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**1 - 6**of**6**### 4 NEW EXISTENCE RESULTS FOR THE MEAN FIELD EQUATION ON COMPACT SURFACES VIA DEGREE THEORY

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### ASYMMETRIC BLOW-UP FOR THE SU(3) TODA SYSTEM

"... ABSTRACT. We consider the so-called Toda system in a smooth planar domain under ho-mogeneous Dirichlet boundary conditions. We prove the existence of a continuum of so-lutions for which both components blow-up at the same point. This blow-up behavior is asymmetric, and moreover one component include ..."

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ABSTRACT. We consider the so-called Toda system in a smooth planar domain under ho-mogeneous Dirichlet boundary conditions. We prove the existence of a continuum of so-lutions for which both components blow-up at the same point. This blow-up behavior is asymmetric, and moreover one component includes also a certain global mass. The proof uses singular perturbation methods.

### MULTIPLICITY RESULTS FOR THE MEAN FIELD EQUATION ON COMPACT SURFACES

"... Abstract. We are concerned with the following class of equations with expo-nential nonlinearities on a compact surface Σ: −∆u = ρ1 h eu´ Σ h e u dVg ..."

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Abstract. We are concerned with the following class of equations with expo-nential nonlinearities on a compact surface Σ: −∆u = ρ1 h eu´ Σ h e u dVg