Abstract
The goal of this paper is to study the multiplicity result of positive solutions of a class of degenerate elliptic equations. On the basis of the mountain pass theorems and the sub– and supersolutions argument for p–Laplacian operators, under suitable conditions on the nonlinearity f(x, s), we show the following problem:
possesses at least two positive solutions for large λ, where Ω is a bounded open subset of R N, N ≥ 2, with C 2 boundary, λ is a positive parameter, Δ p is the p–Laplacian operator with p > 1, α, q are given constants such that p − 1 < α < q, and a(x) is a continuous positive function in \( \ifmmode\expandafter\bar\else\expandafter\=\fi{\Omega } \)Ω.
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References
Tolksdorf, P.: On the Dirichlet problem for quasilinear equations in domains with conial boundary points. Comm. P.D.E., 8, 773–817 (1983)
Tolksdorf, P.: Regularity for more general class of quasilinear elliptic equations. J. Diff. Eqns., 51, 126–150 (1984)
Rabinowitz, P. H.: Pairs of positive solutions of nonlinear elliptic partial differential equations. Indiana Univ. Math. J., 23(2), 173–186 (1973)
Guo, Z. M.: Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problem. Nonlinear Analysis, 18(10), 957–071 (1992)
Takeuchi, S.: Positive solutions of a degenerate elliptic equation with logistic reaction. Proc. Amer. Math. Soc., 129(2), 433–441 (2000)
Takeuchi, S.: Multiplicity result for a degenerate elliptic equation with logistic reaction. J. Diff. Eqns., 173, 138–144 (2001)
Diaz, J.: Nonlinear Partial Differential Equation and Free Boundary Problems, Vol.1, Elliptic Equations, Pitman Research Notes in Math., Vol. 106, Boston, 1985
Garcia Azorero, J. P., J. Manfredi, J., Peral Alonso, I.: Sobolev versus H¨older local minimizers and global multiplicity for some quasilinear elliptic equations. Comm. Contem. Math., 2, 385–404 (2000)
Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conf. Ser. in Math., 65, Amer. Math. Soc., Province, 1986
Guo, D. J., Sun, J. X., Qi, G. J.: Some extensions of the mountain pass lemma. Diffential Integral Equations, 1(3), 351–358 (1988)
Du, Y., Guo, Z. M.: Liouville type results and eventual flatness of positive solutions for p–Laplacian equations. Adv. Diff. Eqns., 7(12), 1479–1512 (2002)
Du, Y., Guo, Z. M.: Boundary blow–up solutions and their applications in quasilinear elliptic equations. J. d’Analyse Math., 89, 277–302 (2003)
Serrin, J., Zou, H.: Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta. Math., 189, 79–142 (2002)
Vazquez, J. L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim., 12, 191–202 (1984)
Sakaguchi, S.: Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problem. Ann. Sc. Norm. Super. Pisa, CL. Sci., 14, 403–421 (1987)
Dinca, G., Jebelean, P., Mawhin, J.: A result of Ambrosetti–Rabinowitz type for p–Laplacian, Qualitative problems for differential equations and control theory, World Sci. Punlishing, Rever Edge, 231–242, 1995
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Dong, W., Chen, J.T. Existence and Multiplicity Results for a Degenerate Elliptic Equation. Acta Math Sinica 22, 665–670 (2006). https://doi.org/10.1007/s10114-005-0696-0
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DOI: https://doi.org/10.1007/s10114-005-0696-0