Abstract
Using the nonsmooth critical point theory we investigate the existence and multiplicity of solutions for a differential inclusion problem with singular coefficients involving the p(x)Laplacian.
Mathematics Subject Classification 2000: 35D05; 35J20; 35J60; 35J70.
Keywords:
p(x)Laplacian; differential inclusion; singularity1 Introduction
In this article, we study the existence and multiplicity of solutions for the differential inclusion problem with singular coefficients involving the p(x)Laplacian of the form
where the following conditions are satisfied:
(P) Ω is a bounded open domain in ℝ^{N}, N ≥ 2,
(A) For i = 1, 2,
here
A typical example of (1.1) is the following problem involving subcritical SobolevHardy exponents of the form
and in this case the assumption corresponding to (A) is the following
The operator div(∇u^{p(x)2 }∇u) is said to be the p(x)Laplacian, and becomes pLaplacian when p(x) ≡ p (a constant). The p(x)Laplacian possesses more complicated nonlinearities than the pLaplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has been received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electrorheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field [1,2]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermorheological viscous flows of nonNewtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [3,4]. Another field of application of equations with variable exponent growth conditions is image processing [5]. The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [611] for an overview of and references on this subject, and to [1221] for the study of the p(x)Laplacian equations and the corresponding variational problems.
Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, the existence of multiple solutions for Dirichlet boundary value problems with discontinuous nonlinearities has been widely investigated in recent years. Chang [22] extended the variational methods to a class of nondifferentiable functionals, and directly applied the variational methods for nondifferentiable functionals to prove some existence theorems for PDE with discontinuous nonlinearities. Later Kourogenis and Papageorgiou [23] obtained some nonsmooth critical point theories and applied these to nonlinear elliptic equations at resonance, involving the pLaplacian with discontinuous nonlinearities. In the celebrated work [24,25], Ricceri elaborated a Ricceritype variational principle and a three critical points theorem for the Gâteaux differentiable functional, respectively. Later, Marano and Motreanu [26,27] extended Ricceri's results to a large class of nondifferentiable functionals and gave some applications to differential inclusion problems involving the pLaplacian with discontinuous nonlinearities.
In [21], by means of the critical point theory, Fan obtain the existence and multiplicity
of solutions for (1.1) under the condition of
This article is organized as follows: In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the generalized gradient of the locally Lipschitz function; In Section 3, we give the variational principle which is needed in the sequel; In Section 4, using the critical point theory, we prove the existence and multiplicity results for problem (1.1).
2 Preliminaries
2.1 Variable exponent Sobolev spaces
Let Ω be a bounded open subset of ℝ^{N}, denote
For
On the basic properties of the space W^{1,p(x)}(Ω) we refer to [7,2830]. Here we display some facts which will be used later.
Denote by S(Ω) the set of all measurable real functions defined on Ω. Two functions in S(Ω) are considered as the same element of S(Ω) when they are equal almost everywhere. For
with the norm
and
with the norm
Denote by
Proposition 2.1. [7,31]The spaces L^{p(x) }(Ω) , W^{1,p(x) }(Ω) and
Proposition 2.2. [7,31]The conjugate space of L^{p(x) }(Ω) is
Proposition 2.3. [7,31]In
So
Proposition 2.4. [7,28,29,31]Assume that the boundary of Ω possesses the cone property and
Let us now consider the weighted variable exponent Lebesgue space.
Let a ∈ S(Ω) and a(x) > 0 for x ∈ Ω. Define
with the norm
then
Proposition 2.5. (see [7,31]) Set ρ(u) = ∫_{Ω }a(x)u(x)^{p(x) }dx. For
(1)
(2)
(3)
(4)
(5)
(6)
Proposition 2.6. (see [21]) Assume that the boundary of Ω possesses the cone property and
then there is a compact embedding
The following proposition plays an important role in the present article.
Proposition 2.7. Assume that the boundary of Ω possesses the cone property and
then there is a compact embedding
Proof. Set
Using Proposition 2.6, we see that the embedding
■
2.2 Generalized gradient of the locally Lipschitz function
Let (X,  · ) be a real Banach space and X* be its topological dual. A function f : X → ℝ is called locally Lipschitz if each point u ∈ X possesses a neighborhood Ω_{u }such that f(u_{1})  f(u_{2}) ≤ Lu_{1 } u_{2} for all u_{1}, u_{2 }∈ Ω_{u}, for a constant L > 0 depending on Ω_{u}. The generalized directional derivative of f at the point u ∈ X in the direction v ∈ X is
The generalized gradient of f at u ∈ X is defined by
which is a nonempty, convex and w*compact subset of X, where 〈·,·〉 is the duality pairing between X* and X. We say that u ∈ X is a critical point of f if 0 ∈ ∂f(u). For further details, we refer the reader to Chang [22].
We list some fundamental properties of the generalized directional derivative and gradient that will be used throughout the article.
Proposition 2.8. (see [22,32]) (1) Let j : X → ℝ be a continuously differentiable function. Then ∂j(u) = {j'(u)}, j^{0}(u; z) coincides with 〈j' (u), z〉_{X }and (f + j)^{0}(u, z) = f^{0}(u; z) + 〈j' (u), z〉_{X }for all u, z ∈ X.
(2) The setvalued mapping u → ∂f(u) is upper semicontinuous in the sense that for each u_{0 }∈ X, ε > 0, v ∈ X, there is a δ > 0, such that for each w ∈ ∂f (u) with w  u_{0} < δ, there is w_{0 }∈ ∂f (u_{0})
(3) (Lebourg's mean value theorem) Let u and v be two points in X. Then there exists a
point w in the open segment joining u and v and
(4) The function
exists, and is lower semi continuous; i.e.,
In the following we need the nonsmooth version of PalaisSmale condition.
Definition 2.1. We say that φ satisfies the (PS)_{c}condition if any sequence {u_{n}} ⊂ X such that φ(u_{n}) → c and m(u_{n}) → 0, as n → +∞, has a strongly convergent subsequence, where m(u_{n}) = inf{u*_{X* }: u* ∈ ∂φ (u_{n})}.
In what follows we write the (PS)_{c}condition as simply the PScondition if it holds for every level c ∈ ℝ for the PalaisSmale condition at level c.
3 Variational principle
In this section we assume that Ω and p(x) satisfy the assumption (P). For simplicity we write
Set
where a_{i }and G_{i }(i = 1, 2) are as in (A).
Define the integral functional
We write
then it is easy to see that J ∈ C^{1}(X, ℝ) and φ = J  Ψ.
Below we give several propositions that will be used later.
Proposition 3.1. (see [19]) The functional J : X → ℝ is convex. The mapping J' : X → X* is a strictly monotone, bounded homeomorphism, and is of (S_{+}) type, namely
Proposition 3.2. Ψ is weaklystrongly continuous, i.e., u_{n }⇀ u implies Ψ(u_{n}) → Ψ(u).
Proof. Define ϒ_{1 }= ∫_{Ω }G_{1}(x, u) dx and ϒ_{2 }= ∫_{Ω }G_{2}(x, u) dx. In order to prove Ψ is weaklystrongly continuous, we only need to prove ϒ_{1 }and ϒ_{2 }are weaklystrongly continuous. Since the proofs of ϒ_{1 }and ϒ_{2 }are identical, we will just prove ϒ_{1}.
We assume u_{n }⇀ u in X. Then by Proposition 2.8.3, we have
where ξ_{n }∈ ∂G_{1}(,τ_{n}(x)) for some τ_{n}(x) in the open segment joining u and u_{n}. From Chang [22] we know that
■
Proposition 3.3. Assume (A) holds and F satisfies the following condition:
(B)
Then φ satisfies the nonsmooth (PS) condition on X.
Proof. Let {u_{n}} be a nonsmooth (PS) sequence, then by (B) we have
and consequently {u_{n}} is bounded.
Thus by passing to a subsequence if necessary, we may assume that u_{n }⇀ u in X as n → ∞. We have
with ε_{n }↓ 0, where ξ_{in}(x) ∈ ∂G_{i}(x, u_{n}) for a.e. x ∈ Ω, i = 1, 2. From Chang [22] or Theorem 1.3.10 of [33], we know that
Therefore we obtain
Remark 3.1. Note that our condition (1.2) is stronger than (1.2) of [21]. Because Ψ' is weaklystrongly continuous in [21], to verify that φ satisfies (PS) condition on X, it is enough to verify that any (PS) sequence is bounded. However, in this paper we do not know whether ξ(u) is weaklystrongly continuous, where ξ(u) ∈ ⇀Ψ. Therefore, it will be very useful to consider this problem.
Below we denote
We shall use the following conditions.
(B_{1}) ∃ c_{0 }> 0 such that G_{2}(x, t) ≥  c_{0 }for x ∈ Ω and t ∈ ℝ.
(B_{2})
Corollary 3.1. Assume (P), (A) and (A_{1}) hold. Then φ satisfies nonsmooth (PS) condition on X provided either one of the following conditions is satisfied.
(1). λ ∈ ℝ and μ = 0.
(2). λ ∈ ℝ, μ = 0 and (B_{1}) holds.
(3). λ ∈ ℝ, μ ∈ ℝ and (B_{2}) holds.
Proof. In case (1) or (2), we have, for x ∈ Ω and t ∈ ℝ,
which shows that the condition (B) with θ = 0 is satisfied.
In case (3), noting that (B_{2}) and (A) imply (B_{1}), by the conclusion (1) and (2) we know φ satisfies (PS) condition if μ ≤ 0. Below assume μ > 0. The conditions (B_{2}) and (A) imply that, for x ∈ Ω and u ∈ X,
so we have
which shows (B) holds. The proof is complete. ■
As X is a separable and reflexive Banach space, there exist (see [[34], Section 17])
For k = 1, 2, . . . , denote
Proposition 3.5. [35]Assume that Ψ : X → ℝ is weaklystrongly continuous and Ψ (0) = 0. Let γ > 0 be given. Set
Then β_{k }→ 0 as k → ∞.
Proposition 3.6. (Nonsmooth Mountain pass theorem, see [23,33]) If X is a reflexive Banach space, φ : X → ℝ is a locally Lipschitz function which satisfies the nonsmooth (PS)_{c}condition, and for some r > 0 and e_{1 }∈ X with e_{1} > r, max{φ(0), φ(e_{1})} ≤·inf{φ(u) : u = r}. Then φ has a nontrivial critical u ∈ X such that the critical value c = φ(u) is characterized by the following minimax principle
where Γ = {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = e_{1}}.
Proposition 3.7. (Nonsmooth Fountain theorem, see [36]) Assume (F_{1}) X is a Banach space, φ : X → ℝ be an invariant locally Lipschitz functional, the subspaces X_{k}, Y_{k }and Z_{k }are defined by (3.3).
If, for every k ∈ ℕ, there exist ρ_{k }> r_{k }> 0 such that
(F_{2})
(F_{3})
(F_{4}) φ satisfies the nonsmooth (PS)_{c }condition for every c > 0, then φ has an unbounded sequence of critical values.
Proposition 3.8. (Nonsmooth dual Fountain theorem, see [37]) Assume (F_{1}) is satisfied and there is a k_{0 }> 0 such that, for each k ≥ k_{0}, there exists ρ_{k }> γ_{k }> 0 such that
(D_{1})
(D_{2})
(D_{3})
(D_{4}) φ satisfies the nonsmooth
Remark 3.2. We say φ that satisfies the nonsmooth
contains a subsequence converging to a critical point of φ.
4 Existence and multiplicity of solutions
In this section, using the critical point theory, we give the existence and multiplicity results for problem (1.1). We shall use the following assumptions:
(S) For i = 1, 2, G_{i}(x, t) = G_{i}(x, t), ∀x ∈ Ω, ∀t ∈ ℝ.
Remark 4.1.
(1) It follows from (A), (A_{2}) and (O_{2}) that
(2)It follows from (A) and (B_{2}) that (see [33, p. 298])
The following is the main result of this article.
Theorem 4.1. Assume (P), (A), (A_{1}) hold.
(1) If (B_{1}) holds, then for every λ ∈ ℝ and μ ≤ 0, problem (1.1) has a solution which is a minimizer of the corresponding functional φ.
(2) If (B_{1}), (A_{2}), (O_{1}), (O_{2}) hold, then for every λ > 0 and μ ≤ 0, problem (1.1) has a nontrivial solution v_{1 }such that v_{1 }is a minimizer of φ and φ(v_{1}) < 0.
(3) If (A_{2}), (B_{2}), (O_{2}) hold, then for every μ > 0, there exists λ_{0}(μ) > 0 such that when λ ≤ λ_{0}(μ), problem (1.1) has a nontrivial solution u_{1 }such that φ(u_{1}) > 0.
(4) If (A_{2}), (B_{2}), (O_{1}), (O_{2}) holds, then for every μ > 0, there exists λ_{0}(μ) > 0 such that when 0 < λ ≤ λ_{0}(μ), problem (1.1) has two nontrivial solutions u_{1 }and v_{1 }such that φ(u_{1}) > 0 and φ(v_{1}) < 0.
(5) If (A_{2}), (B_{2}), (O_{1}), (O_{2}) and (S) holds, then for every μ > 0 and λ ∈ ℝ, problem (1.1) has a sequence of solutions {±u_{k}} such that φ(±u_{k}) → ∞ as k → ∞.
(6) If (A_{2}), (B_{2}), (O_{1}), (O_{2}) and (S) holds, then for every λ > 0 and μ ∈ ℝ, problem (1.1) has a sequence of solutions {±v_{k}} such that φ(±v_{k}) < 0 and φ(±v_{k}) → 0 as k → ∞.
Proof. We will use c, c' and c_{i }as a generic positive constant. By Corollary 3.1, under the assumptions of Theorem 4.1, φ satisfies nonsmooth (PS) condition. We write
then Ψ = Ψ_{1 }+ Ψ_{2}, φ(u) = J(u)  Ψ (u) = J(u)  Ψ_{1}(u)  Ψ_{2}(u). Firstly, we use
(1) Let λ ∈ ℝ and μ ≤ 0. By (A),
By (B_{1}), Ψ_{2}(u) ≤  μc_{0 }∫_{Ω }a_{2}(x) dx = c_{5}. Hence
(2) Let λ > 0, μ ≤ 0 and the assumptions of (2) hold. By the above conclusion (1), φ has a minimize v_{1}. Take
Since
(3) Let μ > 0 and the assumptions of (3) hold. By Remark 4.1.(1), for sufficiently small u
Since
(4) Let μ > 0 and the assumptions of (4) hold. By the conclusion (3), we know that, there exists λ_{0}(μ) > 0 such that when 0 < λ ≤ λ_{0}(μ), problem (1.1) has a nontrivial solution u_{1 }such that φ(u_{1}) > 0. Let γ and α be as in the proof of (3), that is, φ(u) ≥ α/2 > 0 for u ∈ S_{γ}. By (O_{1}), (O_{2}) and the proof of (2), there exists w ∈ X such that w < γ and φ(w) < 0. It is clear that there is v_{1 }∈ B_{γ}, a minimizer of φ on B_{γ}. Thus v_{1 }is a nontrivial solution of (1.1) and φ(v_{1}) < 0.
(5) Let μ > 0, λ ∈ ℝ and the assumptions of (5) hold. By (S), we can use the nonsmooth version Fountain theorem with the antipodal action of ℤ_{2 }to prove (5) (see Proposition 3.7). Denote
Let β_{k}(γ) be as in Proposition 3.5. By Proposition 3.5, for each positive integer n, there exists a positive integer k_{0}(n) such that β_{k}(n) ≤ 1 for all k ≥ k_{0}(n). We may assume k_{0}(n) < k_{0}(n + 1) for each n. We define {γ_{k }: k = 1, 2, . . . , } by
Note that γ_{k }→ ∞ as k → ∞. Then for u ∈ Z_{k }with u = γ_{k }we have
and consequently
i.e., the condition (F_{2}) of Proposition 3.7 is satisfied.
By (A), (A_{1}), (B_{2}) and Remark 4.1.(2), we have
Noting that
(6) Let λ > 0, μ ∈ ℝ and the assumptions of (5) hold. Let us verify the conditions of the Nonsmooth dual Fountain theorem (see Proposition 3.8). By (S), φ is invariant on the antipodal action of ℤ_{2}. For Ψ(u) = ∫_{Ω }F(x, u)dx = Ψ_{1}(u)+ Ψ_{2}(u) let β_{k}(1) be as in Proposition 3.5, that is
By Proposition 3.5, there exists a positive integer k_{0 }such that
which shows that the condition (D_{1}) of Proposition 3.8 is satisfied.
Since
Because
thus the condition (D_{2}) of Proposition 3.8 is satisfied.
Because Y_{k }∩ Z_{k }≠ ∅ and γ_{k }< ρ_{k}, we have
On the other hand, for any u ∈ Z_{k }with u ≤ 1 = ρ_{k}, we have φ(u) = J(u)  Ψ(u) ≥ Ψ(u) ≥ β_{k}(1). Noting that β_{k }→ 0 as k → ∞, we obtain d_{k }→ 0, i.e., (D_{3}) of Proposition 3.8 is satisfied.
Finally let us prove that φ satisfies nonsmooth
Using Proposition 2.8.4, Going to limit in the right side of above equation, we have
so m(u) ≡ 0, i.e., 0 ∈ ∂φ(u), this shows that φ satisfies the nonsmooth
Remark 4.2
Theorem 4.1 includes several important special cases. In particular, in the case of the problem (1.4), i.e., in the case that
all conditions of Theorem 4.1 are satisfied provided (P), (A*), (A_{1}), and (A_{2}) hold.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
GD conceived of the study, and participated in its design and coordination and helped to draft the manuscript. RM participated in the design of the study. All authors read and approved the final manuscript.
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. Research supported by the NSFC (Nos. 11061030, 10971087), 1107RJZA223 and the Fundamental Research Funds for the Gansu Universities.
References

Rüžzička, M: Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin (2000)

Zhikov, VV: Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv. 9, 33–66 (1987)

Antontsev, SN, Shmarev, SI: A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal. 60, 515–545 (2005)

Antontsev, SN, Rodrigues, JF: On stationary thermorheological viscous flows. Ann Univ Ferrara Sez Sci Mat. 52, 19–36 (2006). Publisher Full Text

Chen, Y, Levine, S, Rao, M: Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math. 66(4), 1383–1406 (2006). Publisher Full Text

Diening, L, Hästö, P, Nekvinda, A: Open problems in variable exponent Lebesgue and Sobolev spaces. Drábek, P, Rákosník, J, FSDONA04 Proceedings, 3858, Milovy, Czech Republic (2004). PubMed Abstract  Publisher Full Text

Fan, XL, Zhao, D: On the Spaces L^{p(x) }and W^{m,p(x)}. J Math Anal Appl. 263, 424–446 (2001). Publisher Full Text

Harjulehto, P, Hästö, P: An overview of variable exponent Lebesgue and Sobolev spaces. In: Herron D (ed.) Future Trends in Geometric Function Theory, pp. 85–93. RNC Workshop, Jyväskylä (2003)

Samko, S: On a progress in the theory of Lebesgue spaces with variable exponent maximal and singular operators. Integr Trans Spec Funct. 16, 461–482 (2005). Publisher Full Text

Jikov, VV, Kozlov, SM, Oleinik, OA: Homogenization of Differential Operators and Integral Functionals (Translated from the Russian by Yosifian, GA). Springer, Berlin (1994)

Zhikov, VV: On some variational problems. Russ J Math Phys. 5, 105–116 (1997)

Dai, G: Three symmetric solutions for a differential inclusion system involving the (p(x), q(x))Laplacian in ℝ^{N}. Nonlinear Anal. 71, 1763–1771 (2009). Publisher Full Text

Dai, G: Infinitely many solutions for a Neumanntype differential inclusion problem involving the p(x)Laplacian. Nonlinear Anal. 70, 2297–2305 (2009). Publisher Full Text

Dai, G: Infinitely many solutions for a hemivariational inequality involving the p(x)Laplacian. Nonlinear Anal. 71, 186–195 (2009). Publisher Full Text

Dai, G: Three solutions for a Neumanntype differential inclusion problem involving the p(x)Laplacian. Nonlinear Anal. 70, 3755–3760 (2009). Publisher Full Text

Dai, G: Infinitely many solutions for a differential inclusion problem in ℝ^{N }involving the p(x)Laplacian. Nonlinear Anal. 71, 1116–1123 (2009). Publisher Full Text

Fan, XL, Han, XY: Existence and multiplicity of solutions for p(x)Laplacian equations in R^{N}. Nonlinear Anal. 59, 173–188 (2004)

Fan, XL: On the subsupersolution methods for p(x)Laplacian equations. J Math Anal Appl. 330, 665–682 (2007). Publisher Full Text

Fan, XL, Zhang, QH: Existence of solutions for p(x)Laplacian Dirichlet problems. Nonlinear Anal. 52, 1843–1852 (2003). Publisher Full Text

Fan, XL, Zhang, QH, Zhao, D: Eigenvalues of p(x)Laplacian Dirichlet problem. J Math Anal Appl. 302, 306–317 (2005). Publisher Full Text

Fan, XL: Solutions for p(x)Laplacian Dirichlet problems with singular coefficients. J Math Anal Appl. 312, 464–477 (2005). Publisher Full Text

Chang, KC: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J Math Anal Appl. 80, 102–129 (1981). Publisher Full Text

Kourogenis, NC, Papageorgiou, NS: Nonsmooth crical point theory and nonlinear elliptic equation at .resonance. KODAI Math J. 23, 108–135 (2000). Publisher Full Text

Ricceri, B: A general variational principle and some of its applications. J Comput Appl Math. 113, 401–410 (2000). Publisher Full Text

Ricceri, B: On a three critical points theorem. Arch Math (Basel). 75, 220–226 (2000). Publisher Full Text

Marano, S, Motreanu, D: Infinitely many critical points of nondifferentiable functions and applications to a Neumanntype problem involving the pLaplacian. J Diff Equ. 182, 108–120 (2002). Publisher Full Text

Marano, SA, Motreanu, D: On a three critical points theorem for non differentiable functions and applications to nonlinear boundary value problems. Nonlinear Anal. 48, 37–52 (2002). Publisher Full Text

Diening, L: Riesz potential and Sobolev embeddings on generalized Lebesque and Sobolev Spaces L^{p}^{(·) }and W^{k,p}(·) Math. Nachr. 268, 31–43 (2004). Publisher Full Text

Fan, XL, Shen, JS, Zhao, D: Sobolev embedding theorems for spaces W^{k,p(x) }(Ω). J Math Anal Appl. 262, 749–760 (2001). Publisher Full Text

Samko, SG: HardyLittlewoodSteinWeiss inequality in the Lebesgue spaces with variable exponent. Fract Calc Appl Anal. 6(4), 421–440 (2003)

Kovacik, O, Rakosnik, J: On spaces L^{p(x) }(Ω) and W^{k,p(x)}(Ω). Czechoslovak Math J. 41, 592–618 (1991)

Clarke, FH: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

Gasiéski, L, Papageorgiou, NS: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman and Hall/CRC, Boca Raton (2005)

Zhao, JF: Structure Theory of Banach Spaces (in Chinese). Wuhan University Press, Wuhan (1991)

Garcia Azorero, JP, Peral Alonso, I: Hardy inequalities and some critical elliptic and parabolic problems. J Diff Equ. 144, 441–476 (1998). Publisher Full Text

Dai, G: Nonsmooth version of Fountain theorem and its application to a Dirichlettype differential inclusion problem. Nonlinear Anal. 72, 1454–1461 (2010). Publisher Full Text

Dai, G, Wang, WT, Feng, LL: Nonsmooth version of dual Fountain theorem and its application to a differential inclusion problem. In: Acta Math Sci Ser A Chin Ed 32(2012). 1, 18–28

Kristály, A: Infinitely many solutions for a differential inclusion problem in ℝ^{N}. J Diff Equ. 220, 511–530 (2006). Publisher Full Text