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

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

DOI: https://doi.org/10.1090/S0002-9947-97-01947-8

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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.

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