Abstract
Abstract In this paper, we are concerned with the qualitative analysis of solutions to a general class of nonlinear Schrödinger equations with lack of compactness. The problem is driven by a nonhomogeneous differential operator with unbalanced growth, which was introduced by Azzollini [1]. The reaction is the sum of a nonautonomous power-type nonlinearity with subcritical growth and an indefinite potential. Our main result establishes the existence of at least one nontrivial solution in the case of low perturbations. The proof combines variational methods, analytic tools, and energy estimates.
Highlights
In a recent paper, Azzollini [1] introduced a new class of quasilinear operators with a variational structure
In this paper, we are concerned with the qualitative analysis of solutions to a general class of nonlinear Schrödinger equations with lack of compactness
The problem is driven by a nonhomogeneous differential operator with unbalanced growth, which was introduced by Azzollini [1]
Summary
Azzollini [1] introduced a new class of quasilinear operators with a variational structure He considered nonhomogeneous di erential operators of the type u → div[φ (|∇u(x)| )∇u(x)], where x ∈ RN and φ ∈ C (R+, R+) is a po√tential with unbalanced growth near zero and at in nity. Such a behaviour occurs if φ(t) = ( + t− ), which corresponds to the prescribed mean curvature operator (capillary surface operator), which is de ned by div. Φ(t) behaves like tq/ for small t and tp/ for large t, where potential φ of this type is given by φ(t) = p [( + tq/ )p/q − ]
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