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Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent. (English) Zbl 1254.35083

Summary: We consider the boundary value problem \(\Delta u+u^{p}=0\) in a bounded, smooth domain \(\Omega\) in \(\mathbb R^2\) with homogeneous Dirichlet boundary condition and \(p\) a large exponent. We find topological conditions on \(\Omega\) which ensure the existence of a positive solution \(u_p\) concentrating at exactly \(m\) points as \(p\to\infty\). In particular, for a nonsimply connected domain such a solution exists for any given \(m\geq 1\).

MSC:

35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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References:

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