The existence and multiplicity of \(k\)-convex solutions for a coupled \(k\)-Hessian system. (English) Zbl 1538.35186
Summary: In this paper, we focus on the following coupled system of \(k\)-Hessian equations:
\[
\begin{cases}
S_k(\lambda(D^2u)) = f_1(|x|, -v) & \text{in }B,\\
S_k(\lambda(D^2v)) = f_2(|x|, -u) & \text{in }B,\\
u = v = 0 & \text{on }\partial B.
\end{cases}
\]
Here \(B\) is a unit ball with center 0 and \(f_i\) (\(i = 1, 2\)) are continuous and nonnegative functions. By introducing some new growth conditions on the nonlinearities \(f_1\) and \(f_2\), which are more flexible than the existing conditions for the \(k\)-Hessian systems (equations), several new existence and multiplicity results for \(k\)-convex solutions for this kind of problem are obtained.
Keywords:
system of \(k\)-Hessian equations; \(k\)-convex solutions; existence; multiplicity; fixed-point theoremReferences:
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