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Combining the sharp isoperimetric inequality established by Z. Balogh and A.Krist\'aly [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with ananisotropic symmetrization argument, we establish sharp Morrey-Sobolevinequalities on $n$-dimensional Finsler manifolds having nonnegative $n$-Riccicurvature. A byproduct of this method is a Hardy-Sobolev-type inequality in thesame geometric setting. As applications, by using variational arguments, weguarantee the existence/multiplicity of solutions for certain eigenvalueproblems and elliptic PDEs involving the Finsler-Laplace operator. Our resultsare also new in the Riemannian setting.

Sharp Morrey-Sobolev inequalities and eigenvalue problems on Riemannian-Finsler manifolds with nonnegative Ricci curvature