×

Quasilinear elliptic problems with critical exponents and Hardy terms. (English) Zbl 1210.35021

Summary: Let \(\Omega\ni 0\) be an open bounded domain, \(\Omega\subset\mathbb R^N\) \((N>p^2)\). We are concerned with the multiplicity of positive solutions of
\[ -\Delta_pu- \mu\frac{|u|^{p-2}u}{|x|^p}= \lambda|u|^{p-2}u+ Q(x)|u|^{p^*-2}u, \quad u\in W_0^{1,p}(\Omega), \]
where
\[ -\Delta_pu= -\text{div}\big(|\nabla u|^{p-2}\nabla u\big), \quad 1<p<N, \qquad p^*=\frac{Np}{n-p}, \quad 0<\mu<\bigg( \frac{N-p}{p}\bigg)^p,\;\lambda>0, \]
and \(Q(x)\) is a nonnegative function on \(\overline{\Omega}\). By investigating the effect of the coefficient of the critical nonlinearity, we, by means of variational method, prove the existence of multiple positive solutions.

MSC:

35B33 Critical exponents in context of PDEs
35J61 Semilinear elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] B. Abdellaoui, V. Felli, I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the \(p\); B. Abdellaoui, V. Felli, I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the \(p\) · Zbl 1118.35010
[2] Azorero, J. G.; Alonso, I. P., Existence and nonuniqueness for the \(p\)-Laplacian nonlinear eigenvalues, Comm. Partial Differential Equations, 12, 1389-1430 (1987) · Zbl 0637.35069
[3] Azorero, J. G.; Alonso, I. P., Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323, 877-895 (1991) · Zbl 0729.35051
[4] Azorero, J. G.; Alonso, I. P., Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144, 441-476 (1998) · Zbl 0918.35052
[5] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math., 36, 437-478 (1983) · Zbl 0541.35029
[6] Cao, D.; Han, P., Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations, 205, 521-537 (2004) · Zbl 1154.35346
[7] Cao, D.; Noussair, E. S., Multiple positive and nodal solutions for semilinear elliptic problems with critical exponents, Indiana Univ. Math. J., 44, 1249-1271 (1995) · Zbl 0849.35030
[8] Cao, D.; Peng, S., A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms, J. Differential Equations, 193, 424-434 (2003) · Zbl 1140.35412
[9] Ekeland, I., On the variational principle, J. Math. Anal. Appl., 17, 324-353 (1974) · Zbl 0286.49015
[10] Ekeland, I.; Ghoussoub, N., Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc., 39, 207-265 (2002) · Zbl 1064.35054
[11] Ferrero, A.; Gazzola, F., Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177, 494-522 (2001) · Zbl 0997.35017
[12] Ghoussoub, N.; Yuan, C., Multiple solutions for quasilinear PDEs involving critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352, 5703-5743 (2000) · Zbl 0956.35056
[13] Guedda, M.; Veron, L., Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13, 879-902 (1989) · Zbl 0714.35032
[14] Jannelli, E., The role played by space dimension in elliptic critical problems, J. Differential Equations, 156, 407-426 (1999) · Zbl 0938.35058
[15] Lions, P. L., The concentration compactness principle in the calculus of variationsthe limit case, Rev. Mat. Iberoamericana, 1, 145-201 (1985), 45-121 · Zbl 0704.49005
[16] Ruiz, D.; Willem, M., Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations, 190, 524-538 (2003) · Zbl 1163.35383
[17] Talenti, G., Best constants in Sobolev inequality, Ann. Mat., 110, 353-372 (1976) · Zbl 0353.46018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.