Equations non lineaires avec le p-Laplacienet la fonction exponentielle sur les varietes riemanniennes compactes

AbstractWe solve on a compact Riemannian manifold (Mn,g) the equation Δnψ+a=f(x)eψ when a and f satisfy some conditions which are in some cases necessary and sufficient. On the sphere Sn we obtain similar results for the equation Δkψ+a=f(x)eψ when f is G-invariant where G=O(m)×O(k) with m+k=n+1 and k≥m≥2. Finally we show that the equation Δpu+a(x)up−1=f(x)up∗−1 with p∗=pk/(k−p) has a positive C1,α solution when a and f are G-invariant, if the inf of the functional of the problem satisfies an inequality. Here the exponent p∗ is supercritical: p∗>pn/(n−p).RésuméWe solve on a compact Riemannian manifold (Mn,g) the equation Δnψ+a=f(x)eψ when a and f satisfy some conditions which are in some cases necessary and sufficient. On the sphere Sn we obtain similar results for the equation Δkψ+a=f(x)eψ when f is G-invariant where G=O(m)×O(k) with m+k=n+1 and k≥m≥2. Finally we show that the equation Δpu+a(x)up−1=f(x)up∗−1 with p∗=pk/(k−p) has a positive C1,α solution when a and f are G-invariant, if the inf of the functional of the problem satisfies an inequality. Here the exponent p∗ is supercritical: p∗>pn/(n−p)