Symmetry of intrinsically singular solutions of double phase problems. (Q2093359)

The authors study qualitative properties of singular positive solutions to the following double phase problem \[ (P) \ \ \ \left\{\begin{array}{ll} -\mathrm{div}(p|\nabla u|^{p-2}\nabla u+qa(x)|\nabla u|^{q-2}\nabla u)=f(u), \ \ \ &\text{in} \ \ \Omega \setminus \Gamma,\\[3mm] u=0 &\text{on} \ \ \partial \Omega, \end{array}\right. \] where \(\Omega\) is an open bounded convex set in \(\mathbb{R}^n\), which is symmetric in the \(x_1\)-direction, \(\Gamma\subset\Omega\cap\{x_1=0\}\) is a closed set, \(p\in (\frac{2n}{n+2},2)\), \(q\in (p,2)\), \(a\in C^1(\Omega)\) is a bounded function, which is nonnegative and independent of \(x_1\), and \(f:\mathbb{R}\rightarrow\mathbb{R}\) is a locally Lipschitz function such that \(f(s)>0\), for \(s>0\). The authors introduce the notion of \((p,q)\)-capacity for compact subsets \(K\) of \(\Omega\), defined as \[\mathrm{Cap}_{p,q}(K):=\inf\left\{\int_{\mathbb{R}}(|\nabla\varphi|^p+a(x)|\nabla\varphi|^q)dx: \ \varphi\in C^\infty_0(\Omega), \ \inf_K\varphi\geq 1\right\},\] and prove that if in addition to the above conditions one assumes that \(\mathrm{Cap}_{p,q}(\Gamma)=0\), then every positive solution \(u\in C^1(\overline{\Omega}\setminus\Gamma)\) of problem \((P)\) is symmetric with respect to the hyperplane \(\{x_1=0\}\) and increasing in the \(x_1\)-direction in \(\Omega\cap\{x_1<0\}\). To prove this result, the authors use a variant of the moving plane method for singular solutions introduced in [\textit{B. Sciunzi}, J. Math. Pures Appl. (9) 108, No. 1, 111--123 (2017; Zbl 1371.35114)].