Geodesic
Multiple positive solutions of an elliptic equation with a~convex--concave nonlinearity containing a~sign-changing term
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 167-180.

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We study the existence of multiple positive solutions to a nonlinear Dirichlet problem for the $p$-Laplacian (in a bounded domain in $\mathbb R^N$) with a concave nonlinearity and with a nonlinear perturbation involving a function of the spatial variable whose sign can change the character of concavity. Under two different sets of conditions imposed on the perturbation, we prove the existence of two and three positive solutions, respectively. 
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V. F. Lubyshev. Multiple positive solutions of an elliptic equation with a~convex--concave nonlinearity containing a~sign-changing term. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 167-180. http://geodesic.mathdoc.fr/item/TM_2010_269_a13/

[1] Pokhozhaev S.I., “O metode rassloeniya resheniya nelineinykh kraevykh zadach”, Tr. MIAN, 192, 1990, 146–163 | MR | Zbl

[2] Pokhozhaev S.I., “O metode globalnogo rassloeniya v nelineinykh variatsionnykh zadachakh”, Tr. MIAN, 219, 1997, 286–334 | MR | Zbl

[3] Ambrosetti A., Brézis H., Cerami G., “Combined effects of concave and convex nonlinearities in some elliptic problems”, J. Funct. Anal., 122 (1994), 519–543 | DOI | MR | Zbl

[4] Ambrosetti A., Garcia Azorero J., Peral I., “Multiplicity results for some nonlinear elliptic equations”, J. Funct. Anal., 137 (1996), 219–242 | DOI | MR | Zbl

[5] García Azorero J., Peral Alonso I., “Some results about the existence of a second positive solution in a quasilinear critical problem”, Indiana Univ. Math. J., 43 (1994), 941–957 | DOI | MR | Zbl

[6] García Azorero J.P., Peral Alonso I., Manfredi J.J., “Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations”, Commun. Contemp. Math., 2 (2000), 385–404 | MR | Zbl

[7] Il'yasov Y., “On nonlocal existence results for elliptic equations with convex–concave nonlinearities”, Nonlin. Anal. Theory, Meth. and Appl., 61 (2005), 211–236 | DOI | MR | Zbl

[8] Kyritsi S.T., Papageorgiou N.S., “Pairs of positive solutions for $p$-Laplacian equations with combined nonlinearities”, Commun. Pure and Appl. Anal., 8 (2009), 1031–1051 | DOI | MR | Zbl

[9] Li S., Wu S., Zhou H.-S., “Solutions to semilinear elliptic problems with combined nonlinearities”, J. Diff. Equat., 185 (2002), 200–224 | DOI | MR | Zbl