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Uniqueness results for nonlinear elliptic equations with a lower order term. (English) Zbl 1125.35343

Summary: We consider the functional \[ I(\lambda,u)= \tfrac12 \int_\Omega|\nabla u|^2-\lambda\log \biggl( \frac{1}{|\Omega|} \int_\Omega e^{it}\biggr), \quad u\in H_0^1(\Omega). \] When \(\lambda=8\pi N\) (\(N\) any positive integer), \(l(\lambda,\cdot)\) admits unbounded Palais-Smale sequences. To overcome this lack of compactness, we propose a new deformation lemma.

MSC:

35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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