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Bifurcation results for semilinear elliptic problems in \(\mathbb{R}^N\). (English) Zbl 1067.35027

This paper is devoted to the study of the semilinear bifurcation problem \[ -\Delta\psi -\lambda\psi =a(x)| \psi| ^{p-1}\psi+b(x)| \psi| ^{q-1}\psi \] in \(\mathbb R^N\), with \(\lim_{| x| \rightarrow\infty}\psi(x)=0\), where \(N\geq 1\), \(\lambda\leq 0\), \(1<p<q\leq (N+2)/(N-2)\) if \(N\geq 3\) (and \(q<\infty\) if \(N=1,2\)), \(p<1+4/N\), and \(a,b:\mathbb R^N\rightarrow\mathbb R\) are sign-changing potentials.
The main results of the paper establish, under various assumptions on \(a\) and \(b\), the existence of families of solutions bifurcating from the bottom of the spectrum of \((-\Delta)\). The proofs rely essentially on a nonlinear reduction method that enables the authors to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds.

MSC:

35J60 Nonlinear elliptic equations
35B32 Bifurcations in context of PDEs
47J15 Abstract bifurcation theory involving nonlinear operators
47J30 Variational methods involving nonlinear operators
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
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