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Uniform estimates and blow-up behavior for solutions of $$-\Delta{} u =V(x) e^ u$$ in two dimensions. (English) Zbl 0746.35006
The problem under investigation is (*) $$-\Delta u=fe^ u$$ in $$\Omega$$, $$u\mid_{\partial\Omega}=0$$ in a bounded domain $$\Omega$$
in $$R^ 2$$, where $$f\in L^ p(\Omega)$$ for some $$p$$ in $$1<p\leq\infty$$. If $$u$$ is a solution of (*) with $$e^ u\in L^{p'}(\Omega)$$, where $$p'$$ denotes the conjugate exponent of $$p$$, one result states that $$u\in L^ \infty(\Omega)$$. A detailed investigation is given of the delicate question of uniform estimates for a sequence $$\{u_ n\}$$ satisfying $$- \Delta u_ n=f_ n\exp u_ n$$ in $$\Omega$$, $$u_ n\mid_{\partial\Omega}=0$$.
Theorem. If $$f_ n\geq 0$$ in $$\Omega$$ and $$\| f_ n\|_ p$$, $$\|\exp u_ n\|_{p'}$$ are uniformly bounded, where $$\|\cdot\|_ p$$ denotes the norm in $$L^ p(\Omega)$$, then $$\{u_ n\}$$ is bounded in $$L^ \infty_{loc}(\Omega)$$. Also conditions are found for which $$\{u_ n\}$$ is bounded in $$L^ \infty(\Omega)$$, and examples are constructed with $$\| u_ n\|_ \infty\to\infty$$ as $$n\to\infty$$, i.e., the uniform estimate does not hold up to the boundary of $$\Omega$$ in general.

##### MSC:
 35B45 A priori estimates in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations
##### Keywords:
exponential nonlinearity
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##### References:
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