No CrossRef data available.
Article contents
Existence of solution for a class of activator–inhibitor systems
Part of:
Physiological, cellular and medical topics
Partial differential equations
Elliptic equations and systems
Published online by Cambridge University Press: 12 April 2022
Abstract
We prove the existence of a solution for a class of activator–inhibitor system of type
$- \Delta u +u = f(u) -v$
,
$-\Delta v+ v=u$
in
$\mathbb{R}^{N}$
. The function f is a general nonlinearity which can grow polynomially in dimension
$N\geq 3$
or exponentiallly if
$N=2$
. We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust
References
Ambrosetti, A. and Rabinowitz, P., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381.Google Scholar
Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976) 620–709.CrossRefGoogle Scholar
Alves, C. O., Montenegro, M. and Souto, M. A., Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. PDEs 43 (2012) 537–554.Google Scholar
Azzollini, A., d’Avenia, P. and Pomponio, A., Multiple critical points for a class of nonlinear functionals, Ann. Math. Pura Appl. 190 (2011) 507–523.Google Scholar
Baernstein, A., A unified approach to symmetrization, in Partial differential equations of elliptic type, Symposia Mathematica, vol. XXXV (1994) 47–91.Google Scholar
Bartsch, T. and de la Valeriola, S., Normalized solutions of nonlinear SchrÖdinger equations, Arch. Math. 100 (2013) 75–83.CrossRefGoogle Scholar
Benci, V. and Rabinowitz, P. H., Critical point theorems for indefinite functionals, Invent. Math. 52 (1979) 241–273.CrossRefGoogle Scholar
Berestycki, H., Gallouët, T. and Kavian, O., Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Ser. I Math. 297 (1983) 307–310.Google Scholar
Berestycki, H. and Lions, P.-L., Une méthode locale pour l’existence de solutions positives de problÈmes semi-linéaires elliptiques dans
$\mathbb{R}^N$
, J. Anal. Math. 38 (1980) 144–187.Google Scholar
$\mathbb{R}^N$
, J. Anal. Math. 38 (1980) 144–187.Google ScholarBerestycki, H. and Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983) 313–345.Google Scholar
Cao, D. M., Nonlinear solutions of semilinear elliptic equations with critical exponent in
$R^2$
, Commun. Partial Differ. Equations 17 (1992) 407–435.CrossRefGoogle Scholar
$R^2$
, Commun. Partial Differ. Equations 17 (1992) 407–435.CrossRefGoogle ScholarChen, C. N. and Choi, Y. S., Traveling pulse solutions to FitzHugh–Nagumo equations, Calc. Var. PDEs 54 (2015) 1–45.Google Scholar
Chen, C. N. and Hu, X., Stability criteria for reaction–diffusion systems with skew-gradient structure, Commun. Partial Differ. Equations 33 (2008) 189–208.CrossRefGoogle Scholar
Chen, C. N. and Hu, X., Stability analysis for standing pulse solutions to FitzHugh–Nagumo equations, Calc. Var. PDEs 49 (2014) 827–845.Google Scholar
Chen, C. N. and Séré, E., Multiple front standing waves in the FitzHugh–Nagumo equations, J. Differ. Equations 302 (2021) 895–925.Google Scholar
Chen, C. N. and Tanaka, K., A variational approach for standing waves of FitzHugh–Nagumo type systems, J. Differ. Equations 257 (2014) 109–144.Google Scholar
Coleman, S., Glaser, V. and Martin, A., Action minima among solutions to a class of Euclidean scalar field equations, Commun. Math. Phys. 58 (1978) 211–221.Google Scholar
Dancer, E. N. and Yan, S., A minimization problem associated with elliptic systems of FitzHugh–Nagumo type, Ann. Inst. H. Poincar Anal. Non Linéaire 21 (2004) 237–253.Google Scholar
de Figueiredo, D. G. and Mitidieri, E., A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal. 17 (1986) 837–849.Google Scholar
FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961) 445–466.CrossRefGoogle ScholarPubMed
Hirata, J., Ikoma, N. and Tanaka, K., Nonlinear scalar field equations in
$\mathbb{R}^{N}$
: mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010) 253–276.Google Scholar
$\mathbb{R}^{N}$
: mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010) 253–276.Google ScholarHodgkin, A. and Huxley, A., A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. London 117 (1952) 500–544.Google ScholarPubMed
Jeanjean, L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28 (1997) 1633–1659.CrossRefGoogle Scholar
Jeanjean, L. and Tanaka, K., A remark on least energy solutions in
$\mathbb R^N$
, Proc. Am. Math. Soc. 131 (2002) 2399–2408.CrossRefGoogle Scholar
$\mathbb R^N$
, Proc. Am. Math. Soc. 131 (2002) 2399–2408.CrossRefGoogle ScholarKlaasen, G. A. and Mitidieri, E., Standing wave solutions of a system derived from the FitzHugh–Nagumo equations for nerve conduction, SIAM J. Math. Anal. 17 (1986) 74–83.CrossRefGoogle Scholar
Klaasen, G. A. and Troy, W. C., Standing wave solutions of a system of reaction diffusion equations derived from the FitzHugh–Nagumo equations, SIAM J. Appl. Math. 44 (1984) 96–110.CrossRefGoogle Scholar
Lions, P. L., Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982) 315–334.CrossRefGoogle Scholar
Moroz, V. and Van Schaftingen, J., Existence of ground states for a class of nonlinear Choquard equations, arXiv:1212.2017 (2012).Google Scholar
Nagumo, J., Arimoto, S. and Yoshizawa, S., An active pulse transmission line simulating nerve axon, Proc. IRE 50 (1962) 2061–2070.CrossRefGoogle Scholar
Oshita, Y., On stable nonconstant stationary solutions and mesoscopic patterns for FitzHugh–Nagumo equations in higher dimensions, J. Differ. Equations 188 (2003) 110–134.CrossRefGoogle Scholar
Reinecke, C. and Sweers, G., A positive solution on
$\mathbb{R}^{n}$
to a system of elliptic equations of FitzHugh–Nagumo type, J. Differ. Equations 153 (1999) 292–312.Google Scholar
$\mathbb{R}^{n}$
to a system of elliptic equations of FitzHugh–Nagumo type, J. Differ. Equations 153 (1999) 292–312.Google ScholarSweers, G. and Troy, W. C., On the bifurcation curve for an elliptic system of FitzHugh–Nagumo type, Physica D 177 (2003) 1–22.CrossRefGoogle Scholar
Wei, J. and Winter, M., Clustered spots in the FitzHugh–Nagumo system, J. Differ. Equations 213 (2005) 121–145.CrossRefGoogle Scholar
Wei, J. and Winter, M., Standing waves in the FitzHugh–Nagumo system and a problem in combinatorial geometry, Math. Z. 254 (2006) 359–383.Google Scholar
Yagi, A., Abstract parabolic evolution equations and their applications, Springer Monographs in Mathematics (Springer–Verlag, Berlin, 2010).Google Scholar
Zhang, J. and Zou, W., A Berestycki–Lions theorem revisited, Commun. Contemp. Math. 14 (2012) 1250033-1–1250033-14.CrossRefGoogle Scholar