Multiple solutions for superlinear double phase Neumann problems

We study a double phase Neumann problem with a superlinear reaction which need not satisfy the Ambrosetti-Rabinowitz condition. Using the Nehari manifold method, we show that the problem has at least three nontrivial bounded ground state solutions, all with sign information (positive, negative and nodal).

A Multiplicity Theorem for Superlinear Double Phase Problems

We consider a nonlinear Dirichlet problem driven by the double phase differential operator and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. Using the Nehari manifold, we show that the problem has at least three nontrivial bounded solutions: nodal, positive and by the symmetry of the behaviour at +∞ and −∞ also negative.

A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

We consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

Constant sign and nodal solutions for superlinear double phase problems

We consider a double phase problems with unbalanced growth and a superlinear reaction, which need not satisfy the Ambrosetti–Rabinowitz condition. Using variational tools and the Nehari method, we show that the Dirichlet problem has at least three nontrivial solutions, a positive solution, a negative solution and a nodal solution. The nodal solution has exactly two nodal domains.

On a logarithmic Hartree equation

We study the existence of radially symmetric solutions for a nonlinear planar Schrödinger-Poisson system in presence of a superlinear reaction term which doesn’t satisfy the Ambrosetti-Rabinowitz condition. The system is re-written as a nonlinear Hartree equation with a logarithmic convolution term, and the existence of a positive and a negative solution is established via critical point theory.

Superlinear Robin problems with indefinite linear part

We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential and a superlinear reaction term which need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools we prove two theorems. An existence theorem producing a nontrivial smooth solution and a multiplicity theorem producing a whole unbounded sequence of nontrivial smooth solutions.