Existence, uniqueness, and multiplicity of radially symmetric <i>k</i> -admissible solutions for <i>k</i> -hessian equations

Abstract This paper investigates the existence, uniqueness, and multiplicity of radially symmetric k -admissible solutions for the k -Hessian equation subject to a homogeneous Dirichlet boundary condition: (P) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:msub> <m:mi>S</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>D</m:mi> <m:mn>2</m:mn> </m:msup> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:msup> <m:mi>λ</m:mi> <m:mi>k</m:mi> </m:msup> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mo>−</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> <m:mspace width="2em"/> </m:mtd> <m:mtd> <m:mtext>in</m:mtext> <m:mspace width="thinmathspace"/> <m:mspace width="thinmathspace"/> <m:mi>B</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mtd> <m:mtd> <m:mtext>on</m:mtext> <m:mspace width="thinmathspace"/> <m:mspace width="thinmathspace"/> <m:mi mathvariant="normal">∂</m:mi> <m:mi>B</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mfenced> </m:math> $$\left\{\begin{array}{ll} S_k(D^2 u)=\lambda^k f(-u)\qquad&amp;\text{in}\,\,\ B, \\ u=0 &amp;\text{on}\,\,\ \partial B, \end{array}\right. $$ where B = { x ∈ ℝ n : ∣ x ∣ &lt; 1}, λ is a positive parameter, k ∈ {1, …, n }, f : [0, ∞) → [0, ∞) is continuous. In contrast to previous studies, our main results are established under the essential condition that the radially symmetric k -admissible solutions of (P) are generically non-convex. The main tool is the fixed point theorem in cones.