On a semilinear Schrodinger equation with critical Sobolev exponent

We consider the semilinear Schrodinger equation -Deltau+V(x)u= K(x) \u \ (2*-2 u) + g(x; u), u is an element of W-1,W-2 (R-N), where N greater than or equal to4, V, K, g are periodic in x(j) for 1 less than or equal toj less than or equal toN, K>0, g is of subcritical growth and 0 is in a gap of the spectrum of -Delta +V. We show that under suitable hypotheses this equation has a solution u not equal 0. In particular, such a solution exists if K equivalent to 1 and g equivalent to 0.

On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary

in this paper we investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev exponents on the right-hand side of the equation and in the boundary condition. It is assumed that the coefficients Q and P are smooth. We examine the common effect of the mean curvature of the boundary a deltaOhm and the shape of the graph of the coefficients Q and P on the existence of solutions of problem (1.1). (C) 2003 Published by Elsevier Inc.

On the Schrodinger equation involving a critical Sobolev exponent and magnetic field

We consider the semilinear Schrodinger equation -Delta(A)u + V(x)u = Q(x)vertical bar u vertical bar(2* -2) u. Assuming that V changes sign, we establish the existence of a solution u not equal 0 in the Sobolev space H-A,V(1) + (R-N). The solution is obtained by a min-max type argument based on a topological linking. We also establish certain regularity properties of solutions for a rather general class of equations involving the operator -Delta(A).

On multiple solutions of the exterior neumann problem involving critical sobolev exponent

In this paper we consider the exterior Neumann problem involving a critical Sobolev exponent. We establish the existence of two solutions having a prescribed limit at infinity.

Sharp Sobolev inequality involving a critical nonlinearity on a boundary

We consider the solvability of the Neumann problem for the equation -Delta u + lambda u = 0, partial derivative u/partial derivative v = Q(x)vertical bar u vertical bar(q-2)u on partial derivative Omega, where Q is a positive and continuous coefficient on partial derivative Omega, lambda is a parameter and q = 2(N - 1)/(N - 2) is a critical Sobolev exponent for the trace embedding of H-1(Omega) into L-q(partial derivative Omega). We investigate the joint effect of the mean curvature of partial derivative Omega and the shape of the graph of Q on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter lambda interferes with the spectrum of -Delta with the Neumann boundary conditions. We apply a min-max principle based on the topological linking.

On the nonlinear Neumann problem with critical and supercritical nonlinearities

We investigate the solvability of the Neumann problem (1.1) involving a critical Sobolev exponent. In the first part of this work it is assumed that the coeffcients Q and h are at least continuous. Moreover Q is positive on overline Omega and lambda > 0 is a parameter. We examine the common effect of the mean curvature and the shape of the graphs of the coeffcients Q and h on the existence of low energy solutions. In the second part of this work we consider the same problem with Q replaced by - Q. In this case the problem can be supercritical and the existence results depend on integrability conditions on Q and h.

On positive solutions of nonlinear elliptic equations involving concave and critical nonlinearities

We study the existence of nonnegative solutions of elliptic equations involving concave and critical Sobolev nonlinearities. Applying various variational principles we obtain the existence of at least two nonnegative solutions.

Quasilinear dirichlet problems in RN with critical growth

In this study, a given quasilinear problem is solved using variational methods. In particular, the existence of nontrivial solutions for GP is examined using minimax methods. The main theorem on the existence of a nontrivial solution for GP is detailed.