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The spectrum in \(\mathbb R\) and \(\mathbb R^2\) of nonlinear elliptic equations with positive parameters. (English) Zbl 1336.35263

Let \(\Omega\) be an open, convex and bounded subset of \({\mathbb R}^m\), \(m\geq 2\), with smooth boundary \(\partial\Omega\). The authors study the nonlinear parametric problem
\[ \begin{cases} Lu(x)+\lambda(x,u(x))+\mu g(x,u(x)=0, & x\in\Omega, \\ u(x)=0 , & x\in\partial\Omega, \end{cases} \]
where \(L\) is a formally self-adjoint elliptic partial differential operator of second order and \(\lambda, \mu>0\). The spectrum of this problem is the set of \((\lambda, \mu)\) such that there exists at least one solution of this equation for this pair of positive numbers.
From the authors’s summary: “In this paper we study the spectrum of nonlinear elliptic equations with positive parameters in their nonlinear part. In order to investigate the spectrum in these specific cases, we introduce the monotone method which is an extension of the upper and lower solution methods. Using the Picard iterative process we prove some existence theorems for nonlinear elliptic boundary value problems. We work with both positive and negative solutions.”

MSC:

35P05 General topics in linear spectral theory for PDEs
35J60 Nonlinear elliptic equations
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References:

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