Badiale, Marino; Pomponio, Alessio Bifurcation results for semilinear elliptic problems in \(\mathbb{R}^N\). (English) Zbl 1067.35027 Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 1, 11-32 (2004). 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. Reviewer: Vicenţiu D. Rădulescu (Craiova) Cited in 4 Documents 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 Keywords:bifurcation; entire solution; semilinear elliptic problem; subcritical Sobolev exponent PDFBibTeX XMLCite \textit{M. Badiale} and \textit{A. Pomponio}, Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 1, 11--32 (2004; Zbl 1067.35027) Full Text: DOI arXiv