Abstract: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 $L^p(0,\infty ) (2\leq p\leq \infty )$.
Keywords: nonlinear Dirichlet problem, classical solution, bifurcation point, ordinary differential equation
AMS Subject Classification: 34B15, 34C11