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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

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