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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^*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.