Uniqueness and sign properties of minimizers in a quasilinear indefinite problem

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ 1&lt;q&lt;p $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ a\in C(\overline{\Omega}) $\end{document}</tex-math></inline-formula> be sign-changing, where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded and smooth domain of <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula>. We show that the functional <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ I_{q}(u): = \int_{\Omega}\left( \frac{1}{p}|\nabla u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>has exactly one nonnegative minimizer <inline-formula><tex-math id="M5">\begin{document}$ U_{q} $\end{document}</tex-math></inline-formula> (in <inline-formula><tex-math id="M6">\begin{document}$ W_{0}^{1,p}(\Omega) $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M7">\begin{document}$ W^{1,p}(\Omega) $\end{document}</tex-math></inline-formula>). In addition, we prove that <inline-formula><tex-math id="M8">\begin{document}$ U_{q} $\end{document}</tex-math></inline-formula> is the only possible <i>positive</i> solution of the associated Euler-Lagrange equation, which shows that this equation has at most one positive solution. Furthermore, we show that if <inline-formula><tex-math id="M9">\begin{document}$ q $\end{document}</tex-math></inline-formula> is close enough to <inline-formula><tex-math id="M10">\begin{document}$ p $\end{document}</tex-math></inline-formula> then <inline-formula><tex-math id="M11">\begin{document}$ U_{q} $\end{document}</tex-math></inline-formula> is positive, which also guarantees that minimizers of <inline-formula><tex-math id="M12">\begin{document}$ I_{q} $\end{document}</tex-math></inline-formula> do not change sign. Several of these results are new even for <inline-formula><tex-math id="M13">\begin{document}$ p = 2 $\end{document}</tex-math></inline-formula>.