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The optimal weighted $W^{2, p}$ estimates of elliptic equation with non-compatible conditions
Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights
1. | College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China |
2. | Department of Mathematics, Tongji University, Shanghai, 200092, China |
3. | Department of Applied Mathematics, Xidian University, Xi'an, 710071, China |
$ -\Delta_N u-\mu \frac{1}{(|x|\log(\frac{R}{|x|}))^N}|u|^{N-2}u= \lambda V(x)|u|^{N-2} u + f(x,u), u\in W_0^{1, N}(\Omega), $
where $\Omega$ be a bounded domain containing $0$ in $R^N$, $N \geq 2, 0 < \mu < (\frac{N-1}{N})^N$, and the weight function $V(x)$ may change sign and has nontrivial positive part. Using Moser-Trudinger inequality and nonstandard linking structure introduced by Degiovanni and Lancelotti [6], we prove the existence of a nontrivial solution for any $\lambda\in R$.
References:
[1] |
Adimurthi, M. Ramaswamy and N. Chaudhuri, Improved Hardy-Sobolev inequality and its application, Proceeding of the American Mathematical Society, 130 (2002), 489-505.
doi: doi:10.1090/S0002-9939-01-06132-9. |
[2] |
H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense. Madrid, 10 (1997), 443-469. |
[3] |
M. Cuesta, Eigenvalue problems for the p-laplacian with indefinite weights, Electron. J. Differential Equations, 33 (2001), 1-9. |
[4] |
J. M. do O, Semilinear Dirichlet problems for the N-Laplacian in $R^N$ with nonlinearities in critical growth range, Differential Integral Equations, 9 (1996), 967-979. |
[5] |
J. M. do O, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimensional two, J. Math. Anal. Appl., 345 (2008), 286-304.
doi: doi:10.1016/j.jmaa.2008.03.074. |
[6] |
M. Degiovanni and S. Lancelotti, Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity, Ann. I. H. Poincare-AN, 24 (2007), 907-919. |
[7] |
D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.
doi: doi:10.1007/BF01205003. |
[8] |
E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.
doi: doi:10.1007/BF01390270. |
[9] |
X. Fan and Z. Li, Linking and existence results for perturbations of the p-Laplacian, Nonlinear Anal, 42 (2000), 1413-1420.
doi: doi:10.1016/S0362-546X(99)00161-3. |
[10] |
J. P. Garcia and I. A. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 446-476. |
[11] |
N. Ghoussoub, "Duality and Perturbation Methods in Critical Point Theory," Cambridge Univ. Press, Cambridge, 1993.
doi: doi:10.1017/CBO9780511551703. |
[12] |
N. Ghoussoub and C. Yuan, Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.
doi: doi:10.1090/S0002-9947-00-02560-5. |
[13] |
J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970), 1077-1092.
doi: doi:10.1512/iumj.1971.20.20101. |
[14] |
K. Perera and A. Szulkin, p-Laplacian problems where the nonlinearity crosses an eigenvalue, Discrete Contin. Dyn. Syst., 13 (2005), 743-753.
doi: doi:10.3934/dcds.2005.13.743. |
[15] |
I. Peral and J. L. Vazquez, On the stability or instability of singular solutions with exponential reaction term, Arch. Rational Mech. Anal., 129 (1995) 201-224.
doi: doi:10.1007/BF00383673. |
[16] |
Y. T. Shen, Y. X. Yao and Z. H. Chen, On a nonlinear elliptic problem with critical potential in R, Science in China, Ser A mathematics, 47 (2004), 741-755. |
[17] |
A. Szulkin and M. Willem, Eigenvalue problems with indefinite weights, Stud. Math., 135 (1999), 189-201. |
[18] | |
[19] |
G. Zhang and S. Liu, On a class of elliptic equation with critical potential and indefinite weights in $R^2$, Acta Mathematica Scientia, 28 (2008), 929-936. |
show all references
References:
[1] |
Adimurthi, M. Ramaswamy and N. Chaudhuri, Improved Hardy-Sobolev inequality and its application, Proceeding of the American Mathematical Society, 130 (2002), 489-505.
doi: doi:10.1090/S0002-9939-01-06132-9. |
[2] |
H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense. Madrid, 10 (1997), 443-469. |
[3] |
M. Cuesta, Eigenvalue problems for the p-laplacian with indefinite weights, Electron. J. Differential Equations, 33 (2001), 1-9. |
[4] |
J. M. do O, Semilinear Dirichlet problems for the N-Laplacian in $R^N$ with nonlinearities in critical growth range, Differential Integral Equations, 9 (1996), 967-979. |
[5] |
J. M. do O, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimensional two, J. Math. Anal. Appl., 345 (2008), 286-304.
doi: doi:10.1016/j.jmaa.2008.03.074. |
[6] |
M. Degiovanni and S. Lancelotti, Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity, Ann. I. H. Poincare-AN, 24 (2007), 907-919. |
[7] |
D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.
doi: doi:10.1007/BF01205003. |
[8] |
E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.
doi: doi:10.1007/BF01390270. |
[9] |
X. Fan and Z. Li, Linking and existence results for perturbations of the p-Laplacian, Nonlinear Anal, 42 (2000), 1413-1420.
doi: doi:10.1016/S0362-546X(99)00161-3. |
[10] |
J. P. Garcia and I. A. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 446-476. |
[11] |
N. Ghoussoub, "Duality and Perturbation Methods in Critical Point Theory," Cambridge Univ. Press, Cambridge, 1993.
doi: doi:10.1017/CBO9780511551703. |
[12] |
N. Ghoussoub and C. Yuan, Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.
doi: doi:10.1090/S0002-9947-00-02560-5. |
[13] |
J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970), 1077-1092.
doi: doi:10.1512/iumj.1971.20.20101. |
[14] |
K. Perera and A. Szulkin, p-Laplacian problems where the nonlinearity crosses an eigenvalue, Discrete Contin. Dyn. Syst., 13 (2005), 743-753.
doi: doi:10.3934/dcds.2005.13.743. |
[15] |
I. Peral and J. L. Vazquez, On the stability or instability of singular solutions with exponential reaction term, Arch. Rational Mech. Anal., 129 (1995) 201-224.
doi: doi:10.1007/BF00383673. |
[16] |
Y. T. Shen, Y. X. Yao and Z. H. Chen, On a nonlinear elliptic problem with critical potential in R, Science in China, Ser A mathematics, 47 (2004), 741-755. |
[17] |
A. Szulkin and M. Willem, Eigenvalue problems with indefinite weights, Stud. Math., 135 (1999), 189-201. |
[18] | |
[19] |
G. Zhang and S. Liu, On a class of elliptic equation with critical potential and indefinite weights in $R^2$, Acta Mathematica Scientia, 28 (2008), 929-936. |
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