Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators

We are concerned with the existence of infinitely many radial symmetric solutions for a nonlinear stationary problem driven by a new class of nonhomogeneous differential operators. Our proof relies on the symmetric version of the mountain pass theorem.


Introduction
Given an even functional on an infinite-dimensional Banach space that fulfills natural assumptions, the symmetric mountain pass lemma of P. Rabinowitz [18] establishes the existence of an unbounded sequence of critical values.This result extends to a symmetric framework the initial version of the mountain pass theorem due to A. Ambrosetti and P. Rabinowitz [1].At the same time, the symmetric mountain pass theorem can be viewed as an extension of the Ljusternik-Schnirelmann theorem in the framework of unbounded functionals defined on Banach spaces.As pointed out by H. Brezis and F. Browder [6], the mountain pass theorem "extends ideas already present at Poincaré and Birkhoff".We refer to Y. Jabri [12] and P. Pucci and V. Rȃdulescu [17] for more details on the mountain pass theorem and related applications.
We recall the original statement of the symmetric mountain pass theorem.
Theorem 1.1.Let X be a real infinite-dimensional Banach space and J ∈ C 1 (X, R) a functional satisfying the Palais-Smale condition and the following hypotheses: (i) J (0) = 0 and there are constants ρ, α > 0 such that J |∂Bρ ≥ α; (ii) J is even; and (iii) for all finite-dimensional subspaces X 0 ⊂ X, there exists R = R(X 0 ) > 0 such that J (u) ≤ 0 for all u ∈ X 0 \ B R (X 0 ).Then J has an unbounded sequence of critical values.
Key words and phrases.Anisotropic elliptic problem, nonhomogeneous differential operator, variable exponent, symmetric mountain pass theorem.This result is an efficient tool for proving multiplicity properties in semilinear or quasilinear elliptic problems with odd nonlinearities and Dirichlet boundary condition.The standard application of Theorem 1.1 concerns the following boundary value problem (see Y. Jabri [12, pp. 122-124]) where Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary and f : R → R is a Carathéodory function with the following properties: (iii) there are constants µ > 2 and r > 0 such that for almost every x ∈ Ω and all |u| ≥ r Under these hypotheses, Theorem 1.1 yields the existence of an unbounded sequence of solutions of problem (1).
The present paper was inspired by recent advances in the study of nonlinear stationary problems driven by nonhomogeneous differential operators.Important pioneering contributions to this field are due to T.C. Halsey [11] and V.V. Zhikov [20] who studied the behaviour of non-Newtonian electrorheological fluids and anisotropic materials.These models strongly rely on partial differential equations with variable exponent, which have been intensively studied in the last few decades.We refer to the recent monograph by V. Rȃdulescu and D. Repovš [19] for a comprehensive qualitative analysis of nonlinear PDEs with variable exponent by means of variational and topological methods.These problems (with possible lack of uniform convexity) are essentially described by the differential operator ∆ p(x) u := div (|∇| p(x)−2 ∇u), which changes its growth properties according to the point.More precisely, the variable exponent p(x) describes the geometry of a material that is allowed to change its hardening exponent according to the point.Recently, I.H.Kim and Y.H. Kim [13] introduced a new class of nonhomogeneous differential operators, which extend the standard operators with variable exponent.We refer to S. Baraket, S. Chebbi, N. Chorfi, and V. Rȃdulescu [4] and N. Chorfi and V. Rȃdulescu [8] for contributions in this new abstract setting.
In order to introduce the problem studied in this paper and our main result, we need to recall some basic notions and properties.We refer to V. Rȃdulescu and D. Repovš [19], resp.I.H.Kim and Y.H. Kim [13] for more details.

Lebesgue and Sobolev spaces with variable exponent
For all p ∈ C + (R N ), let L p(x) (R N ) be the Lebesgue space with variable exponent defined by and endowed with the norm ) the following Hölder-type inequality holds: (2) Next, we define the corresponding Sobolev function space with variable exponent by This space is endowed with the norm The critical Sobolev exponent of p ∈ C + (R N ) is defined by The function spaces with variable exponent have some striking properties, namely: has no variable exponent analogue.
(ii) Variable exponent Lebesgue spaces do not have the mean continuity property.More precisely, if p is continuous and nonconstant in an open ball B, then there exists a function u ∈ L p(x) (B) such that u(x + h) ∈ L p(x) (B) for all h ∈ R N with arbitrary small norm.
(iii) The function spaces with variable exponent are never translation invariant.The use of convolution is also limited, for instance the Young inequality We refer to [19] for additional properties.

The main result
Throughout this paper we shall assume that p ∈ C + (R N ) and p is a radial function, that is, We are concerned with the study of the following nonlinear problem where the potential V : R N → [0, +∞) and the nonlinearity f satisfy the following hypotheses: (5) f is odd and lim there exist µ > bp + c and M > 0 such that 0 < µF (u) ≤ uf (u) for all |u| ≥ M , where F (u) := u 0 f (t)dt.In this paper, due to the symmetry assumptions imposed to p and V , we are looking for radial solutions of problem (4).
We say that u ∈ W Let E 0 denote E restricted to the function space W 1,p(x) rad (R N ).By the isometric Palais principle [16] (see also [14,Theorem 1.50]), any critical point of E 0 is also a critical point of E. This shows that finding radially symmetric solutions of problem (4) reduces to finding the nontrivial critical points of the energy functional E 0 .Theorem 4.1.Assume that hypotheses (H1)-(H3), (5), (6), and (7) are fulfilled.Then problem (4) has infinitely many solutions.
As we shall see in the proof of this result, problem (4) still has at least one (radially symmetric) solution, provided that the oddness symmetry hypothesis on f is removed.
Theorem 4.1 extends the pioneering multiplicity result of A. Ambrosetti and P. Rabinowitz [1,Theorem 3.13] in the following directions: (i) the standard (linear) second order uniformly elliptic operator Σ N i,j=1 (a ij (x) u xi ) xj is replaced by the nonhomogeneous differential operator div (φ(x, |∇u|)∇u); (ii) our study is performed in the entire Euclidean space, instead of a bounded domain with smooth boundary.However, in our abstract setting, the lack of compactness of R N is compensated by the compactness of the embedding of the space W

Proof of Theorem 4.1
We first check that E 0 satisfies the geometric hypotheses of the mountain pass theorem.
We observe that hypothesis (7) implies that there are positive constants C 1 and It follows that there exists C 0 > 0 such that ( 11) (R N ) with w p(x) = 1 and λ ∈ R (with |λ| > 1).In order to find an upper estimate for E 0 (λw), we first observe that using hypothesis (H2), we have (12) where C > 0 is a constant depending only on |a| p ′ (x) and the best constant of the continuous embedding W . By (11) we have (13 Estimates ( 12), (13), and ( 14) imply This proves our claim (10).Next, since the space W This fact implies that our initial claim (10) can be improved as follows: for all w ∈ W 1,p(x) rad (R N ) with w p(x) = 1 there exist λ(w) > 0 and η(w) > 0 such that (15), we deduce that there exists λ 0 > 0 depending only on X 0 such that E 0 (λw) ≤ 0 for all |λ| ≥ λ 0 and for all w ∈ X 0 , w p(x) = 1.
Choosing R(X 0 ) = λ 0 , we obtain the statement contained in our Step 2.
Step 3. Any Palais-Smale sequence of E 0 is bounded.
We recall that (u We also recall that E 0 satisfies the Palais-Smale condition if any Palais-Smale sequence (u n ) ⊂ W 1,p(x) rad (R N ) of E 0 is relatively compact.Arguing by contradiction and normalizing, we assume that (up to a subsequence) u n = λ n v n , where λ n = u n p(x) → ∞ and v n p(x) = 1.By (16) we deduce that ( 17) Using hypothesis (H2), relation (17) yields It follows that (18) [λn|vn|≥M] .
By hypothesis (7), we deduce that (19) [λn|vn|≥M] Using hypothesis (H3), we have (20) Combining relations (19) and (20) in relationship with (16), we deduce We conclude that any Palais-Smale sequence of E 0 is relatively compact in W 1,p(x) rad (R N ), hence E 0 satisfies the Palais-Smale condition.Next, we observe that E 0 is even and E 0 (0) = 0.The above steps show that E 0 fulfills the hypotheses of Theorem 1.1.We deduce that problem (4) has infinitely many solutions in W 1,p(x) rad (R N ).
This operator was introduced by A. Azzollini, P. d'Avenia, and A. Pomponio [3] and it is described by a potential with different growth near zero and at infinity (the double-power growth hypotheses).We refer to A. Azzollini [2] and N. Chorfi and V. Rȃdulescu [7] for recent contributions in connection with the abstract setting generated by this operator.
In a general framework, the presence of two variable exponents p 1 (x) and p 2 (x) dictates the geometry of a composite that changes its hardening exponent according to the point.Problems with nonstandard growth conditions of (p, q)-type have been initially studied by P. Marcellini [15] who was interested in the properties of the integral energy functional R N F (x, ∇u)dx, where F : R N × R N → R satisfies unbalanced polynomial growth conditions, namely |ξ| p F (x, ξ) |ξ| q with 1 < p < q.
We believe that the main result of this paper can be extended to "unbalanced" anisotropic differential operators of the type This abstract setting is in close relationship with the recent contributions of G. Mingione et al. [5,9], who studied non-autonomous problems with associated energies of the type