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The Nehari manifold for a semilinear elliptic equation involving a sublinear term. (English) Zbl 1130.35049

The author discusses the problem of existence and multiplicity of non-negative solutions to the problem

$\left\{\begin{array}{cc}-{\Delta }u\left(x\right)=\lambda u\left(x\right)+{b\left(x\right)|u\left(x\right)|}^{\gamma -2}u\left(x\right)\hfill & \text{for}\phantom{\rule{4.pt}{0ex}}u\in {\Omega }\hfill \\ u\left(x\right)=0\hfill & \text{for}\phantom{\rule{4.pt}{0ex}}x\in \partial {\Omega },\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(1\right)$

where ${\Omega }\subset {ℝ}^{N}$ is a smooth bounded domain, $b:{\Omega }\to ℝ$ a smooth function, $\lambda \in ℝ$ and $1<\gamma <2$. When $1<\gamma <2$ the problem $\left(1\right)$ is asymptotically linear and the author establishes results on bifurcation from infinity when $\lambda ={\lambda }_{1}$, the principal eigenvalue of the linear problem

$-{\Delta }u\left(x\right)=\lambda u\left(x\right)\phantom{\rule{1.em}{0ex}}x\in {\Omega };\phantom{\rule{2.em}{0ex}}u\left(x\right)=0\phantom{\rule{1.em}{0ex}}x\in \partial {\Omega }·$

By exploiting the relationship between the Nehari manifold and the fibering maps (maps of the form $t↦J\left(tu\right)$ where $J$ is the Euler functional associated to $\left(1\right)$), the author studies how the Nehari manifold changes as $\lambda$ varies. The bifurcation is then described in terms of the sign of the quantity ${\int }_{{\Omega }}b{\phi }_{1}^{\gamma }\phantom{\rule{0.166667em}{0ex}}dx$ where ${\phi }_{1}$ is the positive eigenfunction of the above linear problem corresponding to ${\lambda }_{1}$.

##### MSC:
 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 (nonlinear operator equations)
##### References:
 [1] Brown, K.J., Zhang, Y.: The Nehari manifold for a semilinear elliptic problem with a sign changing weight function. Jour. Diff. Equations 193, 481-499 (2003) · Zbl 1074.35032 · doi:10.1016/S0022-0396(03)00121-9 [2] Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L.: Variational methods for indefinite superlinear homogeneous elliptic problems. Nonlinear Differential Equations and Applications 2, 553-572 (1995) · Zbl 0840.35035 · doi:10.1007/BF01210623 [3] Binding, P.A., Drabek, P., Huang, Y.X.: On Neumann boundary value problems for some quasilinear elliptic equations. Electronic Journal of Differential Equations 5, 1-11 (1997) [4] Drabek, P., Pohozaev, S.I.: Positive solutions for the p-Laplacian: application of the fibrering method. Proc. Royal Soc. Edinburgh 127, 703-726 (1997) [5] Nehari, Z.: On a class of nonlinear second-order differential equations. Trans. Amer. Math. Soc. 95, 101-123 (1960) · doi:10.1090/S0002-9947-1960-0111898-8 [6] Toland, J.: Asymptotic linearity and nonlinear eigenvalue problems. Quart. J. Math. Oxford Ser. 24(2), 241-250 (1973) · Zbl 0256.47049 · doi:10.1093/qmath/24.1.241