We consider resonance problems at an arbitrary eigenvalue of the Laplacien. We prove the existence of nontrivial solutions for some semilinear elliptic Dirichlet boundary values problems. We also obtain two nontrivial solutions by using Morse theory.

We study a semilinear elliptic equation with an asymptotic linear nonlinearity. Exact multiplicity of solutions are obtained under various conditions on the nonlinearity and the spectrum set. Our method combines a bifurcation approach and Leray–Schauder degree theory.

A semilinear elliptic equation with generalized cubic nonlinearity is studied. Global bifurcation diagrams and the existence of multiple solutions are obtained and in certain cases, exact multiplicity is proved.

We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These are solutions to an inhomogeneous semilinear elliptic PDE that give rise diffuse objects (varifolds) limit hypersurfaces, possibly singularities, whose is determined by "prescribed curvature" function and limiting multiplicity. establish upper bounds for eigenvalues problem, as well more sub...

Understanding the entire solutions of nonlinear elliptic equations in RN such as (0.1) Δu+ f(u) = 0 in R , is a basic problem in PDE research. This is the context of various classical results in literature like the Gidas-Ni-Nirenberg theorems on radial symmetry, Liouville type theorems, or the achievements around De Giorgi’s conjecture. In those results, the geometry of level sets of the soluti...