Concentration Compactness at the Mountain Pass Level in Semilinear Elliptic Problems

Concentration-compactness at the mountain pass level in semilinear elliptic problems

The concentration compactness framework for semilinear elliptic equations without compactness, set originally by P.-L.Lions for constrained minimization in the case of homogeneous nonlinearity, is extended here to the case of general nonlinearities in the standard mountain pass setting of Ambrosetti–Rabinowitz. In these setting, existence of solutions at the mountain pass level c is verified un...

Multiple Solutions For Semilinear Elliptic Boundary Value Problems At Resonance

In recent years several nonlinear techniques have been very successful in proving the existence of weak solutions for semilinear elliptic boundary value problems at resonance. One technique involves a variational approach where solutions are characterized as saddle points for a related functional. This argument requires that the Palais-Smale condition and some coercivity conditions are satisfie...

Compact Embeddings and Indefinite Semilinear Elliptic Problems

Our purpose is to find positive solutions u ∈ D(R ) of the semilinear elliptic problem −∆u = h(x)u for 2 < p. The function h may have an indefinite sign. Key ingredients are a h-dependent concentration-compactness Lemma and a characterization of compact embeddings of D(R ) into weighted Lebesgue spaces.

Optimal Control of Semilinear Elliptic Obstacle Problems

Max-min Characterization of the Mountain Pass Energy Level for a Class of Variational Problems

We provide a max-min characterization of the mountain pass energy level for a family of variational problems. As a consequence we deduce the mountain pass structure of solutions to suitable PDEs, whose existence follows from classical minimization argument. In the literature the existence of solutions for variational PDE is often reduced to the existence of critical points of functionals F havi...