Quantum mechanics

Truncation (statistics)

Singular perturbation

Pure mathematics

Bifurcation

Statistics

Monotonic function

Boundary value problem

Applied mathematics

Lambda

Perturbation (astronomy)

Differential operator

Physics

Nonlinear system

Mathematics

Optics

Mountain pass theorem

Mathematical analysis

Dirichlet problem

Parametric statistics

Term (time)

Calculus of Variations and Partial Differential Equations

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order $$(p-1)$$ near $$+\infty $$ and with a reaction which has the competing effects of a parametric singular term and a $$(p-1)$$-superlinear perturbation which does not satisfy the usual Ambrosetti–Rabinowitz condition. Using variational tools, together with suitable truncation and strong comparison techniques, we prove a “bifurcation-type” theorem that describes the set of positive solutions as the parameter $$\lambda $$ moves on the positive semiaxis. We also show that for every $$\lambda >0$$, the problem has a smallest positive solution $$u^*_\lambda $$ and we demonstrate the monotonicity and continuity properties of the map $$\lambda \mapsto u^*_\lambda $$.

Nonlinear nonhomogeneous singular problems