Bifurcation for an elliptic problem with
nonlinear boundary conditions

ROSA PARDO^*
Universidad Complutense de Madrid, Departamento de Matemática Aplicada, 28040, Madrid, Spain.

Abstract. This paper gives a survey over bifurcation problems for elliptic equations with nonlinear boundary conditions depending on a real parameter. We consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter. As the parameter crosses some critical values, there appear certain resonances in the equation producing solutions that bifurcate from infinity. We study the bifurcation branches, and characterize when they are sub- or supercritical. Furthermore, we apply these results and techniques to obtain Landesman-Lazer type conditions guarantying the existence of solutions in the resonant case and to obtain a uniform Anti-Maximum Principle and several results related to the spectral behavior when the potential at the boundary is perturbed. We also characterize the stability type of the solutions in the unbounded branches.

In the remainder of this paper, we start our analysis on a sublinear oscillatory nonlinearity. We first focus our attention on the loss of Landesman-Lazer type conditions, and even in that situation, we are able to prove the existence of infinitely many resonant solutions and infinitely many turning points.

Next we focus our attention on stability switches. Even in the absence of resonant solutions, we are able to provide sufficient conditions for the existence of sequences of stable solutions, unstable solutions, and turning points.

We also discuss on bifurcation from the trivial solution set, and on a sublinear oscillatory nonlinearity.

Finally, we states a formula for the derivative of a localized Steklov eigenvalue on a subset of the boundary, with respect to tangential variations of that subset.

Keywords: Bifurcation from infinity, stability, instability, multiplicity, resonance, turning points.
MSC2010: 35B32, 35B34, 35B35, 58J55, 35J25, 35J60, 35J65

Bifurcaciín para un problema elíptico con condiciones
de frontera no lineales

Resumen.. Este artículo presenta un estudio sobre bifurcaciín para problemas elípticos con condiciones de frontera no-lineales. Consideramos una ecuaciín elíptica con condiciones de frontera no-lineales dependiendo de un parámetro. Supondremos que el término no lineal es asintíticamente lineal en el infinito. Cuando el parámetro cruza ciertos valores críticos (conocidos como los autovalores de Steklov) aparece un fenímeno de resonancia en la ecuaciín, lo que garantiza la existencia de ramas no acotadas de soluciones. Este fenímeno se conoce como bifurcaciín desde infinito. Estudiamos las ramas de soluciones y caracterizamos cuando son subcríticas (a la izquierda del autovalor) o supercríticas (a la derecha del autovalor). Aplicamos estos resultados para obtener condiciones del tipo Landesman-Lazer, que garantizan la existencia de soluciones para el problema resonante (cuando el parámetro coincide con el autovalor). Obtenemos también un Principio del Anti-Máximo, y resultados relativos al comportamiento espectral, cuando se perturba el potencial en la frontera. Además caracterizamos el tipo de estabilidad de las soluciones en dichas ramas no acotadas.

En el resto del articulo, analizamos no linealidades oscilatorias y sublineales. Centramos nuestra atenciín en la pérdida de condiciones del tipo Landesman- Lazer. Incluso en esta situaciín, demostramos la existencia de una sucesiín de infinitas soluciones del problema resonante y una sucesiín de infinitos puntos de retroceso.

A continuaciín, analizamos los cambios de estabilidad. Incluso en ausencia de soluciones resonantes, proporcionamos condiciones suficientes para la existencia de una sucesiín de infinitas soluciones estables, una sucesiín de infinitas soluciones inestables y una sucesiín de infinitos puntos de retroceso.

También analizamos la bifurcaciín desde la soluciín trivial con una nolinealidad de tipo sublineal y oscilatorio.

Finalmente establecemos una fírmula para la derivada del autovalor de Steklov localizado sobre un subconjunto de la frontera, con respecto a variaciones tangenciales del subconjunto.

Palabras claves: Bifurcaciín en el infinito, estabilidad, inestabilidad, multiplicidad, resonancia, puntos de inflexiín.

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Referencias

[1] Adams R.A., Sobolev Spaces, Academic Press, New York, 1974.

[2] Amann H., Linear and Quasilinear Parabolic Problems. Abstract Linear Theory, Birkäuser Verlag, 1995.

[3] Arcoya D. and Gamez J.L., "Bifurcation Theory and Related Problems: Anti-Maximum Principle and Resonance", Comm. Partial Differential Equations 5 (2001), no. 4, 557–569.

[4] Arcoya D. and Rossi J., "Antimaximum principle for quasilinear problems", Adv. Differential Equations 9 (2004), no. 9-10, 1185–1200.

[5] Arrieta J.M., Carvalho A.N. and Rodríguez-Bernal A., "Parabolic Problems with Nonlinear Boundary Conditions and Critical Nonlinearities", J. Differential Equations 156 (1999), no. 2, 376–406.

[6] Arrieta J.M., Consul N. and Rodríguez–Bernal A., "Stable boundary layers in a diffusion problem with nonlinear reaction at the boundary", Z. Angew. Math. Phys. 55 (2004), no. 1, 1–14.

[7] Arrieta J.M., Pardo R. and Rodríguez–Bernal A., "Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity", Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 2, 225–252.

[8] Arrieta J.M., Pardo R. and Rodríguez-Bernal A., "Equilibria and global dynamics of a problem with bifurcation from infinity", J. Differential Equations 246 (2009), no. 5, 2055– 2080.

[9] Arrieta J.M., Pardo R., and Rodríguez-Bernal A, "Infinite resonant solutions and turning points in a problem with unbounded bifurcation", Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), no. 9, 2885–2896.

[10] Cañada A., "Multiplicity results near the principal eigenvalue for boundary-value problems with periodic nonlinearity", Math. Nachr. 280 (2007), no. 3, 235–241.

[11] Castro A. and Pardo R., "Infinitely many stability switches in a problem with sublinear oscillatory boundary conditions", Preprint.

[12] Castro A. and Pardo R., "Resonant solutions and turning points in an elliptic problem with oscillatory boundary conditions", Pacific J. Math. 257 (2012), no. 1, 75–90.

[13] Clement P. and Peletier L.A., "An anti-maximum principle for second order elliptic operators" , J. Differential Equations 34 (1979), no. 2, 218–229.

[14] Costa D., Jeggle H., Schaaf R. and Schmitt K., "Oscillatory perturbations of linear problems of resonance", Results Math. 14 (1988), no. 3-4, 275–287.

[15] Crandall M.G. and Rabinowitz P.H., "Bifurcation from simple eigenvalues" , J. Functional Analysis 8 (1971), 321–340.

[16] Dillon R. and Othmer H., "A mathematical model for outgrowth and spatial patterning of the vertebrate limb bud" , J. Theor. Biol. 197 (1999), 295–330.

[17] Evans L.C., Partial Differential Equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.

[18] Frank-Kamenetskii D.A., Diffusion and heat transfer in chemical kinetics, Plenum Press, New York, 1969.

[19] Galstyan A., Korman P. and Li Y., "On the oscillations of the solution curve for a class of semilinear equations", J. Math. Anal. Appl. 321 (2006), no. 2, 576–588.

[20] García-Melián J., Rossi J.D. and Sabina de Lis J.C., "Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions", Commun. Contemp. Maths. 11 (2009), no. 4, 585–613.

[21] García-Melián J., Rossi J.D. and Sabina de Lis J.C., "An elliptic system with bifurcation parameters on the boundary conditions", J. Differential Equations 247 (2009), no. 3, 779- 810.

[22] Henry D., Perturbation of the boundary in boundary-value problems of partial differential equations. London Mathematical Society, Lecture Note Series, 318. Cambridge University Press, Cambridge, 2005.

[23] Hunt G.W., Peletier M.A., Champneys A.R., Woods P.D., Wadee M., Ahmer M., Budd C.J. and Lord G.J., "Cellular buckling in long structures", Nonlinear Dynam. 21 (2000) no. 1, 3–29.

[24] Kato T., Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.

[25] Korman P., "An oscillatory bifurcation from infinity, and from zero", NoDEA Nonlinear Differential Equations Appl. 15 (2008), no. 3, 335–345.

[26] Landesman E.M. and Lazer A.C., "Nonlinear Perturbations of linear elliptic problems at resonance", J. Math. Mech. 19 (1970), 609–623.

[27] Lípez-Gímez J. and Sabina de Lis J.C., "First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs" , J. Differential Equations 148 (1998), no. 1, 47–64.

[28] Murray J.D., Mathematical Biology, Vol. II: Spatial models and biomedical applications, Springer-Verlag, New York, 2003.

[29] Pardo R., Pereira A.L. and Sabina de Lis J.C., "The Tangential variation of a localized flux-type eigenvalue problem", J. Differential Equations 252 (2012), no. 3, 2104–2130.

[30] Rabinowitz P.H., "Some global results for nonlinear eigenvalue problems", J. Functional Analysis 7 (1971), 487–513.

[31] Rabinowitz P.H., "On Bifurcation From Infinity", J. Differential Equations 14 (1973), 462–475.

[32] Riddle R.D. and Tabin C.J., "How limbs develop", Scientific American (1999), 54–59.

[33] Simon J., "Differentiation with respect to the domain in boundary value problems", Numer. Funct. Anal. Optim. 2 (1980), no. 7-8, 649–687.

[34] Wolpert L., Principles of Development, Oxford University Press, New York, 2002.

[35] Woods P.D. and Champneys A.R., "Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation", Phys. D. 129 (1999), no. 3-4, 147–170.

^*E-mail: rpardo@mat.ucm.es
Partially suppported by Spanish Ministerio de Economía y Competitividad under Project MTM2012-31298.
Recibido:12 August 2012,, Aceptado: 4 de junio de 2012.