Existence Result for Non-linearly Perturbed Hardy-Schrödinger Problems: Local and Non-local cases

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a smooth bounded domain having zero in its interior $0\in \Omega .$ We fix $0<\alpha \le 2$ and $0\le s<\alpha .$ We investigate a sufficient condition for the existence of a positive solution for the following perturbed problem associated with the Hardy-Schrödinger operator $L_{\gamma ,\alpha ,}:=(-\Delta {)}^{\frac{\alpha }{2}}-\frac{\gamma }{|x{|}^{\alpha }}$ on $\Omega :$ \begin{equation*} \left\{\begin{array}{rl} \displaystyle ({-}{ Δ})^{\fracα{2}}u- γ\frac{u}{|x|^α} - λu= {\frac{u^{2_α^*(s)-1}}{|x|^s}}+ h(x) u^{q-1} & \text{in } Ω\\ u=0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \text{in } \mathbb{R}^n \setminus Ω, \end{array}\right. \end{equation*} where $2_{\alpha }^{\ast }(s):=\frac{2(n-s)}{n-\alpha },$ $\lambda \in \mathbb{R}$, $h\in C^0(\overline{\Omega }),$ $h\ge 0,$ $q\in (2,2_{\alpha }^{\ast })$ with $2_{\alpha }^{\ast }:=2_{\alpha }^{\ast }(0),$ and $\gamma <{\gamma }_H(\alpha ),$ the latter being the best constant in the Hardy inequality on ${\mathbb{R}}^n.$ We prove that there exists a threshold ${\gamma }_{crit}(\alpha )$ in $(-\infty ,{\gamma }_H(\alpha ))$ such that the existence of solutions of the above problem is guaranteed by the non-linear perturbation $(i.e.,h(x)u^{q-1})$ whenever $\gamma \le {\gamma }_{crit}(\alpha ),$ while for ${\gamma }_{crit}(\alpha )<\gamma <{\gamma }_H(\alpha )$, it is determined by a subtle combination of the geometry of the domain and the size of the nonlinearity of the perturbations.

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• Fractional elliptic equations with Hardy potential and critical nonlinearities