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Stable solutions to double phase problems involving a nonlocal term

Part of: Partial differential equations Elliptic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  23 October 2023

Belgacem Rahal  and
Phuong Le Open the ORCID record for Phuong Le [Opens in a new window]
Belgacem Rahal
Affiliation:
Higher Institute of Computer Science and Management of Kairouan, University of Kairouan, Kairouan, Tunisia (rahhalbelgacem@gmail.com; belgacemrahal.isigk@gmail.com) LR18ES16: Analysis, Geometry and Applications, Faculty of Sciences of Monastir, University of Monastir, Monastir, Tunisia
Phuong Le
Affiliation:
Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam (phuongl@uel.edu.vn) Vietnam National University, Ho Chi Minh City, Vietnam
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Abstract

In this paper, we study weak solutions, possibly unbounded and sign-changing, to the double phase problem

\begin{equation*}-\text{div} (|\nabla u|^{p-2} \nabla u + w(x)|\nabla u|^{q-2} \nabla u) = \left(\frac{1}{|x|^{N-\mu}}*f|u|^r\right) f(x)|u|^{r-2}u \quad\text{in}\ \mathbb{R}^N,\end{equation*}
where $q\ge p\ge2$, r > q, $0 \lt \mu \lt N$ and $w,f \in L^1_{\rm loc}(\mathbb{R}^N)$ are two non-negative functions such that $w(x) \le C_1|x|^a$ and $f(x) \ge C_2|x|^b$ for all $|x| \gt R_0$, where $R_0,C_1,C_2 \gt 0$ and $a,b\in\mathbb{R}$. Under some appropriate assumptions on p, q, r, µ, a, b and N, we prove various Liouville-type theorems for weak solutions which are stable or stable outside a compact set of $\mathbb{R}^N$. First, we establish the standard integral estimates via stability property to derive the non-existence results for stable weak solutions. Then, by means of the Pohožaev identity, we deduce the Liouville-type theorem for weak solutions which are stable outside a compact set.

Keywords

double phase problemsstable solutionsLiouville theoremsHartree nonlinearity

MSC classification

Primary: 35J92: Quasilinear elliptic equations with $p$-Laplacian 35Q40: PDEs in connection with quantum mechanics 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35B35: Stability
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 4 , November 2023 , pp. 1119 - 1141
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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Footnotes

The online version of this article has been updated since original publication. A notice detailing the changes has also been published at DOI https://doi.org/10.1017/S0013091523000718

References

Cowan, C. and Fazly, M., On stable entire solutions of semi-linear elliptic equations with weights, Proc. Amer. Math. Soc. 140(6) (2012), 2003–2012. doi:10.1090/S0002-9939-2011-11351-0CrossRefGoogle Scholar
Damascelli, L., Farina, A., Sciunzi, B. and Valdinoci, E., Liouville results for m-Laplace equations of Lane-Emden-Fowler type, Ann. Inst. H. Poincaré C Anal. Non LinéAire 26(4) (2009), 1099–1119. doi:10.1016/j.anihpc.2008.06.001Google Scholar
Dancer, E. N., Du, Y. and Guo, Z., Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations 250(8) (2011), 3281–3310. doi:10.1016/j.jde.2011.02.005CrossRefGoogle Scholar
DiBenedetto, E., ${C^{1+\unicode{x03B1}}}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7(8) (1983), 827–850. doi:10.1016/0362-546X(83)90061-5CrossRefGoogle Scholar
Dupaigne, L., Stable Solutions of Elliptic Partial Differential Equations. Monographs and Surveys in Pure and Applied Mathematics, Volume 143 (Chapman & Hall/CRC, Boca Raton, FL, 2011).CrossRefGoogle Scholar
Farina, A., On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb R^N$, J. Math. Pures Appl. (9) 87(5) (2007), 537–561. doi:10.1016/j.matpur.2007.03.001CrossRefGoogle Scholar
Farina, A., Mercuri, C. and Willem, M., A Liouville theorem for the p-Laplacian and related questions, Calc. Var. Partial Differential Equations 58(4) (2019), . doi:10.1007/s00526-019-1596-yCrossRefGoogle Scholar
Jikov, V. V., Kozlov, S. M. and Oleinik, O. A., Homogenization of Differential Operators and Integral functionals (Springer-Verlag, Berlin, 1994). Translated from the Russian by G. A. Yosifian. doi:10.1007/978-3-642-84659-5CrossRefGoogle Scholar
Le, P., Stable solutions to the static Choquard equation, Bull. Aust. Math. Soc. 102(3) (2020), 471–478. doi:10.1017/s0004972720000519CrossRefGoogle Scholar
Le, P., On classical solutions to the Hartree equation, J. Math. Anal. Appl. 485(2) (2020), . doi:10.1016/j.jmaa.2020.123859CrossRefGoogle Scholar
Le, P., Liouville Results for Double Phase Problems in ${\mathbb{R}^N}$, Qual. Theory Dyn. Syst. 21(3) (2022), . doi:10.1007/s12346-022-00596-9CrossRefGoogle Scholar
Le, P., Liouville theorems for a p-Laplace equation with Hartree type nonlinearity, Vietnam J. Math. 51(2) (2023), 263–276. doi:10.1007/s10013-021-00508-5CrossRefGoogle Scholar
Lieb, E. H., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57(2) (1976/77), 93–105. doi:10.1002/sapm197757293CrossRefGoogle Scholar
Mercuri, C., Moroz, V. and Van Schaftingen, J., Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency, Calc. Var. Partial Differential Equations 55(6) (2016), . doi:10.1007/s00526-016-1079-3CrossRefGoogle Scholar
Rahal, B. and Harrabi, A., Liouville results for m-Laplace equations in half-space and strips with mixed boundary value conditions and finite Morse index, J. Dynam. Differential Equations 30(3) (2018), 1161–1185. doi:10.1007/s10884-017-9593-3CrossRefGoogle Scholar
Rahal, B., Some Liouville theorems for Hénon type equations in half-space with nonlinear boundary value conditions and finite Morse indices, Anal. Math. Phys. 10(4): (2020), . doi:10.1007/s13324-020-00398-9CrossRefGoogle Scholar
Sun, W. and Chang, X., Existence of least energy nodal solutions for a double-phase problem with nonlocal nonlinearity, Appl. Anal. 102(6) (2023), 1752–1764. doi:10.1080/00036811.2021.1999422CrossRefGoogle Scholar
Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51(1) (1984), 126–150. doi:10.1016/0022-0396(84)90105-0CrossRefGoogle Scholar
Wang, C. and Ye, D., Some Liouville theorems for Hénon type elliptic equations, J. Funct. Anal. 262(4) (2012), 1705–1727. doi:10.1016/j.jfa.2011.11.017CrossRefGoogle Scholar
Zhao, X., Liouville theorem for Choquard equation with finite Morse indices, Acta Math. Sci. Ser. B (Engl. Ed.) 38(2) (2018), 673–680. doi:10.1016/S0252-9602(18)30773-2Google Scholar
Zhikov, V. V., Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50(4) (1986), .Google Scholar
Zhikov, V. V., On Lavrentiev’s phenomenon, Russian J. Math. Phys. 3(2) (1995), 249–269.Google Scholar