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We consider the nonlinear Dirichlet problem $$-u^{\text{'}\text{'}}-r\left(x\right){\left|u\right|}^{\sigma }u=\lambda u\text{in}(0,\infty ),u\left(0\right)=0\text{and}\underset{x\to \infty }{lim}u\left(x\right)=0,$$ 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 $L^p(0,\infty )(2\le p\le \infty )$.

We prove existence and bifurcation results for a semilinear eigenvalue problem in ${ℝ}^N$ $(N\ge 2)$, where the linearization — $$ has no eigenvalues. In particular, we show that under rather weak assumptions on the coefficients $\lambda =0$ is a bifurcation point for this problem in $H^1,H^2$ and $L^p$ $(2\le p\le \infty )$.