Lee, Lyapunov inequalities for one-dimensional-Laplacian problems with a singular weight function, J. Inequal. Appl., 2010 (2010), 1-9.  doi: 10.1155/2010/865096.  Google Scholar

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