zbMATH — the first resource for mathematics

\(H^s\) 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.

35P15 Estimates of eigenvalues in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35R11 Fractional partial differential equations
Full Text: DOI arXiv
[1] Aikawa, H.; Kipleläinen, T.; Shanmugalingam, N.; Zhong, X., Boundary Harnack principle for \(p\)-harmonic functions in smooth Euclidean spaces, Potential Anal., 26, 281-301, (2007) · Zbl 1121.35060
[2] Barrios, B.; Colorado, E.; Servadei, R.; Soria, F.: A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear, 2014) · Zbl 1350.49009
[3] Brasco, L.; Franzina, G.: Convexity properties of Dirichlet integrals and Picone-type inequalities. Kodai Math. J. (2014a, to appear) · Zbl 1315.47054
[4] Brasco, L., Lindgren, E., Parini, E.: The fractional Cheeger problem (2014b, preprint) · Zbl 1301.49115
[5] Brezis, H.; Nirenberg, L., \(H\)\^{1} versus \(C\)\^{1} minimizers, C. R. Acad. Sci. Paris., 317, 465-472, (1993) · Zbl 0803.35029
[6] Cabré, X.; Sire, Y., Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. Henri Poincaré (C) Nonlinear Anal., 31, 23-53, (2014) · Zbl 1286.35248
[7] Cabré, X.; Sire, Y.: Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Trans. Am. Math. Soc. (2014, to appear) · Zbl 1252.46023
[8] Caffarelli, L.A., Nonlocal equations, drifts and games, Nonlinear Partial Differ. Equ. Abel Symposia., 7, 37-52, (2012) · Zbl 1266.35060
[9] Caffarelli, L.; Silvestre, L., An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32, 1245-1260, (2007) · Zbl 1143.26002
[10] Caffarelli, L.; Roquejoffre, J.-M.; Sire, Y., Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12, 1151-1179, (2010) · Zbl 1221.35453
[11] Chen, W.; Li, C.; Ou, B., Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59, 330-343, (2006) · Zbl 1093.45001
[12] Di Castro, A., Kuusi, T., Palatucci, G.: Local behavior of fractional \(p\)-minimizers (2014, preprint) · Zbl 1355.35192
[13] Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 521-573, (2012) · Zbl 1252.46023
[14] Fiscella, A., Servadei, R., Valdinoci, E.: Density properties for fractional Sobolev spaces (2014, preprint) · Zbl 1346.46025
[15] Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Mat. Univ. Parma. 5 (2014) · Zbl 1327.35286
[16] Garcìa Azorero, J.P.; Peral Alonso, I.; Manfredi, J.J., Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., 2, 385-404, (2000) · Zbl 0965.35067
[17] Iannizzotto, A., Liu, S., Perera, K., Squassina, M.: Existence results for fractional p-Laplacian problems via Morse theory (2014, preprint) · Zbl 06567151
[18] Li, S.; Perera, K.; Su, J., Computation of critical groups in elliptic boundary-value problems where the asymptotic limits may not exist, Proc. R. Soc. Edinburgh Sec. A., 131, 721-732, (2001) · Zbl 1114.35321
[19] Lindgren, E.; Lindqvist, P., Fractional eigenvalues, Calc. Var. PDE., 49, 795-826, (2014) · Zbl 1292.35193
[20] Liu, J.; Liu, S., The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc., 37, 592-600, (2005) · Zbl 1122.35033
[21] Mingione, G., Gradient potential estimates, J. Eur. Math. Soc., 13, 459-486, (2011) · Zbl 1217.35077
[22] Ros-Oton, X.; Serra, J., The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101, 275-302, (2014) · Zbl 1285.35020
[23] Ros-Oton, X.; Serra, J., The pohožaev identity for the fractional Laplacian, Arch. Rat. Mech. Anal., 213, 587-628, (2014) · Zbl 1361.35199
[24] Ros-Oton, X., Serra, J.: Nonexistence results for nonlocal equations with critical and supercritical nonlinearities. Commun. Partial Differ. Equ. (2014, to appear) · Zbl 1315.35098
[25] Servadei, R.; Valdinoci, E., Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389, 887-898, (2012) · Zbl 1234.35291
[26] Servadei, R.; Valdinoci, E., Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33, 2105-2137, (2013) · Zbl 1303.35121
[27] Servadei, R.; Valdinoci, E., Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29, 1091-1126, (2013) · Zbl 1275.49016
[28] Servadei, R., Valdinoci, E.: The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. (2014, to appear) · Zbl 1323.35202
[29] Servadei, R.; Valdinoci, E., Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58, 1-261, (2014) · Zbl 1292.35315
[30] Silvestre, L., Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60, 67-112, (2007) · Zbl 1141.49035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.