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Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth. (English) Zbl 1205.35118
The authors consider the following elliptic boundary value problem \[ \begin{cases} -\Delta u= f(x,u)+\mu g(x, u)\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(f,g: \overline\Omega\times\mathbb{R}\to \mathbb{R}\), \(f(x,0)= g(x,0)= 0\), and \(\mu\) is a real parameter. They prove existence and multiplicity solutions for (1). To this end the authors use variational approach.

35J65 Nonlinear boundary value problems for linear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49K27 Optimality conditions for problems in abstract spaces
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