On quasi-linear equation problems involving critical and singular nonlinearities
-
1927
Downloads
-
3776
Views
Authors
Yanbin Sang
- Department of Mathematics, School of Science, North University of China, Taiyuan, Shanxi 030051, China.
Xiaorong Luo
- Department of Mathematics, School of Science, North University of China, Taiyuan, Shanxi 030051, China.
Abstract
We consider the singular boundary value problem
\[ \left\{\begin{array}{l}
-\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=h(x)\frac{u^{-\gamma}}{|x|^{b(1-\gamma)}}+\mu \frac{u^{p^*-1}}{|x|^{bp^*}} \ \ \ \text{in} \ \Omega\backslash\{0\}, \cr u>0\ \ \ \ \ \text{in}\ \Omega\backslash\{0\}, \cr u=0\ \ \ \ \ \text{on} \ \partial\Omega, \end{array}\right.
\]
where \(\Omega\subset \mathbb{R}^N(N\geq 3)\) is a bounded
domain such that \(0\in \Omega\),
\(0<\gamma<1\), \(0\leq a<\frac{N-p}{p}\), \(a\leq b<a+1\), \(p^* :=p^* (a,b)=\frac{Np}{N-(1+a-b)p}\), and
\(h(x)\) is a given function.
Based on different assumptions, using variational methods and Ekeland's principle, we admit that this problem possesses two positive solutions.
keywords
Share and Cite
ISRP Style
Yanbin Sang, Xiaorong Luo, On quasi-linear equation problems involving critical and singular nonlinearities, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4966--4982
AMA Style
Sang Yanbin, Luo Xiaorong, On quasi-linear equation problems involving critical and singular nonlinearities. J. Nonlinear Sci. Appl. (2017); 10(9):4966--4982
Chicago/Turabian Style
Sang, Yanbin, Luo, Xiaorong. "On quasi-linear equation problems involving critical and singular nonlinearities." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4966--4982
Keywords
- Critical exponent
- \(p\)-Laplacian operator
- extremal value
- singular nonlinearity
MSC
References
-
[1]
B. Abdellaoui, V. Felli, I. Peral, Existence and nonexistence for quasilinear equations involving the p-Laplacian, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 9 (2006), 445–484.
-
[2]
C. O. Alves, A. El Hamidi , Nehari manifold and existence of positive solutions to a class of quasilinear problems , Nonlinear Anal., 60 (2005), 611–624.
-
[3]
J.-P. Aubin, I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York (1984)
-
[4]
H. Brézis, E. Lieb, A relation between pointwise convergence of functional and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490.
-
[5]
F. Catrina, Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229–258.
-
[6]
Y.-P. Chen, J.-Q. Chen, Existence of multiple positive weak solutions and estimates for extremal values to a class of elliptic problems with Hardy term and singular nonlinearity, J. Math. Anal. Appl., 429 (2015), 873–900.
-
[7]
Y.-P. Chen, J.-Q. Chen, Multiple positive solutions for a semilinear equation with critical exponent and prescribed singularity, Nonlinear Anal., 130 (2016), 121–137.
-
[8]
J.-Q. Chen, E. M. Rocha , Positive solutions for elliptic problems with critical nonlinearity and combined singularity, Math. Bohem., 135 (2010), 413–422.
-
[9]
S.-Q. Cong, Y.-Z. Han, Compatibility conditions for the existence of weak solutions to a singular elliptic equation, Bound. Value Probl., 2015 (2015), 11 pages.
-
[10]
F. J. S. A. Corrêa, A. S. S. Corrêa, G. M. Figueiredo, Positive solution for a class of pq-singular elliptic equation, Nonlinear Anal. Real World Appl., 16 (2014), 163–169.
-
[11]
Z.-Y. Deng, Y.-S. Huang , On G-symmetric solutions of the quasilinear elliptic equations with singular weights and critical exponents , Scientia Sinica Mathematica., 42 (2012), 1053–1066.
-
[12]
Z.-Y. Deng, Y.-S. Huang, On positive G-symmetric solutions of a weighted quasilinear elliptic equation with critical Hardy- Sobolev exponent, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 1619–1633.
-
[13]
N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703–5743.
-
[14]
J. Giacomoni, I. Schindler, P. Takáč , Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 6 (2007), 117–158.
-
[15]
P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms, Nonlinear Anal., 61 (2005), 735–758.
-
[16]
Y. Jalilian , On the existence and multiplicity of solutions for a class of singular elliptic problems, Comput. Math. Appl., 68 (2014), 664–680.
-
[17]
D.-S. Kang, On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms, Nonlinear Anal., 68 (2008), 1973–1985.
-
[18]
D.-S. Kang , Positve solutions to the weighted critical quasilinear problems, Appl. Math. Compt., 213 (2009), 432–439.
-
[19]
N. H. Loc, K. Schmitt, Boundary value problems for singular elliptic equations, Rocky Mt. J. Math., 41 (2011), 555–572.
-
[20]
Y.-b. Sang , An exact estimate result for a semilinear equation with critical exponent and prescribed singularity, J. Math. Anal. Appl., 447 (2017), 128–153.
-
[21]
Y.-J. Sun, S.-J. Li, A nonlinear elliptic equation with critical exponent: estimates for extremal values, Nonlinear Anal., 69 (2008), 1856–1869.
-
[22]
Y.-J. Sun, S.-P. Wu, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal., 260 (2011), 1257–1284.
-
[23]
G. Tarantello , On nonhomogeneous elliptic equations involving critical Sobolev exponent , Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281–304.
-
[24]
S. Terracini, On positive entire solutions to a class of equations with singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241–264.
-
[25]
L. Wang, Q.-L. Wei, D.-S. Kang , Multiple positive solutions for p-Laplace elliptic equations involving concave-convex nonlinearities and a Hardy-type term, Nonliear Anal., 74 (2011), 626–638.
-
[26]
B. J. Xuan, The eightvalue problem for a singular quasilinear elliptic equation, Electron. J. Differ. Equat., 16 (2004), 1–11.
