In this paper, we are concerned with the singularly perturbed double phase problem: [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text], [Formula: see text] with [Formula: see text] is the [Formula: see text]-Laplacian operator, [Formula: see text] and [Formula: see text] possess global extremum points, and [Formula: see text] is asymptotically [Formula: see text]-linear at infinity. We establish the existence, nonexistence, multiplicity and concentration of solutions. Due to the appearance of asymptotically [Formula: see text]-linear nonlinearities, the method of Nehari manifold dependent on an auxiliary set is developed. In addition, refined analysis techniques and global compactness arguments are introduced to analyze the asymptotic behavior of solutions. This paper completes the results of Zhang, Zhang and Rădulescu [Math. Z. 301 (2022) 4037–4078], where [Formula: see text] is [Formula: see text]-superlinear at infinity.

In this paper, we address the existence and qualitative features of ground states associated with the [Formula: see text]-Laplacian Schrödinger–Poisson system: [Formula: see text] under the mass constraint [Formula: see text] The nonlinearity [Formula: see text] is assumed to be mass super-critical, while the parameter satisfies [Formula: see text]. In the case [Formula: see text] and [Formula: see text], we establish the existence of ground states as well as infinitely many radially symmetric solutions. The proof is based on constructing an appropriate bounded Palais–Smale sequence combined with genus theory. Moreover, by employing the Pohozaev manifold framework, we derive a nonexistence result in the regime [Formula: see text] and [Formula: see text]. In contrast, for the case [Formula: see text] with [Formula: see text], variational methods enable us to prove the existence of solutions.

This paper investigates the existence and multiplicity of multi-peak positive solutions for a class of [Formula: see text]-Laplacian Schrödinger equations with logarithmic non-linearity in [Formula: see text]: [Formula: see text] Here, [Formula: see text] is a small parameter and the potential [Formula: see text] has multiple wells. The problem is characterized by the combined difficulties of the non-homogeneous [Formula: see text]-Laplacian and a logarithmic term which prevents standard variational approaches. By combining a penalization method with variational techniques, we prove that for any non-empty set of potential wells, there exists a family of positive solutions. These solutions exhibit multiple peaks which concentrate at the minima of the potential as [Formula: see text]. This establishes the existence of at least [Formula: see text] distinct multi-peak solutions, where [Formula: see text] is the number of wells, providing the first systematic result on the construction of such solutions for this class of quasilinear logarithmic problems.

We consider the front propagation problems for the hyperbolic mean curvature motion in central force fields. Problems of this type describe the long time behavior for the electrically charged membrane with a constant charge per area and a radially symmetric potential moving with the mean curvature motion law. In this paper, we establish the existence of generalized periodic traveling waves with a large speed of propagation under appropriate assumptions on the central periodic force fields.

The aim of this paper is to discuss the existence of normalized solutions to the following nonlocal double phase problems driving by the discrete fractional Laplacian: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] if [Formula: see text], [Formula: see text] if [Formula: see text], and [Formula: see text]([Formula: see text] or [Formula: see text], [Formula: see text] or [Formula: see text]) is the discrete fractional [Formula: see text]-Laplacian. By variational methods, we discuss the existence of non-negative normalized homoclinic solutions under the conditions that the nonlinear term satisfies sublinear growth or superlinear growth conditions. In particular, we establish the compactness of the associated energy functional of the problem without weights. Our paper is the first time to deal with the existence of normalized solutions for discrete double phase problems.

We prove the existence of generalized solutions of the Monge–Kantorovich equations with fractional [Formula: see text]-gradient constraint, [Formula: see text], associated to a general, possibly degenerate, linear fractional operator of the type, [Formula: see text] with integrable data, in the space [Formula: see text], which is the completion of the set of smooth functions with compact support in a bounded domain [Formula: see text] for the [Formula: see text]-norm of the distributional Riesz fractional gradient [Formula: see text] in [Formula: see text] (when [Formula: see text], [Formula: see text] is the classical gradient). The transport densities arise as generalized Lagrange multipliers in the dual space of [Formula: see text] and are associated to the variational inequalities of the corresponding transport potentials under the constraint [Formula: see text]. Their existence is shown by approximating the variational inequality through a penalization of the constraint and nonlinear regularization of the linear operator [Formula: see text]. For this purpose, we also develop some relevant properties of the spaces [Formula: see text], including the limit case [Formula: see text] and the continuous embeddings [Formula: see text], for [Formula: see text]. We also show the localization of the nonlocal problems ([Formula: see text]), to the local limit problem with classical gradient constraint when [Formula: see text], for which most results are also new for a general, possibly degenerate, partial differential operator [Formula: see text] with coefficients only integrable and bounded gradient constraint.

Let [Formula: see text] and [Formula: see text] be bounded operators in a Banach space. For the equation [Formula: see text] we derive sharp explicit delay-dependent exponential stability conditions via the commutator [Formula: see text]. The stability conditions in terms of the commutator enable us to construct a feedback with delay, which stabilizes a system. Gives us a possibility construct Stabilization of Systems with Retarded Feedback in a Banach Space. Our results are new even in the finite-dimensional case.

We consider a class of nonhomogeneous elliptic equations with fractional Laplacian and nonlinear gradient terms, namely [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] is the nonlinearity, [Formula: see text] the potential and [Formula: see text] is a forcing term. Some examples of nonlinearities dealt with are [Formula: see text], [Formula: see text] and [Formula: see text], covering large values of [Formula: see text], and particularly variational supercritical powers for [Formula: see text] and super-[Formula: see text] ones for [Formula: see text] (superquadratic if [Formula: see text]). Moreover, we are able to consider some exponential growths, [Formula: see text] belonging to certain classes of power series, or [Formula: see text] satisfying some conditions in the Lipschitz spirit. We obtain results on existence, uniqueness, symmetry, and other qualitative properties in a new framework, namely modulation-type spaces based on Lorentz spaces. For that, we need to develop properties and estimates in those spaces such as complex interpolation, Hölder-type inequality, estimates for product, convolution and Riesz potential operators, among others. In order to handle the nonlinearity, other ingredients are estimates for composition operators in our setting.

In this note, we explore the nature of Lens spaces to study the first width of those spaces, more precisely, we use the existence of a sharp sweep out associated to a Clifford torus to provide a simple and pretty application of the Willmore conjecture for the computation of the [Formula: see text]-width of Lens spaces.

We prove the existence of at least three weak solutions for the fourth-order problem with indefinite weight involving the Leray–Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [Appl. Anal. 89 (2010) 1–10].