On a class of ( p , q ) -Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

M.S. Shahrokhi-Dehkordi

Communications in Mathematics (2017)

  • Volume: 25, Issue: 1, page 13-20
  • ISSN: 1804-1388

Abstract

top
Let Ω ⊂ ℝ n be a bounded starshaped domain and consider the ( p , q ) -Laplacian problem - Δ p u - Δ q u = λ ( 𝐱 ) | u | p ☆ - 2 u + μ | u | r - 2 u where μ is a positive parameter, 1 < q ≤ p < n , r ≥ p ☆ and p ☆ : = n p n - p is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the ( p , q ) -Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

How to cite

top

Shahrokhi-Dehkordi, M.S.. "On a class of $(p,q)$-Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain." Communications in Mathematics 25.1 (2017): 13-20. <http://eudml.org/doc/294093>.

@article{Shahrokhi2017,
abstract = {Let $\Omega \subset \mathbb \{R\}^n$ be a bounded starshaped domain and consider the $(p,q)$-Laplacian problem \begin\{align*\} -\Delta \_p u-\Delta \_q u = \lambda (\{\bf x\} )\vert u\vert ^\{p^\star -2\} u+\mu |u|^\{r-2\} u \end\{align*\} where $\mu $ is a positive parameter, $1 < q \le p < n$, $r\ge p^\{\star \}$ and $p^\{\star \}:=\frac\{np\}\{n-p\}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p, q)$-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.},
author = {Shahrokhi-Dehkordi, M.S.},
journal = {Communications in Mathematics},
keywords = {Quasi-linear elliptic problem; $(p,q)$-Laplacian operator; Critical Sobolev-Hardy exponent; Starshaped domain},
language = {eng},
number = {1},
pages = {13-20},
publisher = {University of Ostrava},
title = {On a class of $(p,q)$-Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain},
url = {http://eudml.org/doc/294093},
volume = {25},
year = {2017},
}

TY - JOUR
AU - Shahrokhi-Dehkordi, M.S.
TI - On a class of $(p,q)$-Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain
JO - Communications in Mathematics
PY - 2017
PB - University of Ostrava
VL - 25
IS - 1
SP - 13
EP - 20
AB - Let $\Omega \subset \mathbb {R}^n$ be a bounded starshaped domain and consider the $(p,q)$-Laplacian problem \begin{align*} -\Delta _p u-\Delta _q u = \lambda ({\bf x} )\vert u\vert ^{p^\star -2} u+\mu |u|^{r-2} u \end{align*} where $\mu $ is a positive parameter, $1 < q \le p < n$, $r\ge p^{\star }$ and $p^{\star }:=\frac{np}{n-p}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p, q)$-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.
LA - eng
KW - Quasi-linear elliptic problem; $(p,q)$-Laplacian operator; Critical Sobolev-Hardy exponent; Starshaped domain
UR - http://eudml.org/doc/294093
ER -

References

top
  1. Benci, V., Cerami, G., Existence of positive solutions of the equation - Δ u + a ( x ) u = u ( N + 2 ) / ( N - 2 ) in ℝ N , J. Funct. Anal., 88, 1990, 91-117, (1990) MR1033915
  2. Benci, V., D'Avenia, P., Fortunato, D., Pisani, L., 10.1007/s002050000101, Arch. Ration. Mech. Anal., 154, 4, 2000, 297-324, (2000) Zbl0973.35161MR1785469DOI10.1007/s002050000101
  3. Benci, V., Micheletti, A. M., Visetti, D., 10.1006/jdeq.2001.4155, J. Differ. Equ., 184, 2, 2002, 299-320, (2002) Zbl1157.35348MR1929880DOI10.1006/jdeq.2001.4155
  4. Candito, P., Marano, S. A., Perera, K., 10.1007/s00030-015-0353-y, Nonlinear Differ. Equ. Appl., 22, 2015, 1959-1972, (2015) Zbl1328.35053MR3415031DOI10.1007/s00030-015-0353-y
  5. Cherfils, L., Iĺyasov, Y., On the stationary solutions of generalized reaction diffusion equations with p & q -Laplacian, Commun. Pure Appl. Anal., 4, 1, 2005, 9-22, (2005) Zbl1210.35090MR2126276
  6. Derrick, G. H., 10.1063/1.1704233, J. Math. Phys., 5, 1964, 1252-1254, (1964) MR0174304DOI10.1063/1.1704233
  7. Fife, P. C., 10.1007/978-3-642-93111-6, Lecture Notes in Biomathematics, 28, 1979, Springer, Berlin, (1979) Zbl0403.92004MR0527914DOI10.1007/978-3-642-93111-6
  8. Figueiredo, G. M., 10.1016/j.jmaa.2011.02.017, J. Math. Anal. Appl., 378, 2011, 507-518, (2011) MR2773261DOI10.1016/j.jmaa.2011.02.017
  9. Filippucci, R., Pucci, P., Robert, F., 10.1016/j.matpur.2008.09.008, J. Math. Pures Appl, 91, 2009, 156-177, (2009) MR2498753DOI10.1016/j.matpur.2008.09.008
  10. Ghoussoub, N., Yuan, C., 10.1090/S0002-9947-00-02560-5, Trans. Amer. Math. Soc., 352, 2000, 5703-5743, (2000) Zbl0956.35056MR1695021DOI10.1090/S0002-9947-00-02560-5
  11. Guedda, M., Véron, L., 10.1016/0362-546X(89)90020-5, Nonlinear Anal., 13, 1989, 879-902, (1989) Zbl0714.35032MR1009077DOI10.1016/0362-546X(89)90020-5
  12. Guo, Q., Han, J., Niu, P., 10.1016/j.na.2012.05.021, Nonlinear Analysis, 75, 2012, 5765-5786, (2012) Zbl1250.35083MR2948296DOI10.1016/j.na.2012.05.021
  13. Kang, D., 10.1016/j.jmaa.2007.10.058, J. Math. Anal. Appl., 341, 2008, 764-782, (2008) Zbl1137.35378MR2398246DOI10.1016/j.jmaa.2007.10.058
  14. Li, G. B., Liang, X., 10.1016/j.na.2009.01.066, Nonlinear Anal., 71, 2009, 2316-2334, (2009) MR2524439DOI10.1016/j.na.2009.01.066
  15. Li, Y., Ruf, B., Guo, Q., Niu, P., 10.1007/s10231-011-0213-2, Annali di Matematica, 192, 2013, 93-113, (2013) Zbl1292.35104MR3011325DOI10.1007/s10231-011-0213-2
  16. López, R., Constant Mean Curvature Surfaces with Boundary, 2013, Springer Monographs in Mathematics, (2013) Zbl1329.53099MR3098467
  17. Marano, S. A., Papageorgiou, N. S., 10.1016/j.na.2012.09.007, Nonlinear Anal., 77, 2013, 118-129, (2013) Zbl1260.35036MR2988766DOI10.1016/j.na.2012.09.007
  18. Shahrokhi-Dehkordi, M. S., Taheri, A., Quasiconvexity and uniqueness of stationary points on a space of measure preserving maps, Journal of Convex Analysis, 17, 1, 2010, 69-79, (2010) Zbl1186.49033MR2642716
  19. Sun, M., 10.1016/j.jmaa.2011.08.030, J. Math. Anal. Appl., 386, 2, 2012, 661-668, (2012) Zbl1229.35089MR2834776DOI10.1016/j.jmaa.2011.08.030
  20. Wilhelmsson, H., 10.1103/PhysRevA.36.965, Phys. Rev. A, 36, 2, 1987, 965-966, (1987) MR0901723DOI10.1103/PhysRevA.36.965
  21. Yin, H., Yang, Z., 10.1016/j.jmaa.2011.04.090, J. Math. Anal. Appl., 382, 2011, 843-855, (2011) Zbl1222.35083MR2810836DOI10.1016/j.jmaa.2011.04.090

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.