For semilinear Gellerstedt equations with Tricomi, Goursat, or Dirichlet boundary conditions, we prove Pohožaev-type identities and derive nonexistence results that exploit an invariance of the linear part with respect to certain nonhomogeneous dilations. A critical-exponent phenomenon of power type in the nonlinearity is exhibited in these mixed elliptic-hyperbolic or degenerate settings where...

Moduli spaces M of self-dual SU(2) connections (“instantons”) over a compact Riemannian 4.manifold (hl, g) carry a natural L2 metric g, which is generally incomplete. For instantons of Pontryagin index 1 over a compact, simply connected, oriented, positive-definite base manifold, the completion M is Donaldson’s compactification; in fact the boundary of the completion is an isometric copy of (M,...

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W k p (R)(1 6 p 6 ∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W k p (R) will be investigated. When the dilation matrix is isotropic, a char...

Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except a) are measurable in both time and one spatial variable, and VMO in the other spatial variables. The coefficient a is measurable in time and VMO in the spatial variables. The unique solvability of equations in the whole space is applied to solving Dirichlet and oblique derivative pr...

Superconvergence phenomena are demonstrated for Galerkin approximations of solutions of second order parabolic and hyperbolic problems in a single space variable. An asymptotic expansion of the Galerkin solution is used to derive these results and, in addition, to show optimal order error estimates in Sobolev spaces of negative index in multiple dimensions.

We deal with Orlicz-Sobolev embeddings in open subsets of R. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given Orlicz space. Moreover, the optimal Orlicz-Sobolev space is exhibited whenever it exists. Parallel questions are addressed for Orlicz-Sobolev embeddings into Orlicz spaces ...

In this work, by using variational methods we study the existence of nontrivial positive solutions for a class p-Kirchhoff type problems with critical Sobolev exponent.