Existence of positive solutions for some problems with nonlinear diffusion

Cañada, A.; Drábek, P.; Gámez, J.

doi:10.1090/S0002-9947-97-01947-8

Existence of positive solutions for some problems with nonlinear diffusion
HTML articles powered by AMS MathViewer

by A. Cañada, P. Drábek and J. L. Gámez PDF

Trans. Amer. Math. Soc. 349 (1997), 4231-4249 Request permission

Abstract:

In this paper we study the existence of positive solutions for problems of the type \begin{equation*} \begin {aligned} -\Delta _pu(x) &=u(x)^{q-1}h(x,u(x)), && x\in \Omega , \\ u(x)&=0, && x\in \partial \Omega , \end{aligned} \end{equation*} 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.

References

Aomar Anane, Simplicité et isolation de la première valeur propre du $p$-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 16, 725–728 (French, with English summary). MR 920052
Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7
Haïm Brezis and Louis Nirenberg, $H^1$ versus $C^1$ local minimizers, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 5, 465–472 (English, with English and French summaries). MR 1239032
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
A. Cañada and J. L. Gámez, Elliptic systems with nonlinear diffusion in population dynamics, Differential Equations Dynam. Systems 3 (1995), no. 2, 189–204. MR 1386519
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. MR 853732
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. MR 877987, DOI 10.1007/BFb0072688
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. MR 1071298, DOI 10.1016/0022-247X(90)90163-A
Mitsuharu Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal. 76 (1988), no. 1, 140–159. MR 923049, DOI 10.1016/0022-1236(88)90053-5
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. MR 953752
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. MR 869102, DOI 10.1080/03605308708820484
Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. MR 727034, DOI 10.1016/0022-0396(84)90105-0
Neil S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747. MR 226198, DOI 10.1002/cpa.3160200406