Minimum action solutions of nonhomogeneous Schrödinger equations

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 inde nite 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.


Introduction
In a recent paper, 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 ∈ R N and φ ∈ C (R+, R+) is a potential with unbalanced growth near zero and at in nity. For instance, 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 ∇u + |∇u| . (1.1) More generally, φ(t) behaves like t q/ for small t and t p/ for large t, where < p < q < N. An example of potential φ of this type is given by This potential generates the di erential operator div ( + |∇u| q ) (p−q)/q |∇u| q− ∇u .
Another important example includes the (p, q)-Laplace operator u → ∆p u + ∆q u, which is generated by φ(t) = t p/ + t q/ for all t ≥ . Due to the unbalanced growth of φ, the associated functional is a double-phase energy. The study of nonautonomous variational integrals with double growth has been initiated by Marcellini [15][16][17]. We also point out the pioneering work by Zhikov [27], in relationship with phenomena arising in nonlinear elasticity and strongly anisotropic materials in the context of homogenisation. These functionals revealed to be important also in the study of duality theory and in the context of the Lavrentiev phenomenon [28]. We recall that Zhikov considered the variational integral where the modulating potential a(·) dictates the geometry of the composite made by two di erential materials, with hardening exponents p and q, respectively. In this paper we study a nonlinear Schrödinger equation driven by the operator de ned in (1.1) and with lack of compactness. A feature of this paper is the presence of a power-type subcritical reaction and an indefinite potential. Related problems have been studied by Azzollini, d'Avenia and Pomponio [2] and Chor and Rădulescu [12].
The study developed in the present paper is motivated by the central role played by the Schrödinger equation in quantum theory, Newton conservation laws in classical mechanics, Bose-Einstein condensates and nonlinear optics, stability of Stokes waves in water, propagation of the electric eld in optical bers, self-focusing and collapse of Langmuir waves in plasma physics, deep water waves and freak waves (or rogue waves) in the ocean, etc.

The main result
In this paper, we are concerned with the study of the following quasilinear Schrödinger equation with lack of compactness − div[φ (|∇u| )∇u] + φ (u )u = a(x) |u| r− u + λf (x) in R N , (2.2) where N ≥ and λ is a real parameter.
We describe in what follows the main hypotheses on the above considered problem. Let p, q and r be real numbers such that As in Azzollini, d'Avenia and Pomponio [2], we assume that the potential φ ∈ C (R+, R+) satis es the following hypotheses: (φ ) there exists µ ∈ ( , ) such that tφ (t) ≤ rµφ(t) for all t ≥ ; (φ ) the mapping t → φ(t ) is strictly convex.
These hypotheses imply that Since our hypotheses allow that φ approaches 0, problem (2.2) is both degenerate and non-uniformly elliptic.
We assume that the weight a : We describe in what follows the abstract setting corresponding to problem (2.2). Let · r denote the Lebesgue norm for all ≤ r ≤ ∞ and C ∞ c (R N ) the space of all C ∞ functions with a compact support.

De nition 1. We de ne the function space
In the same way we can de ne the space L p (Ω) + L q (Ω) for an arbitrary open set Ω ⊂ R N . However, the space On L p (R N ) + L q (R N ) we can also consider the equivalent norm · * given by According to [3,Lemma 2.9], for all f ∈ L p (R N ) ∩ L q (R N ) and u ∈ L p (R N ) + L q (R N ), the following Höldertype inequality holds: We de ne the function space X as the completion of C ∞ c (R N ) in the norm According to the terminology introduced by Fučik, Nečas, Souček and Souček [13, p. 117] we can say that u is an eigenfunction of the nonlinear problem (2.2) corresponding to the eigenvalue λ.
The main result of this paper establishes the following existence property in the case of 'low perturbations'. We point out that the study of these problems was initiated by Azzollini, d'Avenia and Pomponio [2]. They studied the problem The potential φ satis es the same hypotheses as above, while < p < q < N, < α ≤ p * q /p , and max{q, α} < s < p * . The lack of compactness due to the unboundedness of the Euclidean space is handled in [2] by restricting the study to the case of radially symmetric weak solutions. In the framework developed in [2], a central role is played by the compact embedding of a related function space with radial symmetry into a related Lebesgue space. This abstract setting does not hold in this work. Due to the lack of symmetry of the problem, we develop a general approach that cannot be reduced to the radial case as in [2].
Finally, we point out that with similar arguments we can treat the more general problem where h : R N × R → R is a Carathéodory function satisfying the following conditions:

Auxiliary results
The energy functional associated to problem (2.2) is J : X → R de ned by By standard arguments we obtain that J is well-de ned and is of class C ; we refer to Azzollini [2, Theorem 2.5] for more details. Moreover, for all u, v ∈ X its Gâteaux directional derivative is given by An important role in our arguments will be played by the following version of the mountain pass theorem, which is due to Brezis and Nirenberg [9]. Theorem 2. Let X be a real Banach space and assume that J : X → R is a C -functional that satis es the following geometric hypotheses:

Moreover, if J satis es the Palais-Smale condition at the level c, then c is a critical value of J.
We prove in what follows that the mountain pass geometry described in the above hypotheses (i) and (ii) are ful lled.

Lemma 1.
For any R > large enough, there exists ζ ∈ X such that J(ζ ) < and ζ ≥ R.
Proof. Fix ψ ∈ X \ { }. We nd ζ of the form ζ = tψ for t > large enough. By our hypotheses, we have where c and c are positive numbers.
By assumption (2.3) we conclude that J(tψ) < for t large enough.
The following result establishes a low perturbation property, which remains valid only for small perturbations from the origin of the parameter λ. Roughly speaking, this result shows that the mountain pass geometry does not change in the case of 'low' perturbations.

Lemma 2.
There exists positive numbers a and ρ such that J(u) ≥ a for all u ∈ X with u p,q = ρ.
Proof. By hypothesis (φ ) we have for all u ∈ X Thus, there exists c > such that for all u ∈ X (3.7) Next, using assumption (2.3), we nd a > and ρ > such that J(u) ≥ a for all u ∈ X with u p,q = ρ. We x some small ρ > with this property. Taking λ ∈ R such that |λ| ≤ a /( c ρ) and using (3.7), we conclude that J(u) ≥ a := a / > for all u ∈ X with u p,q = ρ.
The proof of Lemma 2 is now complete.
Proof. We have (3.10) By (2.4) we obtain for all n and some positive constant c (3.11) where µ ∈ ( , ). Returning to (3.11) we deduce that for all n Combining relations (3.12) and (3.13) we conclude that the sequence (un) ⊂ X is bounded.

Proof of the main result
In this section we give the proof of Theorem 1. By Lemma 3 and [3, Corollary 2.11] we can assume that, up to a subsequence, un U in X. (4.14)

Lemma 4. The function U given in (4.14) is a solution of problem (2.2).
Proof. Let η ∈ C ∞ c (R N ) and denote ω := supp (η). By Lemma 3 and [3, Proposition 2.17(ii)] we can assume, going eventually to a subsequence, that un → U in L r (ω). (4.15) De ne the functionals J , J : X → R by Thus, by relation (3.9), we have By density arguments (using the de nition of X), we deduce that relation (4.19) holds for all η ∈ X, that is, U is a solution of (2.2). The proof of Lemma 4 is now complete.
We conclude the proof of Theorem 1 by establishing that U ≠ . By relation (3.9) we deduce that there is a positive integer N such that for all n ≥ N we have (4.20) Since the mapping t → φ(t ) is convex (by hypothesis (φ )) we have and φ(u n ) − φ (u n )u n ≤ . (4.22) Let c be the real number de ned in (3.8 J(γ(t)) < ε.
Let a be the real positive number given by Lemma 2 and take < ε < a. Fix γ ∈ F joining the origin and ζ given by Lemma 1 with R big enough, hence γ ( ) = and γ ( ) = ζ . It follows that γ ( ) = and γ ( ) > R. By continuity, there exists t ∈ ( , ) such that γ (t ) = R, hence This contradiction shows that our claim (4.23) is true.
We rst assume that r > . Returning to (4.20) and using (4.22) and (4.23), it follows that for all n big enough we have Arguing by contradiction, we assume that U = . We rst assume that r ≤ . Then by the Hölder-type inequality (2.4) and (4.24) we obtain < c ≤ |λ| f L p ∩L q un * .
If either λ = or (up to a subsequence) un * → as n → ∞, then we have a contradiction. If not, we can take |λ| small enough (as requested in Theorem 1) in order to nd a contradiction. So, U ≠ . We now consider the case r > . Using again relation (4.24) we obtain for all n su ciently large < c ≤ c R N a(x)|un| r dx + c |λ| un * , (4.25) where c and c are positive constants. We argue again by contradiction and assume that U = . Then with an argument as above we obtain that for all n large enough we have < c ≤ c R N a(x)|un| r dx.
Using now (4.15) and the assumption U = we obtain a contradiction. We can give a direct argument in order to show that U ≠ . Indeed, by (4.25), it follows that (un) does not converge strongly to 0 in L r (R N ). From now on, with the same argument as in Gazzola and Rădulescu [14, pp. 55-56], we obtain that U ≠ .
We conclude that U is a nontrivial solution of problem (2.2) and the proof is complete.
Comments. (i) Related arguments can be applied in order to show that the same low-perturbation result remains true if hypotheses (a) is replaced by conditions (a ) a ∈ L ∞ loc (R N \ { }) and essinf x∈R N a(x) > ; (a ) lim x→ a(x) = lim |x|→∞ a(x) = +∞.