Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1130.35049
Brown, K.J.
The Nehari manifold for a semilinear elliptic equation involving a sublinear term. (English)
[J] Calc. Var. Partial Differ. Equ. 22, No. 4, 483-494 (2005). ISSN 0944-2669; ISSN 1432-0835/e

The author discusses the problem of existence and multiplicity of non-negative solutions to the problem $$\cases -\Delta u(x)=\lambda u(x)+b(x)\vert u(x)\vert ^{\gamma-2}u(x)& \text{for }u\in\Omega\\ u(x)=0& \text{for }x\in\partial\Omega,\endcases\eqno(1)$$ where $\Omega\subset \Bbb R^N$ is a smooth bounded domain, $b:\Omega\to \Bbb R$ a smooth function, $\lambda\in \Bbb R$ and $1<\gamma<2$. When $1<\gamma<2$ the problem $(1)$ is asymptotically linear and the author establishes results on bifurcation from infinity when $\lambda=\lambda_{1}$, the principal eigenvalue of the linear problem $$-\Delta u(x)=\lambda u(x)\quad x\in\Omega;\qquad u(x)=0\quad x\in\partial\Omega.$$ By exploiting the relationship between the Nehari manifold and the fibering maps (maps of the form $t\mapsto J(tu)$ where $J$ is the Euler functional associated to $(1)$), the author studies how the Nehari manifold changes as $\lambda$ varies. The bifurcation is then described in terms of the sign of the quantity $\int_{\Omega}b\phi_{1}^{\gamma}\,dx$ where $\phi_{1}$ is the positive eigenfunction of the above linear problem corresponding to $\lambda_{1}$.
[Piero Montecchiari (Ancona)]

MSC 2000:: *35J60 Nonlinear elliptic equations
35J20 Second order elliptic equations, variational methods
35J25 Second order elliptic equations, boundary value problems
47J15 Abstract bifurcation theory
47J30 Variational methods

Keywords: Nehari manifold; fibering maps; semilinear sublinear elliptic equations

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]