Geodesic
Existence and bifurcation results for a class of nonlinear boundary value problems in $(0,\infty )$
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 297-305

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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 $L^p(0,\infty )\, (2\leq p\leq \infty )$.
Classification : 34A47, 34B15, 34C11, 34C23
Keywords: nonlinear Dirichlet problem; classical solution; bifurcation point; ordinary differential equation 
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     author = {Rother, Wolfgang},
     title = {Existence and bifurcation results for a class  of nonlinear boundary value problems in $(0,\infty )$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {297--305},
     publisher = {mathdoc},
     volume = {32},
     number = {2},
     year = {1991},
     mrnumber = {1137791},
     zbl = {0749.34016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1991__32_2_a11/}
}
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Rother, Wolfgang. Existence and bifurcation results for a class  of nonlinear boundary value problems in $(0,\infty )$. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 297-305. http://geodesic.mathdoc.fr/item/CMUC_1991__32_2_a11/