# Article

 Title: Existence and bifurcation results for a class of nonlinear boundary value problems in $(0,\infty )$  (English) Author: Rother, Wolfgang Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 32 Issue: 2 Year: 1991 Pages: 297-305 . Category: math . Summary: We consider the nonlinear Dirichlet problem $$-u'' -r(x)|u|^\sigma u= \lambda u \text{ in } (0,\infty ), \, u(0)=0 \text{ and } \lim _{x\rightarrow \infty } u(x)=0,$$ and develop conditions for the function $r$ such that the considered problem has a positive classical solution. Moreover, we present some results showing that $\lambda =0$ is a bifurcation point in $W^{1,2} (0,\infty )$ and in $L^p(0,\infty )\, (2\leq p\leq \infty )$. Keyword: nonlinear Dirichlet problem Keyword: classical solution Keyword: bifurcation point Keyword: ordinary differential equation MSC: 34A47 MSC: 34B15 MSC: 34C11 MSC: 34C23 idZBL: Zbl 0749.34016 idMR: MR1137791 . Date available: 2009-01-08T17:44:20Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/116971 . Reference: [1] Adams R.A.: Sobolev Spaces.Academic Press, New York, 1975. Zbl 1098.46001, MR 0450957 Reference: [2] Berger M.S.: On the existence and structure of stationary states for a nonlinear Klein-Gordon equation.J. Funct. Analysis 9 (1972), 249-261. Zbl 0224.35061, MR 0299966 Reference: [3] Brezis H., Kato T.: Remarks on the Schrödinger operator with singular complex potentials.J. Math. pures et appl. 58 (1979), 137-151. Zbl 0408.35025, MR 0539217 Reference: [4] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order.SpringerVerlag, Berlin, Heidelberg, New York, 1983. Zbl 1042.35002, MR 0737190 Reference: [5] Hörmander L.: Linear Partial Differential Operators.Springer-Verlag, Berlin, Heidelberg, New York, 1976. MR 0404822 Reference: [6] Stuart C.A.: Bifurcation for Dirichlet problems without eigenvalues.Proc. London Math. Soc. (3) 45 (1982), 169-192. Zbl 0505.35010, MR 0662670 .

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