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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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