Ελλειπτικές εξισώσεις με υπερκρίσιμο εκθέτη σε συμπαγείς πολλαπλότητες με σύνορο

Abstract

The present Thesis is incorporated in the research area of Nonlinear Analysis, especially solvability of Nonlinear Elliptic PDE’s with supercritical exponent. The nonlinear nature of the equations makes it impossible to be solved by means of compact imbeddings. Taking advantage of the symmetry properties of the manifold we overcome the obstacle as well as we succeed in solving equations of this type possessing supercritical exponent. In the first part of the Thesis we calculate the first best constant in the general Sobolev inequality and in the general Sobolev trace inequality on the solid torus, we study the phenomenon of concentration and solve problems (P1) and (P2). (P1) #pu + a(x)up?1 = f(x)uq?1, u > 0 on T, u|@T = 0, q = 2p 2 ? p > 6, 3 2 < p < 2. and (P2) #pu + a(x)up?1 = #f(x)uq?1, u > 0 on T, |ru|p?2 @u @# + b(x)up?1 = #g(x)u?q?1 on @T, q = 2p 2 ? p > 6, ?q = p 2 ? p > 4, 3 2 < p < 2, # scalar. In the second part we calculate the first best constant in the general Sobolev inequality and in the general Sobolev trace inequality on a smooth, compact, n?dimensional Riemannian manifold (M, g), n # 3, with boundary, which is invariant under the action of a subgroup G of the isometry group Is(M, g) of M, the orbits of which have infinity cardinality. We also present brief solutions of problems (P3) and (P4). (P3) #pu + a(x)up?1 = f(x)uq?1, u > 0 on M, u|@M = 0, 2(n ? k) n < p < n ? k, q = (n ? k)p n ? k ? p and (P4) #pu + a(x)up?1 = f(x)uq?1, u > 0 on M, |ru|p?2 @u @# + b(x)up?1 = h(x)u?q?1 on @M, 2(n ? k) n < p < n ? k, q = (n ? k)p n ? k ? p , ?q = (n ? k ? 1)p n ? k ? p .