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On an elliptic equation with exponential growth. (English) Zbl 0887.35055

The authors study existence and multiplicity of solutions of the homogeneous Dirichlet problem for the equation \(-\Delta_p u=\lambda V(x)e^u\) in a bounded domain \(\Omega\subset\mathbb R^N\), where \(-\Delta_p\) is the \(p\)-Laplacian, \(p>1\), \(\lambda>0\), \(V\in L^q(\Omega)\), \(q\geq1\) if \(p>N\), \(q>N/p\) if \(p\leq N\). The existence for \(\lambda\) small is shown using the Banach fixed point theorem. If, in addition, \(V\) changes sign and \(p\geq N\) then the Mountain Pass Theorem is used in order to obtain a second solution. The authors find also sufficient conditions for existence and nonexistence of positive solutions. Moreover, they study the existence of a minimal solution for \(V\geq0\) and \(p<N\) and they analyze radial solutions in the case \(V(x)=|x|^{-\alpha}\), \(\alpha<p\), \(\Omega\) being the unit ball.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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