Multiple positive solutions for a \((p,q)\)-Laplace equation involving singular and critical terms. (English) Zbl 1425.35066

Summary: In this paper, we study a \((p,q)\)-Laplace equation with singular and critical terms and establish the existence of multiple positive solutions via use of variational methods.


35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J75 Singular elliptic equations
35B33 Critical exponents in context of PDEs
35A15 Variational methods applied to PDEs
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[1] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. · Zbl 0273.49063
[2] N. Benouhiba and Z. Belyacine, On the solutions of the \((p, q)\)-Laplacian problem at resonance, Nonlin. Anal. 77 (2013), 74-81. · Zbl 1255.35036
[3] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergece of functionals, Proc. Amer. Math. Soc. 88 (1983), 486-490.
[4] P. Candito, S. Carl and R. Livrea, Multiple solutions for quasilinear elliptic problems via critical points in open sublevels and truncation principles, J. Math. Anal. Appl. 395 (2012), 156-163. · Zbl 1246.35094
[5] D.G. Costa and C.A. Magalhães, Existence results for perturbations of the \(p\)-Laplacian, Nonlin. Anal. 24 (1995), 409-418. · Zbl 0818.35029
[6] M. Cuesta and H.R. Quoirin, A weighted eigenvalue problem for the \(p\)-Laplacian plus a potential, Nonlin. Diff. Eqs. Appl. 16 (2009), 469-491. · Zbl 1174.35089
[7] M. Degiovanni and S. Lancelotti, Linking solutions for \(p\)-Laplace equations with nonlinearity at critical growth, J. Funct. Anal. 256 (2009), 3643-3659. · Zbl 1170.35418
[8] L.F.O. Faria, O.H. Miyagaki and M. Tanaka, Existence of a positive solution for problems with \((p, q)\)-Laplacian and convection term in \(\mathbb{R}^N\), Bound. Value Prob. 158 (2016), paper no. 158. · Zbl 1347.35122
[9] M.E. Filippakis and N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the \(p\)-Laplacian, J. Diff. Eqs. 245 (2008), 1883-1922. · Zbl 1151.35036
[10] N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and hardy exponents, Trans. Amer. Math. Soc. 352 (2000), 5703-5743. · Zbl 0956.35056
[11] G.B. Li and C.Y. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of \(p\)-Laplacian type without the Ambrosetti-Rabinowitz condition, Nonlin. Anal. 72 (2010), 4602-4613. · Zbl 1190.35104
[12] G.B. Li and G. Zhang, Multiple solutions for the \((p, q)\)-Laplacian problem with critical exponent, Acta Math. Sci. 29 (2009), 903-918. · Zbl 1212.35125
[13] S.B. Liu, Existence of solutions to a superlinear \(p\)-Laplacian equation, Electr. J. Diff. Eqs. 66 (2001), 1-6. · Zbl 1011.35062
[14] S.A. Maranoa and N.S. Papageorgiou, Constant-sign and nodal solutions of coercive \((p, q)\)-Laplacian problems, Nonlin. Anal. 77 (2013), 118-129. · Zbl 1260.35036
[15] D. Motreanua, V.V. Motreanua and N.S. Papageorgio, A multiplicity theorem for problems with the \(p\)-Laplacian, Nonlin. Anal. 68 (2008), 1016-1027.
[16] N.S. Papageorgiou and E.M. Rochab, Existence of three nontrivial solutions for asymptotically \(p\)-linear noncoercive \(p\)-Laplacian equations, Nonlin. Anal. 74 (2011), 5314-5326. · Zbl 1219.35069
[17] L.S. Shi and X.J. Chang, Multiple solutions to \(p\)-Laplacian problems with concave nonlinearities, J. Math. Anal. Appl. 363 (2010), 155-160. · Zbl 1180.35261
[18] L. Wang, Q.L. Wei and D.S. Kang, Multiple positive solutions for \(p\)-Laplace elliptic equations involving concave convex nonlinearities and a Hardy-type term, Nonlin. Anal. 74 (2011), 626-638. · Zbl 1203.35122
[19] B.J. Xuan, Existence results for a superlinear \(p\)-Laplacian equation with indefinite weights, Nonlin. Anal. 54 (2003), 949-958. · Zbl 1103.35326
[20] H.H. Yin and Z.D. Yang, Multiplicity of positive solutions to a \(p-q\)-Laplacian equation involving critical nonlinearity, Nonlin. Anal. 75 (2012), 3021-3035. · Zbl 1235.35123
[21] G.Q. Zhang, Ground state solution for quasilinear elliptic equation with critical growth in \(\mathbb{R}^N\), Nonlin. Anal. 75 (2012), 3178-3187. · Zbl 1237.35070
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