| Abstract: |
| We are interested in the convexity of superlevel sets of positive solutions for a class of quasilinear elliptic equations of the form:
\begin{equation}
\begin{cases}
-{\rm div}(a(u) \nabla u)+\frac{1}{2}a`(u)| \nabla u|^2=
f(u) & \text{in $\Omega$,} \\
u=0 & \text{on $\partial\Omega$,}
\end{cases}
\end{equation}
in bounded strictly convex domains. Our first result states that any positive solution has convex superlevel sets when the problem has a sublinear structure. We further show that any positive semi-stable solution has strictly convex superlevel sets. The existence of positive semi-stable solutions are also investigated for various types of nonlinear terms and quasilinear terms. |
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