Variational Approaches to Nonlinear, Nonlocal and Singular Elliptic Problems

Dear Colleagues,

Modern nonlinear analysis and mathematical physics draw extensively on the fruitful interplay between variational methods and nonlinear differential equations. Variational analysis provides a rigorous framework for identifying critical points of energy functionals, while nonlinear elliptic equations arise in the mathematical description of a wide range of phenomena in quantum mechanics, differential geometry, nonlinear optics and field theory. In particular, the study of normalized solutions, for which the -norm is prescribed, has become increasingly important in the analysis of physical systems subject to a fixed-mass constraint, including Bose–Einstein condensates and solitary waves. This Special Issue aims to explore the rich interaction between variational techniques, constrained optimization and the qualitative theory of nonlinear elliptic equations.

Despite the substantial progress achieved in recent years, many fundamental questions concerning the existence, multiplicity, qualitative properties and orbital stability of normalized solutions remain open, especially in the presence of competing nonlinearities, nonlocal operators such as fractional Laplacians or singular perturbations. Further analytical challenges arise in problems involving Schrödinger operators with point interactions, where the singular nature of the defect requires suitable functional settings and refined variational techniques. Related difficulties occur for quasilinear and nonhomogeneous operators, including the Born–Infeld operator, whose nonlinear structure and intrinsic gradient constraints call for analytical tools that go beyond the standard elliptic framework.

These features often push classical variational methods to their limits and make it particularly challenging to describe concentration and multi-peak profiles, deal with loss of compactness, establish regularity and determine the global behavior of solution branches. This Special Issue seeks to bring together recent advances employing innovative variational, topological and analytical methods to address these and other fundamental questions in nonlinear elliptic theory.

Suggested themes include:

• Variational methods and constrained critical point theory;
• Existence, multiplicity and stability of normalized solutions;
• Nonlinear Schrödinger equations and Bose–Einstein condensation;
• Fractional, nonlocal and nonhomogeneous elliptic operators;
• Schrödinger operators and Laplacians with point interactions;
• Born–Infeld and prescribed mean-curvature-type operators;
• Singular perturbations, semiclassical analysis and blow-up phenomena;
• Concentration, multi-peak solutions and loss of compactness;
• Sharp functional and geometric inequalities and concentration–compactness principles;
• Bifurcation theory and topological methods for elliptic equations.

We invite submissions of original research articles and high-quality review papers presenting significant theoretical advances as well as innovative analytical applications in these areas.

Prof. Dr. Alessio Pomponio
Guest Editor

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