Intervals of bifurcation points for semilinear elliptic problems
José Carmona, Antonio J. Martínez Aparicio, Pedro J. Martínez−Aparicio · Advances in Nonlinear Analysis · 2025
Abstract In this article, we study the behavior of multiple continua of solutions to the semilinear elliptic problem − Δ u = λ f ( u ) , in Ω , u = 0 , on ∂ Ω , \left\{\begin{array}{ll}-\Delta u=\lambda f\left(u),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0,\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω \Omega is a bounded open subset of R N {{\mathbb{R}}}^{N} and f f is a nonnegative continuous real function with multiple zeros. We analyze both the behavior of unbounded continua of solutions having norm between consecutive zeros of f f , and the asymptotic behavior of the multiple unbounded continua in the case in which f f has a countable infinite set of positive zeros. In both cases, we pay special attention to the multiplicity results they give rise to. For the model cases f ( t ) = t r ( 1 + sin t ) f\left(t)={t}^{r}\left(1+\sin t) and f ( t ) = t r