Abundance of entire solutions to nonlinear elliptic equations by the variational method

Abstract

We study entire bounded solutions to the equation ∆u − u + u 3 = 0 in R 2 . Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in a unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. The method is also applicable for more general equations in any dimension.

Solutions with various structures for semilinear equations in $\mathbb R^n$ driven by fractional Laplacian

“…Other classes of solutions constructed in our paper, apparently, were not studied earlier. In the local case, similar solutions are considered in [12]. However, the solutions constructed in 6.3 are new even for s = 1.…”

“…was investigated by many authors. For the overview of research methods see, e.g., the recent paper [12] and references therein.…”

“…In [12], a variational approach was suggested. It is based on the concentration-compactness principle by P.-L. Lions and on the reflections.…”

“…In this paper, we modify the method from [12] to equations with fractional Laplacian. We stress that a nonlocal generalization of other methods using for constructing solutions of (2) is not an easy task.…”

“…The equation (29) for u in Ω R follows from (12). Now we assume that the polyhedron Ω has the following property: the space R n can be filled with its reflections, colored checkerwise.…”