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Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth
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Scattering for the two dimensional NLS with (full) exponential nonlinearity
A nonlocal concave-convex problem with nonlocal mixed boundary data
1. | Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées, Université Abou Bakr Belkaïd, Tlemcen, Tlemcen 13000, Algeria |
2. | Département de Mathématiques, Université Ibn Khaldoun, Tiaret, Tiaret 14000, Algeria |
3. | University of Melbourne, School of Mathematics and Statistics, Peter Hall Building, Parkville, Melbourne VIC 3010, Australia |
4. | School of Mathematics and Statistics, 35 Stirling Highway, Crawley, Perth WA 6009, Australia |
5. | Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50,20133 Milan, Italy |
6. | Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1,27100 Pavia, Italy |
$(P_{\lambda}) \equiv\left\{\begin{array}{rcll}(-\Delta)^s u& = &\lambda u^{q}+u^{p}&{\text{ in }}\Omega,\\ u&>&0 &{\text{ in }} \Omega, \\ \mathcal{B}_{s}u& = &0 &{\text{ in }} \mathbb{R}^{N}\backslash \Omega,\end{array}\right.$ |
$0<q<1<p$ |
$N>2s$ |
$λ> 0$ |
$Ω \subset \mathbb{R}^{N}$ |
$(-Δ)^su(x) = a_{N,s}\;P.V.∈t_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy,$ |
$a_{N,s}$ |
$\mathcal{B}_{s}u = uχ_{Σ_{1}}+\mathcal{N}_{s}uχ_{Σ_{2}}.$ |
$Σ_{1}$ |
$Σ_{2}$ |
$\mathbb{R}^{N}\backslash Ω$ |
$Σ_{1} \cap Σ_{2} = \emptyset$ |
$\overline{Σ}_{1}\cup \overline{Σ}_{2} = \mathbb{R}^{N}\backslash Ω.$ |
$\mathcal{N}_{s}u$ |
$\mathcal{B}_{s}u$ |
$P_{λ}$ |
$λ$ |
$p$ |
References:
[1] |
S. Alama,
Semilinear elliptic equation with sublinear indefinite nonlinearities, Adv. Differential Equation, 4 (1999), 813-842.
|
[2] |
A. Ambrosetti,
Critical points and nonlinear variational problems, Mem. Soc. Math. France (N.S.), 49 (1992), 1-139.
|
[3] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
|
[4] |
A. Ambrosetti and P.H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
|
[5] |
D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd, edition, Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, 2009. |
[6] |
J. G. Azorero and I. Peral,
Multiplicity of solutions for elliptic problems with critical exponent or with a non-symmetric term, Trans. Am. Math. Soc, 323 (1991), 877-895.
|
[7] |
B. Barrios, E. Colorado, R. Servadei and F. Soria,
A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.
|
[8] |
B. Barrios and M. Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, arXiv: 1607.01505. |
[9] |
B. Barrios, M. Medina and I. Peral, Some remarks on the solvability of non-local elliptic problems with the Hardy potential, Commun. Contemp. Math, 16 (2014), 1350046, 29 pp. |
[10] |
H. Brezis and S. Kamin,
Sublinear elliptic equations in $\mathbb{R}^{N}$, Manuscripta Math., 74 (1992), 87-106.
|
[11] |
C. Bucur and M. Medina, A fractional elliptic problem in $\mathbb{R}^{N}$ with critical growth and convex nonlinearities, arXiv: 1609.01911. |
[12] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer; Unione Matematica Italiana, Bologna, 2016. |
[13] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
|
[14] |
L. Caffarelli and L. Silvestre,
Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.
|
[15] |
E. Colorado and I. Peral,
Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal, 199 (2003), 468-507.
|
[16] |
M. Cozzi, Qualitative Properties of Solutions of Nonlinear Anisotropic PDEs in Local and Nonlocal Settings, PhD thesis, 2015. |
[17] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
|
[18] |
S. Dipierro, M. Medina, I. Peral and E. Valdinoci,
Bifurcation results for a fractional elliptic equation with critica exponent in $\mathbb{R}^N$, Manuscripta Math., 153 (2017), no.1-230.
|
[19] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^{N}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, 2017. |
[20] |
S. Dipierro, X. Ros-Oton and E. Valdinoci,
Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam, 33 (2017), 377-416.
|
[21] |
N. Ghoussoub and D. Preiss,
A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 6 (1989), 321-330.
|
[22] |
M. Grossi and F. Pacella,
Positive solutions of nonlinear elliptic equations with critical Sobolev exponent and mixed boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A., 116 (1990), 23-43.
|
[23] |
N. S. Landkof, Foundations of Modern Potential Theory, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag. |
[24] |
T. Leonori, I. Peral, A. Primo and F. Soria,
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.
|
[25] |
A. C. Ponce, Elliptic PDEs, Measures and Capacities, Tracts in Mathematics 23, European Mathematical Society (EMS), Zurich, 2016. |
[26] |
X. Ros-Oton,
Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.
|
[27] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
|
[28] |
R. Servadei and E. Valdinoci,
Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl, 389 (2012), 887-898.
|
[29] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[30] |
R. Servadei and E. Valdinoci,
Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.
|
[31] |
G. Stampacchia,
Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.
|
[32] |
M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergeb. Math. Grenzgeb. (3), Springer-Verlag, Berlin Heidelberg, 1990. |
show all references
References:
[1] |
S. Alama,
Semilinear elliptic equation with sublinear indefinite nonlinearities, Adv. Differential Equation, 4 (1999), 813-842.
|
[2] |
A. Ambrosetti,
Critical points and nonlinear variational problems, Mem. Soc. Math. France (N.S.), 49 (1992), 1-139.
|
[3] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
|
[4] |
A. Ambrosetti and P.H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
|
[5] |
D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd, edition, Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, 2009. |
[6] |
J. G. Azorero and I. Peral,
Multiplicity of solutions for elliptic problems with critical exponent or with a non-symmetric term, Trans. Am. Math. Soc, 323 (1991), 877-895.
|
[7] |
B. Barrios, E. Colorado, R. Servadei and F. Soria,
A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.
|
[8] |
B. Barrios and M. Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, arXiv: 1607.01505. |
[9] |
B. Barrios, M. Medina and I. Peral, Some remarks on the solvability of non-local elliptic problems with the Hardy potential, Commun. Contemp. Math, 16 (2014), 1350046, 29 pp. |
[10] |
H. Brezis and S. Kamin,
Sublinear elliptic equations in $\mathbb{R}^{N}$, Manuscripta Math., 74 (1992), 87-106.
|
[11] |
C. Bucur and M. Medina, A fractional elliptic problem in $\mathbb{R}^{N}$ with critical growth and convex nonlinearities, arXiv: 1609.01911. |
[12] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer; Unione Matematica Italiana, Bologna, 2016. |
[13] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
|
[14] |
L. Caffarelli and L. Silvestre,
Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.
|
[15] |
E. Colorado and I. Peral,
Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal, 199 (2003), 468-507.
|
[16] |
M. Cozzi, Qualitative Properties of Solutions of Nonlinear Anisotropic PDEs in Local and Nonlocal Settings, PhD thesis, 2015. |
[17] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
|
[18] |
S. Dipierro, M. Medina, I. Peral and E. Valdinoci,
Bifurcation results for a fractional elliptic equation with critica exponent in $\mathbb{R}^N$, Manuscripta Math., 153 (2017), no.1-230.
|
[19] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^{N}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, 2017. |
[20] |
S. Dipierro, X. Ros-Oton and E. Valdinoci,
Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam, 33 (2017), 377-416.
|
[21] |
N. Ghoussoub and D. Preiss,
A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 6 (1989), 321-330.
|
[22] |
M. Grossi and F. Pacella,
Positive solutions of nonlinear elliptic equations with critical Sobolev exponent and mixed boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A., 116 (1990), 23-43.
|
[23] |
N. S. Landkof, Foundations of Modern Potential Theory, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag. |
[24] |
T. Leonori, I. Peral, A. Primo and F. Soria,
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.
|
[25] |
A. C. Ponce, Elliptic PDEs, Measures and Capacities, Tracts in Mathematics 23, European Mathematical Society (EMS), Zurich, 2016. |
[26] |
X. Ros-Oton,
Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.
|
[27] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
|
[28] |
R. Servadei and E. Valdinoci,
Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl, 389 (2012), 887-898.
|
[29] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[30] |
R. Servadei and E. Valdinoci,
Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.
|
[31] |
G. Stampacchia,
Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.
|
[32] |
M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergeb. Math. Grenzgeb. (3), Springer-Verlag, Berlin Heidelberg, 1990. |
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