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.


35B45 A priori estimates in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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