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Bifurcation from the essential spectrum of superlinear elliptic equations. (English) Zbl 0621.35009
Consider the bifurcation of the nonlinear elliptic eigenvalue problem: \[ (1)\quad -\Delta u=f(x,u)+\lambda u,\quad x\in {\mathbb{R}}^ N,\quad (\lambda,u)\in {\mathbb{R}}\times H^ 1({\mathbb{R}}^ N),\quad N\geq 1,\quad u\not\equiv 0, \] where \(f(x,u)=o(u)\) as \(u\to 0\). Since the problem (1) is set on \({\mathbb{R}}^ N\), the standard Lyapunov-Schmidt reduction method does not work. For \(f(x,u)=q(x)| u| ^{\sigma}u\) with q(x)\(\to 0\) as \(| x| \to +\infty\), by means of Sobolev compact embedding, C. A. Stuart [Proc. Lond. Math. Soc., III. Ser. 45, 169-192 (1982; Zbl 0505.35010)] proved that \(\lambda =0\) is a bifurcation point for (1). Naturally, this method fails to treat bifurcation for a general nonlinearity f(x,u) of (1). Using an improvement of the concentration- compactness principle, we prove that \(\lambda =0\) is a bifurcation point for (1) under quite general assumptions on f(x,u).

MSC:
35B32 Bifurcations in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35A15 Variational methods applied to PDEs
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