We study the existence of solutions of the following nonlinear Schrödinger equation $$ -Δu+V(x)u-\frac{(N-2)^2}{4|x|^2}u=f(x,u) $$ where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ are periodic with respect to $x\in\mathbb{R}^N.$ We assume that $V$ has positive essential infimum, $f$ satisfies weak growth conditions and $N\geq 3$. The approach to the problem uses variational methods with nonstandard functional setting. We obtain the existence of the ground state solution using the new profile decomposition.
In this paper we are interested in finding so called standing waves for nonlinear Schrödinger equation with inverse-square potential with critical coefficient:
where i 2 = -1 and N ≥ 3.
Looking for solution of the form Ψ (t, x) = e -iλt u(x) leads us to the following equation:
(1.2)
where V : R N → R and f : R N × R are periodic with respect to x ∈ R N and N ≥ 3. Moreover we assume that f has superlinear and subcritical growth. The Schrödinger equation plays an important role in many physical models. For example, in nonlinear optics it describes the propagation of light through optical structures with periodic structure (such as photonic crystals). In the presence of defects in the material, the inverse-square potential (Hardy-type)
µ |x| 2 appears in the equation. Such singular potentials also arise in many other areas as quantum mechanics, nuclear and molecular physics and even quantum cosmology. Reader who would like to know more about physical motivations of Hardy-type potentials may check [8,10,12] and references therein.
In the mathematical perspective the inverse-square potentials were widely investigated by many authors. In most of the works the Hardy type potential is controlled by a constant µ < (N -2) . For instance such subcritical problem was investigated by Guo and Mederski in [13] in a strongly indefinite case. Other exemplary contributions with subcritical nonlinearity are due to Li, Li and Tang in [15] with Berestycki-Lions type conditions, as well as Zhang and Zhang in [26] for the system of Schrödinger equations. For critical nonlinearity, i.e., nonlinearity with growth with power equal to the critical Sobolev exponent 2 * := 2N N -2 there are results by Deng, Jin, Peng [6], Felli and Pistoia [9], Smets in [22] where he obtained nonexistence of solutions, and Terracini in [24]. Critical case when -as in (1.2) -the critical constant µ = (N -2) 2 4 introduces many additional difficulties. The natural norm for our problem is not equivalent to the classical one in H 1 (R N ), which motivates the definition of the space X 1 (R N ) to be the completion of H 1 (R N ) under the natural norm to our problem. A crucial difficulty is that X 1 (R N ) is not invariant under the translations, which makes compactness arguments significantly more delicate. The above mentioned nonstandard functional space embeds continuously into H s (R N ) for every s ∈ (0, 1). This embedding is due to the inequality proven by Frank in [11], whence the space in the context of partial differential equations with Hardy-type potentials was firstly defined independently by Suzuki in [21] and by Trachanas and Zographopulos in [25]. Later, Mukherjee, Nam, and Nguyen also defined and used this space in [17]. Most recently, the Berestycki-Lions scalar-field equation in this setting was studied by Bieganowski and Strzelecki [3].
In this paper the nonlinearity satisfies superlinear and subcritical growth conditions, see (F1)-(F3) below. The idea for generating Cerami sequences is based on the use of [1,Theorem 5.1], in which the Nehari manifold appears and the assumption (F4) is classical for such methods.
In this setting, neither the Palais-Smale nor Cerami compactness condition is satisfied, so compactness must be recovered by diffrent means. The classical method for obtaining strong convergence of the Cerami and Palais-Smale sequences is the so-called profile decomposition, which was introduced by Gérard [14] and Nawa [18]. However, these theorems cannot be applied in our problem due to the fact that singular potential is not translation-invariant. Therefore, inspired by [3], we establish a new profile decomposition adapted to the nontranslation-invariant structure of X 1 (R N ). This part is particularly technically complicated since we show that any possible weak limit of a translated Cerami sequence must be a critical point of the limiting functional (without the singular potential) defined only on H 1 (R N ). By comparing energies as in [13], we exclude such nontrivial limits, which ultimately forces strong convergence.
In what follows through the paper we assume the following conditions (the notation A ≲ B should be understood as A ≤ C × B, for some suitable constant C which is independent of A and B):
|u| is nondecreasing on (-∞, 0) and on (0, ∞). Our main result is the following theorem.
Theorem 1.1. Suppose that (V), (F1)-(F4) hold. Then, there exists a ground state solution to (1.2).
The paper is organized as follows. In Section 2 we introduce the functional framework associated with (1.2) and recall several preliminary results in the space X 1 (R N ). Section 3 is devoted to the variational construction of Cerami sequences. In Section 4 we establish the new profile decomposition adapted to the structure of X 1 (R N ). Finally, in Section 5 we complete the proof of Theorem 1.1 by showing the strong convergence of the Cerami sequence and verifying that the limit is a ground state.
In what follows, | • | k
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