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 $L^p(0,\infty )\, (2\leq p\leq \infty )$.
@article{116971,
author = {Wolfgang Rother},
title = {Existence and bifurcation results for a class of nonlinear boundary value problems in $(0,\infty )$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {32},
year = {1991},
pages = {297-305},
zbl = {0749.34016},
mrnumber = {1137791},
language = {en},
url = {http://dml.mathdoc.fr/item/116971}
}
Rother, Wolfgang. Existence and bifurcation results for a class of nonlinear boundary value problems in $(0,\infty )$. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) pp. 297-305. http://gdmltest.u-ga.fr/item/116971/
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