Title

Branches of positive solutions for subcritical elliptic equations

Document Type

Article

Publication Date

2015

Abstract

Using L∞" style="position: relative;" tabindex="0" id="MathJax-Element-1-Frame">L∞ a-priori bounds for positive solutions to a class of subcritical elliptic problems in bounded C 2 domains, we prove the existence of a branch of positive solutions bifurcating from (λ1,0)" style="position: relative;" tabindex="0" id="MathJax-Element-2-Frame">(λ1,0), where λ1" style="position: relative;" tabindex="0" id="MathJax-Element-3-Frame">λ1 is the first eigenvalue of the Dirichlet eigenvalue problem. We also provide sufficient conditions guarantying that either for any λ<λ1" style="position: relative;" tabindex="0" id="MathJax-Element-4-Frame">λ<λ1 there exists at least a positive solution, or for any continuum (λ,uλ)" style="position: relative;" tabindex="0" id="MathJax-Element-5-Frame">(λ,uλ) of positive solution, there exists a λ∗<0" style="position: relative;" tabindex="0" id="MathJax-Element-6-Frame">λ∗<0 such that λ∗<λ<λ1" style="position: relative;" tabindex="0" id="MathJax-Element-7-Frame">λ∗<λ<λ1 and the corresponding solutions are unbounded in the H 1(Ω)-norm as λ→λ∗" style="position: relative;" tabindex="0" id="MathJax-Element-8-Frame">λ→λ∗.