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\(H^s\) versus \(C^0\)-weighted minimizers. (English) Zbl 1339.35201
Let \(N\geq 2,\Omega\) a bounded domain in \(\mathbb R^N\) with \(C^{1,1}\) boundary, \(s\in ]0,1[\), \(a>0\) \(q\in [1,\frac{2N}{N-2s}]\), \(f:\Omega\times\mathbb R\rightarrow\mathbb R\) a Carathéodory mapping such that \(|f(x,t)|\leq a(1+|t|^{q-1})\) for a.e. \(x\in\Omega\) and for all \(t\in \mathbb R\), \(H^s(\mathbb R^N)=\{u\in L^2(\mathbb R^N)\): \[ |u|_{H^s(\Omega)}\equiv (\int_{\mathbb R^{2N}} \frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}\,\mathrm dx\mathrm dy)^{1/2}<+\infty\}, \; X(\Omega)=\{u\in H^s(\mathbb R^N):u=0\text{ a.e. in }\mathbb R^N\backslash\Omega\}, \] \(\Phi(u)=\frac{|u|_{H^s(\Omega)}^2}{2}-\int_{\Omega}\int_{0}^{u(x)} f(x,t)\,\mathrm dt\mathrm dx\) for every \(u\in X(\Omega)\), \(g\in H^s(\mathbb R^N)_+\); \(\delta:\overline{\Omega}\rightarrow\mathbb R_+\), \(\delta(x)=\mathrm{dist}(x,\mathbb R^N\backslash\Omega)\), \(C_{\delta}^{0}(\overline{\Omega})=\{u \in C^{0}(\overline{\Omega})\): \(\frac{u}{\delta^s}\) admits a continuous extension to \(\overline{\Omega}\}\).
Theorem 1. If \(u\in H^s(\mathbb R^N)\) is such that \(u\geq g\) in \(\mathbb R^N\backslash\Omega\) and \[ \int_{\mathbb R^{2N}}\frac{(u(x)-u(y))\;(v(x)-v(y))}{|x-y|^{N+2s}},\mathrm {d}x\mathrm dy\geq 0 \] for all \(v\in X(\Omega)_{+}\) then \(u \geq 0\) in \(\Omega\) and u admits a l.s.c. representative in \(\Omega\); if \(u \neq0\) then \(u(x)>0\) for all \(x\in\Omega\).
Theorem. 2. If \(u\in X(\Omega)\) and \[ \int_{\mathbb R^{2N}} \frac{(u(x)-u(y))\;(v(x)-v(y))}{|x-y|^{N+2s}}\mathrm{d}x\mathrm{d}y = \int_{\Omega}f(x,u(x))v(x)\mathrm{d}x \] for all \(v\in X(\Omega)\), then \(u\in L^{\infty}(\Omega)\); moreover, if \(q<\frac{2N}{N-2s}\), then there exists \(M \in C(R_{+})\), only depending on \(N\), \(s\), \(a\), \(q\) and \(\Omega\), such that \(|u|_{\infty} \leq M(|u|_{\frac{2N}{N-2s}})\). Thm. 3 Let \(u_{0}\in X(\Omega)\); then there exists \(\rho>0\) such that \(\Phi(u_{0}+v) \geq \Phi(u_{0})\) for all \(v \in X(\Omega) \cap C_{\delta}^{0}(\overline{\Omega})\) with \(|{{v} \over{\delta^{s}}}|_{\infty} \leq \rho\) if and only if there exists \(\varepsilon>0\) such that \(\Phi(u_{0}+v) \geq \Phi(u_{0})\) for all \(v\in X(\Omega)\) with \(|v|_{H^{s}(\Omega)} \leq \varepsilon\). Under further conditions some existence and multiplicity results for \(u\) of thm. 2 are given.

MSC:
35P15 Estimates of eigenvalues in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35R11 Fractional partial differential equations
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