Existence and bifurcation results for a class of nonlinear boundary value problems in $(0,\infty )$

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