Abstract
We study exact multiplicity of positive solutions for a class of Dirichlet problems on a ball. We consider nonlinearities generalizing cubic, allowing both f(0) = 0 and non-positone cases. We use bifurcation approach. We rst prove our results for a special case, and then show that the global picture persists as we vary the roots.
Highlights
We study positive solutions of the Dirichlet problem for the semilinear elliptic equation on the unit ball (1.1)∆u + λf (u) = 0 for |x| < 1, u = 0 on |x| = 1, i.e, a Dirichlet problem on a ball in Rn, depending on a positive parameter λ
In that case it easy to prove that any nontrivial solution of the linearized equation (2.2) is positive, see Lemma 2.1 below, and our result follows along the lines of [8].) Our assumption is (f4) If ρ > θ
To see that the turn is to the right, we observe that the function τ (s), defined in Crandall-Rabinowitz theorem, satisfies τ (0) = τ (0) = 0 and
Summary
We assume that the function f ∈ C 2(R+) satisfies the following conditions (f1) f (0) ≤ 0, (f2) f (b) = f (c) = 0 for some 0 < b < c, and f (u) < 0 on (0, b); c 0 f (u)du In that case it easy to prove that any nontrivial solution of the linearized equation (2.2) is positive, see Lemma 2.1 below, and our result follows along the lines of [8].) Our assumption is (f4) If ρ > θ The following lemma provides a condition for the positivity of w.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Electronic Journal of Qualitative Theory of Differential Equations
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
