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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 $$\cases -\Delta u(x)=\lambda u(x)+b(x)\vert u(x)\vert ^{\gamma-2}u(x)& \text{for }u\in\Omega\\ u(x)=0& \text{for }x\in\partial\Omega,\endcases\eqno(1)$$ 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 $\lambda_{1}$.

35J60Nonlinear elliptic equations
35J20Second order elliptic equations, variational methods
35J25Second order elliptic equations, boundary value problems
47J15Abstract bifurcation theory
47J30Variational methods (nonlinear operator equations)
Full Text: DOI
[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) · Zbl 0886.35060
[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) · Zbl 0880.35045
[5] Nehari, Z.: On a class of nonlinear second-order differential equations. Trans. Amer. Math. Soc. 95, 101-123 (1960) · Zbl 0097.29501 · 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