| 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 $L^p(0,\infty )\, (2\leq p\leq \infty )$. (English) |
| 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 |
| . |
