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Existence, multiplicity and stability results for positive solutions of nonlinear \(p\)-Laplacian equations. (English) Zbl 1106.35019

In the present study the authors are mainly interested in the positive solutions of the following boundary value problem of nonlinear \(p\)-Laplacian equations \[ -\Delta_pu=\lambda f(u),\text{ in }\Omega\quad u=0,\text{ on }\partial\Omega,\tag{1} \] where \(\lambda>0\), \(p>1\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) and \(f\) is a given nonnegative, nondecreasing function. The author studies both the existence of solution to (1) and extends part of the Crandall-Rabinowitz bifurcation theory to this problem. Moreover, the stability of solutions of the parabolic counterparts of (1) is investigated.

MSC:

35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
92D25 Population dynamics (general)

Keywords:

bifurcation
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References:

[10] doi:10.1007/978-1-4612-0895-2 · doi:10.1007/978-1-4612-0895-2
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