Existence of positive solutions for some problems with nonlinear diffusion.(English)Zbl 0884.35039

Summary: We study the existence of positive solutions for problems of the type $-\Delta_pu(x) =u(x)^{q-1} h\bigl(x,u(x) \bigr),\;x\in\Omega, \quad u(x)=0,\;x\in \partial \Omega,$ where $$\Delta_p$$ is the $$p$$-Laplace operator and $$p,q>1$$. If $$p=2$$, such problems arise in population dynamics. Making use of different methods (sub- and super-solutions and a variational approach), we treat the cases $$p=q$$, $$p<q$$ and $$p>q$$, respectively. Also, some systems of equations are considered.

MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 47J25 Iterative procedures involving nonlinear operators
Full Text:

References:

 [1] Aomar Anane, Simplicité et isolation de la première valeur propre du \?-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 16, 725 – 728 (French, with English summary). · Zbl 0633.35061 [2] Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349 – 381. · Zbl 0273.49063 [3] Haïm Brezis and Louis Nirenberg, \?\textonesuperior versus \?\textonesuperior local minimizers, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 5, 465 – 472 (English, with English and French summaries). · Zbl 0803.35029 [4] A. Cañada and J.L. Gámez, Some new applications of the method of lower and upper solutions to elliptic problems, Appl. Math. Lett. 6 (1993), 41-45 CMP 95:17 · Zbl 0791.35049 [5] A. Cañada and J. L. Gámez, Elliptic systems with nonlinear diffusion in population dynamics, Differential Equations Dynam. Systems 3 (1995), no. 2, 189 – 204. · Zbl 0870.35037 [6] J. I. Díaz, Nonlinear partial differential equations and free boundaries. Vol. I, Research Notes in Mathematics, vol. 106, Pitman (Advanced Publishing Program), Boston, MA, 1985. Elliptic equations. · Zbl 0595.35100 [7] Jesús Hernández, Qualitative methods for nonlinear diffusion equations, Nonlinear diffusion problems (Montecatini Terme, 1985) Lecture Notes in Math., vol. 1224, Springer, Berlin, 1986, pp. 47 – 118. [8] Anthony Leung and Guangwei Fan, Existence of positive solutions for elliptic systems — degenerate and nondegenerate ecological models, J. Math. Anal. Appl. 151 (1990), no. 2, 512 – 531. · Zbl 0722.35040 [9] Mitsuharu Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal. 76 (1988), no. 1, 140 – 159. · Zbl 0662.35047 [10] Mitsuharu Ôtani and Toshiaki Teshima, On the first eigenvalue of some quasilinear elliptic equations, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), no. 1, 8 – 10. · Zbl 0662.35080 [11] Maria Assunta Pozio and Alberto Tesei, Support properties of solutions for a class of degenerate parabolic problems, Comm. Partial Differential Equations 12 (1987), no. 1, 47 – 75. · Zbl 0629.35071 [12] Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126 – 150. · Zbl 0488.35017 [13] Neil S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721 – 747. · Zbl 0153.42703
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.