## Positive solutions for a quasilinear elliptic problem involving sublinear and superlinear terms.(English)Zbl 1345.35049

The authors of the present paper study a quasilinear elliptic problem involving a $$p$$-Laplacian operator ($$1<p<N,$$ where $$N\in\mathbb N$$ is the space dimension) and three different nonlinearities and some other parameters. The problem is set either in the whole space (with the corresponding vanishing asymptotic property at infinity) or in bounded smooth domains (with the corresponding homogeneous Dirichlet condition) and they are looking for positive solutions.
On one hand, for some values of one of the parameters the authors show the existence of a $$C^1$$ (weak) solution of the problem. The method of the proof relies on the lower-upper solutions method, combined with a technique of monotone regularization of the nonlinearities. On the other hand, for some other values of this parameter, using some consequences of Picone’s identity, they show the nonexistence of the solutions.
The considered problems are general enough, in the sense that for different choices of the parameters the authors recover some well-known results from the literature. Also, their approach allows them to consider nonlinearities with combined effects of concave and convex terms, besides allowing the presence of singularities. The question of uniqueness of the solutions in the first case is not treated.

### MSC:

 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 35J67 Boundary values of solutions to elliptic equations and elliptic systems 35B09 Positive solutions to PDEs 35B08 Entire solutions to PDEs 35J75 Singular elliptic equations
Full Text:

### References:

 [1] W. Allegretto and Y. Huang, A Picone’s Identity for the p-Laplacian and applications, Nonlinear Anal. 32 (7) (1998), 819-830. · Zbl 1163.57302 [2] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (2) (1994), 519-543. · Zbl 0805.35028 [3] G. Anello, Multiplicity and asymptotic behavior of nonnegative solutions for elliptic problems involving nonlinearities indefinite in sign, Nonlinear Anal. 75 (2012), 3618-3628. · Zbl 1242.35109 [4] S. N. Armstrong and B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations 36 (11) (2011), 2011-2047. · Zbl 1230.35030 [5] J. F. Bonder and L. M. Del Pezzo, An optimization problem for the first eigenvalue of the p-Laplacian plus a potential, Commun. Pure Appl. Anal. 5 (2006), 675-690. · Zbl 0796.60049 [6] L. Caffarelli, R. Hardt and L. Simon, Minimal surfaces with isolated singularities, Manuscripta Math. 48 (1984), 1-18. · Zbl 0568.53033 [7] A. Callegari and A. Nashman, A nonlinear singular boundary-value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math. 38 (1980), 275-281. · Zbl 0453.76002 [8] S. Carl and K. Perera, Generalized solutions of singular p-Laplacian problems in $$\mathbb{R}^N$$, Nonlinear Stud. 18 (1) (2011), 113-124. · Zbl 1217.35083 [9] F. C. Cirstea and V. Radulescu, Existence and uniqueness of positive solutions to a semilinear elliptic problem in $$\mathbb{R}^N$$, J. Math. Anal. Appl. 229 (1999), 417-425. · Zbl 1313.30175 [10] F. C. Cirstea, M. Ghergu and V. Radulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type, J. Math. Pures Appl. 84 (2005), 493-508. · Zbl 1230.92003 [11] D. P. Covei, Existence and uniqueness of positive solutions to a quasilinear elliptic problem in $$\mathbb{R}^N$$, Electron. J. Differential Equations 139 (2005), 1-15. · Zbl 1245.35050 [12] J. I. Díaz and J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Math. Acad. Sci. Paris t. 305 Série I (1987), 521-524. [13] W. Fen and X. Liu, Existence of entire solutions of a singular semilinear elliptic problem, Acta Math. Sin. 20 (2004), 983-988. · Zbl 1130.35037 [14] D. G. Figueiredo, J. P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc. 8 (2) (2006), 269-286. · Zbl 1245.35048 [15] D. G. Figueiredo, J. P. Gossez and P. Ubilla, Local “superlinearity” and “sublinearity” for the p-Laplacian, J. Funct. Anal. 257 (2009), 721-752. · Zbl 1178.35176 [16] W. Fulks and J. S. Maybee, A singular nonlinear equation, Osaka J. Math. 12 (1960), 1-19. · Zbl 0097.30202 [17] J. García, I. Peral and J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2 (3) (2000), 385-404. · Zbl 0965.35067 [18] J. García-Melián and J. Sabrina de Lis, Maximum and comparison principles for operators involving the p-Laplacian, J. Math. Anal. Appl. 218 (1998), 49-65. · Zbl 0897.35015 [19] L. Gasiński and N. S. Papageorgiou, Nonlinear elliptic equations with singular terms and combined nonlinearities, Ann. Henri Poincaré 13 (2012), 481-512. · Zbl 0921.35054 [20] Y. Furusho and Y. Murata, Principal eigenvalue of the p-Laplacian in $$\mathbb{R}^N$$, Nonlinear Anal. 30 (8) (1997), 4749-4756. · Zbl 0890.35099 [21] J. V. Gonçalves and C. A. Santos, Positive solutions for a class of quasilinear singular equations, Electron. J. Differential Equations 56 (2004), 1-15. · Zbl 1109.35309 [22] J. V. Gonçalves, A. L. Melo and C. A. Santos, On existence of $$L^\infty$$-ground states for singular elliptic equations in the presence of a strongly nonlinear term, Adv. Nonlinear Stud. 7 (2007), 475-490. · Zbl 1142.35030 [23] N. Hoang and K. Schmitt, Boundary value problems for singular elliptic equations, Rocky Mountain J. Math. 41 (2) (2011), 555-572. · Zbl 1218.35107 [24] T. Kura, The weak supersolution-subsolution method for second order quasilinear elliptic equations, Hiroshima Math. J. 19 (1989), 1-36. · Zbl 0735.35056 [25] A. V. Lair and A. W. Shaker, Entire solution of a singular semilinear elliptic problem, J. Math. Anal. Appl. 200 (1996), 498-505. · Zbl 0860.35030 [26] A. V. Lair and A. W. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl. 211 (1997), 371-385. · Zbl 1287.94022 [27] E. K. Lee, R. Shivaji and J. Ye, Classes of infinite semipositone systems, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), 853-865. · Zbl 1182.35132 [28] G. M. Lieberman, Boundary regularity for solutions of degenerate ellliptic equations, Nonlinear Anal. 12 (11) (1998), 1203-1219. · Zbl 0675.35042 [29] A. Mohammed, Ground state solutions for singular semi-linear elliptic equations, Nonlinear Anal. 71 (2009), 1276-1280. · Zbl 1167.35371 [30] A. Mohammed, On ground state solutions to mixed type singular semi-linear elliptic equations, Adv. Nonlinear Stud. 10 (2010), 231-244. · Zbl 1200.35136 [31] M. Ôtani and T. Teshima, On the first eigenvalue of some quasilinear elliptic equations, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), 8-10. · Zbl 1211.35111 [32] H. R. Quoirin and P. Ubilla, On some indefinite and non-powerlike elliptic equations, Nonlinear Anal. 79 (2013), 190-203. · Zbl 1261.35051 [33] C. A. Santos, Non-existence and existence of entire solutions for a quasi-linear problem with singular and super-linear terms, Nonlinear Anal. 72 (2010), 3813-3819. · Zbl 1189.35104 [34] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations 8 (7) (1983), 773-817. · Zbl 0515.35024 [35] S. Yijing and L. Shujie, Structure of ground state solutions of singular semilinear elliptic equations, Nonlinear Anal. 55 (2003), 399-417. · Zbl 1122.35341 [36] Z. Zhang, A remark on the existence of positive solutions of a sublinear elliptic problem, Nonlinear Anal. 67 (2007), 147-153. · Zbl 0880.35043
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.