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Mat. Sb. (N.S.), 1982, Volume 117(159), Number 2, Pages 251–265 (Mi msb2202)  

This article is cited in 14 scientific papers (total in 15 papers)

On the solvability of quasilinear elliptic equations of arbitrary order

S. I. Pokhozhaev


Abstract: Quasilinear elliptic equations of arbitrary order $2m\geqslant2$ with smooth linear operators under general linear boundary conditions are considered in the space $W_p^{2m}(\Omega)$, $p>1$.
Theorems are given presenting a priori estimates of $\|u\|_{2m,p}$ in terms of $\|u\|_{k,\infty}\equiv\sum\limits_{|\gamma|\leqslant k}\sup\limits_\Omega|D^\gamma u(x)|$ with some $k$, $0\leqslant k\leqslant 2m-1$, and in terms of $\|u\|_{m,2}$.
For these cases, a critical power law growth is obtained for the nonlinear operator relative to the relevant derivatives. Counterexamples are constructed to show that this critical characteristic growth law cannot be improved without additional assumptions.
On the basis of this theory, existence theorems for certain quasilinear elliptic problems are established under the condition that there exists an a priori estimate for $\|u\|_{k,\infty}$ (in the appropriate family of such problems). An existence theorem is also obtained for the solvability of the Dirichlet boundary value problem for some quasilinear elliptic equations of arbitrary order.
Examples are given.
Bibliography: 11 titles.

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English version:
Mathematics of the USSR-Sbornik, 1983, 45:2, 257–271

Bibliographic databases:

UDC: 517.946
MSC: Primary 35J60, 35B45; Secondary 35J40
Received: 09.04.1981

Citation: S. I. Pokhozhaev, “On the solvability of quasilinear elliptic equations of arbitrary order”, Mat. Sb. (N.S.), 117(159):2 (1982), 251–265; Math. USSR-Sb., 45:2 (1983), 257–271

Citation in format AMSBIB
\Bibitem{Pok82}
\by S.~I.~Pokhozhaev
\paper On the solvability of quasilinear elliptic equations of arbitrary order
\jour Mat. Sb. (N.S.)
\yr 1982
\vol 117(159)
\issue 2
\pages 251--265
\mathnet{http://mi.mathnet.ru/msb2202}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=644772}
\zmath{https://zbmath.org/?q=an:0511.35014|0491.35020}
\transl
\jour Math. USSR-Sb.
\yr 1983
\vol 45
\issue 2
\pages 257--271
\crossref{https://doi.org/10.1070/SM1983v045n02ABEH002598}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. João Batista de Mendonça Xavier, “A priori estimates for the equation − Δu = f(x, u, Du)”, Nonlinear Analysis: Theory, Methods & Applications, 22:12 (1994), 1501  crossref  mathscinet  zmath
    2. O. V. Besov, V. I. Il'in, L. D. Kudryavtsev, V. P. Kurdyumov, S. M. Nikol'skii, L. V. Ovsyannikov, V. A. Sadovnichii, “Stanislav Ivanovich Pokhozhaev (on his sixtieth birthday)”, Russian Math. Surveys, 51:2 (1996), 363–369  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. G. G. Laptev, “A priori estimates of strong solutions of semilinear parabolic equations”, Math. Notes, 64:4 (1998), 488–495  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Laptev G., “Existence of Strong Solutions of Second-Order Semilinear Parabolic Systems”, Differ. Equ., 34:12 (1998), 1639–1645  mathnet  mathscinet  zmath  isi
    5. Laptev G., “A Priori Estimates and Existence of Strong Solutions of Semilinear Parabolic Systems”, Differ. Equ., 34:4 (1998), 516–521  mathnet  mathscinet  zmath  isi
    6. G. G. Laptev, “An Interpolation Method for Deriving a priori Estimates for Strong Solutions to Second-Order Semilinear Parabolic Equations”, Proc. Steklov Inst. Math., 227 (1999), 173–185  mathnet  mathscinet  zmath
    7. Pohozaev, SI, “Critical Nonlinearities in Partial Differential Equations”, Milan Journal of Mathematics, 77:1 (2009), 127  crossref  mathscinet  zmath  isi  elib
    8. S. N. Timergaliev, I. R. Mavleev, “Solvability of the boundary value problem for a partial quasilinear differential equation of the fourth order”, Russian Math. (Iz. VUZ), 54:12 (2010), 45–50  mathnet  crossref  mathscinet  elib
    9. Pokhozhaev S.I., “Critical Nonlinearities in Partial Differential Equations”, Russ. J. Math. Phys., 20:4 (2013), 476–491  crossref  mathscinet  zmath  isi
    10. T. K. Yuldashev, “Obobschennaya razreshimost smeshannoi zadachi dlya nelineinogo integro-differentsialnogo uravneniya vysokogo poryadka s vyrozhdennym yadrom”, Izv. IMI UdGU, 50 (2017), 121–132  mathnet  crossref  elib
    11. T. K. Yuldashev, “Nachalnaya zadacha dlya kvazilineinogo integro-differentsialnogo uravneniya v chastnykh proizvodnykh vysshego poryadka s vyrozhdennym yadrom”, Izv. IMI UdGU, 52 (2018), 116–130  mathnet  crossref  elib
    12. T. K. Yuldashev, K. Kh. Shabadikov, “Smeshannaya zadacha dlya nelineinogo psevdoparabolicheskogo uravneniya vysokogo poryadka”, Matematicheskii analiz, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 156, VINITI RAN, M., 2018, 73–83  mathnet
    13. T. K. Yuldashev, K. Kh. Shabadikov, “Nachalnaya zadacha dlya kvazilineinogo differentsialnogo uravneniya v chastnykh proizvodnykh vysshego poryadka”, Matematicheskii analiz, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 156, VINITI RAN, M., 2018, 106–116  mathnet
    14. T. K. Yuldashev, “Integro-differentsialnoe uravnenie s dvumernym operatorom Uizema vysokoi stepeni”, Matematicheskii analiz, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 156, VINITI RAN, M., 2018, 117–125  mathnet
    15. T. K. Yuldashev, “Obratnaya kraevaya zadacha dlya integro-differentsialnogo uravneniya tipa Bussineska s vyrozhdennym yadrom”, Materialy mezhdunarodnoi nauchnoi konferentsii Aktualnye problemy prikladnoi matematiki i fiziki Kabardino-Balkariya, Nalchik, 1721 maya 2017 g., Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 149, VINITI RAN, M., 2018, 129–140  mathnet  mathscinet
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