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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.

MSC:
 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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