Abstract and Applied Analysis

Multiplicity of Positive Solutions for a $p$-$q$-Laplacian Type Equation with Critical Nonlinearities

Abstract

We study the effect of the coefficient $f(x)$ of the critical nonlinearity on the number of positive solutions for a $p$-$q$-Laplacian equation. Under suitable assumptions for $f(x)$ and $g(x)$, we should prove that for sufficiently small $\lambda >0$, there exist at least $k$ positive solutions of the following $p$-$q$-Laplacian equation, $-{\mathrm{\Delta }}_{p}u-{\mathrm{\Delta }}_{q}u=f(x)|u{|}^{{p}^{\ast}-2}u+\lambda g(x)|u{|}^{r-2}u\text{\hspace\{0.17em\}\hspace\{0.17em\}in\hspace\{0.17em\}\hspace\{0.17em\}}\mathrm{\Omega }$, $u=0\text{\hspace\{0.17em\}\hspace\{0.17em\}on\hspace\{0.17em\}\hspace\{0.17em\}}\partial \mathrm{\Omega ,}$ where $\mathrm{\Omega }\subset {\mathbf{R}}^{N}$ is a bounded smooth domain, $N>p$, $1, $p^{\ast}=Np/(N-p)$ is the critical Sobolev exponent, and ${\Delta }_{s}u=\text{d}\text{i}\text{v}(|\nabla u{|}^{s-2}\nabla u$ is the $s$-Laplacian of $u$.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 829069, 9 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412273298

Digital Object Identifier
doi:10.1155/2014/829069

Mathematical Reviews number (MathSciNet)
MR3193552

Zentralblatt MATH identifier
07023153

Citation

Hsu, Tsing-San; Lin, Huei-Li. Multiplicity of Positive Solutions for a $p$ - $q$ -Laplacian Type Equation with Critical Nonlinearities. Abstr. Appl. Anal. 2014 (2014), Article ID 829069, 9 pages. doi:10.1155/2014/829069. https://projecteuclid.org/euclid.aaa/1412273298

References

• V. Benci, A. M. Micheletti, and D. Visetti, “An Eigenvalue problem for a quasilinear elliptic field equation,” Journal of Differential Equations, vol. 184, no. 2, pp. 299–320, 2002.
• M. Wu and Z. Yang, “A class of $p$-$q$-Laplacian type equation with potentials eigenvalue problem in ${\mathbf{R}}^{N}$, Bound,” Boundary Value Problems, vol. 2009, Article ID 185319, 2009.
• C. He and G. Li, “The existence of a nontrivial solution to the $p$-$q$-Laplacian problem with nonlinearity asymptotic to ${u}^{p-1}$ at infinity in ${\mathbf{R}}^{N}$,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 5, pp. 1100–1119, 2008.
• L. Gongbao and Z. Guo, “Multiple solutions for the $p$-$q$-Laplacian problem with critical exponent,” Acta Mathematica Scientia B, vol. 29, no. 4, pp. 903–918, 2009.
• H. Yin and Z. Yang, “A class of $p$-$q$-Laplacian type equation with concave-convex nonlinearities in bounded domain,” Journal of Mathematical Analysis and Applications, vol. 382, no. 2, pp. 843–855, 2011.
• H. Yin and Z. Yang, “Multiplicity of positive solutions to a $p$-$q$-Laplacian equation involving critical nonlinearity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 6, pp. 3021–3035, 2012.
• H. Brézis and L. Nirenberg, “Positive solutions of nonlinear elliptic equations involving critical sobolev exponent,” Communications on Pure and Applied Mathematics, vol. 36, no. 4, pp. 437–477, 1983.
• C. O. Alves, J. M. do Ó, and O. H. Miyagaki, “On perturbations of a class of a periodic $m$-Laplacian equation with critical growth,” Nonlinear Analysis: Theory, Methods & Applications, vol. 45, no. 7, pp. 849–863, 2001.
• A. Ambrosetti, H. Brézis, and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,” Journal of Functional Analysis, vol. 122, no. 2, pp. 519–543, 1994.
• V. Benci and G. Cerami, “The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems,” Archive for Rational Mechanics and Analysis, vol. 114, no. 1, pp. 79–93, 1991.
• O. Rey, “A multiplicity result for a variational problem with lack of compactness,” Nonlinear Analysis: Theory, Methods & Applications, vol. 13, no. 10, pp. 1241–1249, 1989.
• C. O. Alves and Y. H. Ding, “Multiplicity of positive solutions to a $p$-Laplacian equation involving critical nonlinearity,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 508–521, 2003.
• T.-S. Hsu, “Multiplicity results for $p$-Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions,” Abstract and Applied Analysis, vol. 2009, Article ID 652109, 24 pages, 2009.
• P. Han, “Multiple solutions to singular critical elliptic equations,” Israel Journal of Mathematics, vol. 156, no. 1, pp. 359–380, 2006.
• G. Talenti, “Best constant in Sobolev inequality,” Annali di Matematica Pura ed Applicata, vol. 110, no. 1, pp. 353–372, 1976.
• P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.
• D.-M. Cao and H.-S. Zhou, “Multiple positive solutions of nonhomogeneous semilinear elliptic equations in ${\mathbf{R}}^{N}$,” Proceedings of the Royal Society of Edinburgh A: Mathematics, vol. 126, no. 2, pp. 443–463, 1996.
• G. Tarantello, “On nonhomogeneous elliptic equations involving critical sobolev exponent,” Annales de l'Institut Henri Poincaré: Analyse Non Linéaire Open Archive, vol. 9, no. 3, pp. 281–304, 1992.
• H. Brézis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals,” Proceedings of the American Mathematical Society, vol. 88, no. 3, pp. 486–490, 1983.
• P. Drábek and Y. X. Huang, “Multiplicity of positive solutions for some quasilinear elliptic equation in ${\mathbf{R}}^{N}$ with critical Sobolev exponent,” Journal of Differential Equations, vol. 140, no. 1, pp. 106–132, 1997.
• M. Struwe, Variational Methods, Springer, Heidelberg, Germany, 2nd edition, 1996.
• M. Willem, Minimax Theorems, Birkhäuser, 1996. \endinput