Optimal solvability for a nonlocal problem at critical growth
Let \(\Omega\) be a bounded domain of \(\mathbb R^n\), \(a\in L_{n/(sp)}(\Omega)\) for \(n>sp\), and \(s\in (0,1)\). The authors consider \[ \begin{aligned} &(-\Delta_p)^su+a|u|^{p-2}u=\mu|u|^{p_s^*-2}u\text{ in }\Omega \\ &u=0\text{ in }\mathbb R^n\setminus\Omega\end{aligned} \leqno{(*)} \] such that \(p^*_s\) represents the fractional critical Sobolev exponent, and \(\mu\) is some real number. The associated minimization problem of \((*)\) is defined by \[ S_{p,s}(a) =\inf_{u\in D_0^{s,p}(\Omega)}\left\{[u]_{D^{s,p}_0(\Omega)}+\int_{\mathbb R^n}a(x)|u(x)|^pdx:\|u\|_{L^{p_s^*} (\mathbb R^n)}=1\right\} \] where \[ D^{s,p}_0(\Omega)=\left\{u\in L_{p^*_s}(\mathbb R^n): [u]_{D^{s,p}(\mathbb R^n)}<\infty\text{ and }u_{\restriction_{\mathbb R^n\setminus\Omega}}=0\right\} \] equipped with the norm \[ \|u\|_{D_0^{s,p}(\Omega)}=[u]_{D^{s,p}(\mathbb R^n)}=\left(\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy\right)^{1/p}. \] Subsequently, the authors state two main results related on whether \(S_{p,s}(a)\) has or no a solution. The first one states that \(S_{p,s}(a)\) has not a solution whenever \(a\) is positive and has a solution whenever \(n>sp^2\) (resp. \(n\in (sp,sp^2]\)) and \(\max(-a,0)\) is greater than some positive constant almost everywhere on some ball (Theorem 1.1). The second result states that if \(S_{p,s}(a)\) is strictly negative, then it has (resp. has not) a solution if \(\mu\) is positive (resp. strictly negative) (Theorem 1.2).
scientific article; zbMATH DE number 6806997
scientific article