Ma, To Fu; Sanchez, Luis Three solutions of a quasilinear elliptic problem near resonance. (English) Zbl 0958.35052 Math. Slovaca 47, No. 4, 451-457 (1997). Using critical point theory, the authors prove a multiplicity result near the first eigenvalue \(\lambda_1\) for quasilinear elliptic problems of the form \[ -\Delta_pu-\lambda_1|u|^{p-2}u+\varepsilon |u|^{p-2}u=f(x,u)+h(x) \] with Dirichlet conditions. It is proved that, for sufficiently small \(\varepsilon >0\), the problem above has at least three solutions when \(f\) and \(h\) satisfy a Landesman-Lazer condition. Reviewer: Jean Mawhin (Louvain) Cited in 7 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 58E30 Variational principles in infinite-dimensional spaces 35A15 Variational methods applied to PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) Keywords:\(p\)-Laplacian; critical point theory PDFBibTeX XMLCite \textit{T. F. Ma} and \textit{L. Sanchez}, Math. Slovaca 47, No. 4, 451--457 (1997; Zbl 0958.35052) Full Text: EuDML References: [1] AMBROSETTI A.-RABINOWITZ P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349-381. · Zbl 0273.49063 [2] ANANE A.: Simplicite et isolation de la premiére valeur propre du p-Laplacien avec poids. C. R. Acad. Sci. Paris Ser. I Math. 305 (1987), 725-728. · Zbl 0633.35061 [3] BADIALE M.-LUPO D.: Some remarks on a multiplicity result by Mawhin and Schmitt. Bull. Acad. R. Belgique Cl. Sci. 65 (1989), 210-224. · Zbl 0706.34020 [4] CHIAPPINELLI R.-MAWHIN J.-NUGARI R.: Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities. Nonlinear Anal. T. M. A. 18 (1992), 1099-1112. · Zbl 0780.35038 [5] HIRANO N.: Multiple solutions for quasilinear elliptic equations. Nonlinear Anal. T. M. A. 15 (1990), 625-638. · Zbl 0723.35034 [6] LUPO D.-RAMOS M.: Some multiplicity results for two-point boundary value problems near resonance. Rend. Sem. Mat. Univ. Politec. Torino 48 (1990), 125-135. · Zbl 0764.34017 [7] MAWHIN J.-SCHMITT K.: Nonlinear eigenvalue problems with the parameter near resonance. Ann. Polon. Math. LI (1990), 241-248. · Zbl 0724.34025 [8] RAMOS M.-SANCHEZ L.: A variational approach to multiplicity in elliptic problems near resonance. Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 385-394. · Zbl 0869.35041 [9] SANCHEZ L.: Boundary value problems for some fourth order ordinary differential equations. Applicable Anal. 38 (1990), 161-177. · Zbl 0682.34020 [10] ZONGMING, GUO: Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems. Nonlinear Anal. T. M. A. 18 (1992), 957-971. · Zbl 0782.35053 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.