Show Summary Details
More options …

IMPACT FACTOR 2017: 1.029
5-year IMPACT FACTOR: 1.147

CiteScore 2017: 1.29

SCImago Journal Rank (SJR) 2017: 1.588
Source Normalized Impact per Paper (SNIP) 2017: 0.971

Mathematical Citation Quotient (MCQ) 2017: 1.03

Online
ISSN
2169-0375
See all formats and pricing
More options …
Volume 18, Issue 3

# A Diffusion Equation with a Variable Reaction Order

Jorge García-Melián
• Departamento de Análisis Matemático and IUEA, Universidad de La Laguna, P. O. Box 456, 38200 La Laguna, Tenerife, Spain
• Email
• Other articles by this author:
/ Julio D. Rossi
• Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pab. 1 (1428), Buenos Aires, Argentina
• Email
• Other articles by this author:
/ José C. Sabina de Lis
• Corresponding author
• Departamento de Análisis Matemático and IUEA, Universidad de La Laguna, P. O. Box 456, 38200 La Laguna, Tenerife, Spain
• Email
• Other articles by this author:
Published Online: 2017-09-02 | DOI: https://doi.org/10.1515/ans-2017-6030

## Abstract

This paper deals with the problem

$\left\{\begin{array}{cccc}\hfill -\mathrm{\Delta }u& =\lambda {u}^{q\left(x\right)},\hfill & x\hfill & \hfill \in \mathrm{\Omega },\\ \hfill u& =0,\hfill & x\hfill & \hfill \in \partial \mathrm{\Omega },\end{array}$

where $\mathrm{\Omega }\subset {ℝ}^{N}$ is a bounded smooth domain, $\lambda >0$ is a parameter and the reaction order $q\left(x\right)$ is a Hölder continuous positive function satisfying $q\left(x\right)>1$ for all $x\in \mathrm{\Omega }$. The relevant feature here is that q is assumed to achieve the value one on $\partial \mathrm{\Omega }$. By assuming that q is subcritical, our main result states the existence of a positive solution for all $\lambda >0$. We also study its asymptotic behavior as $\lambda \to 0$ and as $\lambda \to \mathrm{\infty }$. It should be noticed that the fact that $q=1$ somewhere in $\partial \mathrm{\Omega }$ gives rise to serious difficulties when looking for critical points of the functional associated with the problem above. This work is a continuation of [13] where q is assumed to take values both greater and smaller than one in Ω, but is constrained to satisfy $q\left(x\right)>1$ on $\partial \mathrm{\Omega }$.

MSC 2010: 35J20; 35B45

## References

• [1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709.

• [2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381.

• [3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. Google Scholar

• [4]

H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), no. 6, 601–614.

• [5]

X. Cabré and Y. Martel, Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal. 156 (1998), no. 1, 30–56.

• [6]

D. G. de Figueiredo, P.-L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9) 61 (1982), no. 1, 41–63. Google Scholar

• [7]

M. Delgado, J. López-Gómez and A. Suárez, Combining linear and nonlinear diffusion, Adv. Nonlinear Stud. 4 (2004), no. 3, 273–287. Google Scholar

• [8]

X. Fan and D. Zhao, On the spaces ${L}^{p\left(x\right)}\left(\mathrm{\Omega }\right)$ and ${W}^{m,p\left(x\right)}\left(\mathrm{\Omega }\right)$, J. Math. Anal. Appl. 263 (2001), no. 2, 424–446.

• [9]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomath. 28, Springer, Berlin, 1979. Google Scholar

• [10]

J. García-Melián, J. D. Rossi and J. C. Sabina de Lis, Existence, asymptotic behavior and uniqueness for large solutions to $\mathrm{\Delta }u={e}^{q\left(x\right)u⁣*}$, Adv. Nonlinear Stud. 9 (2009), no. 2, 395–424. Google Scholar

• [11]

J. García-Melián, J. D. Rossi and J. C. Sabina de Lis, Large solutions for the Laplacian with a power nonlinearity given by a variable exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 3, 889–902.

• [12]

J. García-Melián, J. D. Rossi and J. C. Sabina de Lis, An application of the maximum principle to describe the layer behavior of large solutions and related problems, Manuscripta Math. 134 (2011), no. 1–2, 183–214.

• [13]

J. García-Melián, J. D. Rossi and J. C. Sabina de Lis, A variable exponent diffusion problem of concave-convex nature, Topol. Methods Nonlinear Anal. 47 (2016), no. 2, 613–639.

• [14]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883–901.

• [15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1983. Google Scholar

• [16]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${𝐑}^{N}$, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 787–809. Google Scholar

• [17]

P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), no. 4, 441–467.

• [18]

J. López-Gómez, Varying stoichiometric exponents. I. Classical steady-states and metasolutions, Adv. Nonlinear Stud. 3 (2003), no. 3, 327–354. Google Scholar

• [19]

J. López-Gómez and A. Suárez, Combining fast, linear and slow diffusion, Topol. Methods Nonlinear Anal. 23 (2004), no. 2, 275–300.

• [20]

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin, 1983. Google Scholar

• [21]

W. M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math. 82, SIAM, Philadelphia, 2011. Google Scholar

• [22]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. Google Scholar

• [23]

P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser, Basel, 2007. Google Scholar

• [24]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, 1986. Google Scholar

• [25]

V. D. Rădulescu, Nonlinear elliptic equations with variable exponent: Old and new, Nonlinear Anal. 121 (2015), 336–369.

• [26]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Grundlehren Math. Wiss. 258, Springer, New York, 1994. Google Scholar

• [27]

M. Struwe, Variational Methods, 4th ed., Ergeb. Math. Grenzgeb. (3) 34, Springer, Berlin, 2008. Google Scholar

Revised: 2017-08-16

Accepted: 2017-08-16

Published Online: 2017-09-02

Published in Print: 2018-08-01

Funding Source: Ministerio de Ciencia e Innovación

Award identifier / Grant number: MTM2014-52822-P

Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2014-52822-P

The authors were supported by Ministerio de Ciencia e Innovación and Ministerio de Economía y Competitividad under project MTM2014-52822-P.

Citation Information: Advanced Nonlinear Studies, Volume 18, Issue 3, Pages 555–566, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.