Boundary Value Problems This Provisional PDF corresponds to the article as it appeared upon acceptance Fully formatted PDF and full text (HTML) versions will be made available soon Existence of solutions for a differential inclusion problem with singular coefficients involving the $p(x)$-Laplacian Boundary Value Problems 2012, 2012:11 doi:10.1186/1687-2770-2012-11 Guowei Dai (daiguowei@nwnu.edu.cn) Ruyun Ma (mary@nwnu.edu.cn) Qiaozhen Ma (maqzh@nwnu.edu.cn) ISSN Article type 1687-2770 Research Submission date November 2011 Acceptance date February 2012 Publication date February 2012 Article URL http://www.boundaryvalueproblems.com/content/2012/1/11 This peer-reviewed article was published immediately upon acceptance It can be downloaded, printed and distributed freely for any purposes (see copyright notice below) For information about publishing your research in Boundary Value Problems go to http://www.boundaryvalueproblems.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2012 Dai et al ; licensee Springer This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Existence of solutions for a differential inclusion problem with singular coefficients involving the p(x)-Laplacian Guowei Dai∗, Ruyun Ma and Qiaozhen Ma Department of Mathematics, Northwest Normal University, Lanzhou 730070, P.R China ∗ Corresponding author: daiguowei@nwnu.edu.cn Email addresses: RM: mary@nwnu.edu.cn QM: maqzh@nwnu.edu.cn Abstract Using the non-smooth critical point theory we investigate the existence and multiplicity of solutions for a differential inclusion problem with singular coefficients involving the p(x)-Laplacian Keywords: p(x)-Laplacian; differential inclusion; singularity Mathematics Subject Classification 2000: 35D05; 35J20; 35J60; 35J70 1 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    −div(| u|p(x)−2 u) ∈ λa (x)∂G (x, u) + µa (x)∂G (x, u) in Ω,  1 2 (1.1)   u=0  on ∂Ω, where the following conditions are satisfied: (P) Ω is a bounded open domain in RN , N ≥ 2, p ∈ C(Ω), < p− := inf Ω p(x) ≤ p+ := supΩ p(x) < +∞, λ, µ ∈ R (A) For i = 1, 2, ∈ Lri (x) (Ω), (x) > for x ∈ Ω, Gi (x, u) is measurable with respect to x (for every u ∈ R) and locally Lipschitz with respect to u (for a.e x ∈ Ω), ∂Gi : Ω × R → R is the Clarke sub-differential of Gi and |ξi | ≤ c1 + c2 |t|qi (x)−1 for x ∈ Ω, − − t ∈ R and ξi ∈ ∂Gi , where ci is a positive constant, ri , qi ∈ C(Ω), ri > 1, qi > 1, ri (x) > qi (x) for all x ∈ Ω, and qi (x) < ri (x) − qi (x) ∗ p (x), ri (x) here ∗ p (x) =        ∀x ∈ Ω, N p(x) N −p(x) if p(x) < N, ∞ if p(x) ≥ N + (A1 ) q1 < p− − (A2 ) q2 > p+ (1.2) (1.3) A typical example of (1.1) is the following problem involving subcritical Sobolev-Hardy exponents of the form    −div(| u|p(x)−2 u) ∈ λ ∂G (x, u) + µ ∂G (x, u)  |x|s1 (x) |x|s2 (x)   u=0  in Ω, (1.4) on ∂Ω, and in this case the assumption corresponding to (A) is the following (A)∗ ∈ Ω, for i = 1, 2, ∂Gi : Ω × R → R is the Clarke sub-differential of Gi and |ξi | ≤ c1 + c2 |t|qi (x)−1 for x ∈ Ω, t ∈ R and ξi ∈ ∂Gi , where ci is a positive constant, − si , qi ∈ C(Ω), ≤ s− ≤ s+ < N, qi > 1, and i i qi (x) < N − si (x)qi (x) ∗ p (x), N ∀x ∈ Ω (1.5) 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 p-Laplacian; 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 electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electro-magnetic field [1, 2] Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal baro-tropic 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 [6–11] for an overview of and references on this subject, and to [12–21] 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 non-differentiable functionals, and directly applied the variational methods for non-differentiable 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 p-Laplacian with discontinuous nonlinearities In the celebrated work [24, 25], Ricceri elaborated a Ricceri-type variational principle and a three critical points theorem for the Gˆteaux differentiable functional, respectively Later, Marano and Motreanu a [26, 27] extended Ricceri’s results to a large class of non-differentiable functionals and gave some applications to differential inclusion problems involving the p-Laplacian 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 Gi (x, ·) ∈ C (R) and gi = Gi satisfying the Carath´odory condition for i = 1, 2, x ∈ Ω The aim of the present article is to generalize e the main results of [21] to the case of the functional of problem (1.1) is nonsmooth This article is organized as follows: In Section 2, we present some necessary prelimi4 nary 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) Preliminaries 2.1 Variable exponent Sobolev spaces Let Ω be a bounded open subset of RN , denote L∞ (Ω) = {p ∈ L∞ (Ω) : ess inf Ω p(x) ≥ + 1} For p ∈ L∞ (Ω), denote + p− = p− (Ω) = ess inf p(x), p+ = p+ (Ω) = ess sup p(x) x∈Ω x∈Ω On the basic properties of the space W 1,p(x) (Ω) we refer to [7, 28–30] 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 p ∈ L∞ (Ω), define the spaces Lp(x) (Ω) and W 1,p(x) (Ω) by +     Lp(x) (Ω) = u ∈ S(Ω) : |u(x)|p(x) dx < ∞   Ω with the norm |u|Lp(x) (Ω) = |u|p(x)   = inf λ > :  u(x) λ p(x)   dx ≤ ,  Ω and W 1,p(x) (Ω) = u ∈ Lp(x) (Ω) : | u| ∈ Lp(x) (Ω) with the norm u 1,p(x) Denote by W0 W 1,p(x) (Ω) = |u|Lp(x) (Ω) + | u|Lp(x) (Ω) ∞ (Ω) the closure of C0 (Ω) in W 1,p(x) (Ω) Hereafter, we always assume that p− > 1,p(x) Proposition 2.1 [7, 31] The spaces Lp(x) (Ω) , W 1,p(x) (Ω) and W0 (Ω) are sepa- rable and reflexive Banach spaces (x) Proposition 2.2 [7, 31] The conjugate space of Lp(x) (Ω) is Lp p0 (x) (x) = For any u ∈ Lp(x) (Ω) and v ∈ Lp 1,p(x) Proposition 2.3 [7, 31] In W0 (Ω) , Ω (Ω) , where p(x) + |uv| dx ≤ |u|p(x) |v|p0 (x) (Ω) the Poincar´ inequality holds, that is, there e exists a positive constant c such that |u|Lp(x) (Ω) ≤ c | u|Lp(x) (Ω) 1,p(x) So | u|Lp(x) (Ω) is an equivalent norm in W0 1,p(x) , ∀u ∈ W0 (Ω) (Ω) Proposition 2.4 [7, 28, 29, 31] Assume that the boundary of Ω possesses the cone property and p ∈ C(Ω) If q ∈ C(Ω) and ≤ q(x) < p∗ (x) for x ∈ Ω, then there is a compact embedding W 1,p(x) (Ω) → Lq(x) (Ω) Let us now consider the weighted variable exponent Lebesgue space Let a ∈ S(Ω) and a(x) > for x ∈ Ω Define     p(x) La(x) (Ω) = u ∈ S(Ω) : a(x) |u(x)|p(x) dx < ∞   Ω with the norm   |u|Lp(x) (Ω) = |u|(p(x),a(x)) = inf a(x)  a(x) λ>0: u(x) λ p(x)   dx ≤ ,  Ω p(x) then La(x) (Ω) is a Banach space The following proposition follows easily from the definition of |u|Lp(x) (Ω) a(x) Proposition 2.5 (see [7, 31]) Set ρ(u) = p(x) Ω a(x) |u(x)|p(x) dx For u, uk ∈ La(x) (Ω) , we have u (1) For u = 0, |u|(p(x),a(x)) = λ ⇔ ρ( λ ) = (2) |u|(p(x),a(x)) < (= 1; > 1) ⇔ ρ(u) < (= 1; > 1) − + p (3) If |u|(p(x),a(x)) > 1, then |u|p (p(x),a(x)) ≤ ρ (u) ≤ |u|(p(x),a(x)) + − p (4) If |u|(p(x),a(x)) < 1, then |u|p (p(x),a(x) ≤ ρ (u) ≤ |u|(p(x),a(x)) (5) limk→∞ |uk |(p(x),a(x)) = ⇐⇒ limk→∞ ρ(uk ) = (6) |uk |(p(x),a(x)) → ∞ ⇐⇒ ρ(uk ) → ∞ Proposition 2.6 (see [21]) Assume that the boundary of Ω possesses the cone property and p ∈ C(Ω) Suppose that a ∈ Lr(x) (Ω), a(x) > for x ∈ Ω, r ∈ C(Ω) and r− > If q ∈ C(Ω) and ≤ q(x) < r(x) − ∗ p (x) := p∗ (x), a(x) r(x) ∀x ∈ Ω, q(x) then there is a compact embedding W 1,p(x) (Ω) → La(x) (Ω) The following proposition plays an important role in the present article (2.1) Proposition 2.7 Assume that the boundary of Ω possesses the cone property and p ∈ C(Ω) Suppose that a ∈ Lr(x) (Ω), a(x) > for x ∈ Ω, r ∈ C(Ω) and r(x) > q(x) for all x ∈ Ω If q ∈ C(Ω) and ≤ q(x) < r(x) − q(x) ∗ p (x), r(x) ∀x ∈ Ω, (2.2) q(x) then there is a compact embedding W 1,p(x) (Ω) → L(a(x))q(x) (Ω) Proof Set r1 (x) = r(x) , q(x) − then r1 > and (a(x))q(x) ∈ Lr1 (x) (Ω) Moreover, from (2.2) we can get ≤ q(x) < r1 (x) − ∗ p (x), r1 (x) ∀x ∈ Ω q(x) Using Proposition 2.6, we see that the embedding W 1,p(x) (Ω) → L(a(x))q(x) (Ω) is compact 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 Ωu such that |f (u1 ) − f (u2 )| ≤ L u1 − u2 for all u1 , u2 ∈ Ωu , for a constant L > depending on Ωu The generalized directional derivative of f at the point u ∈ X in the direction v ∈ X is f (u, v) = lim sup (f (w + tv) − f (w)) w→u,t→0 t The generalized gradient of f at u ∈ X is defined by ∂f (u) = {u∗ ∈ X ∗ : u∗ , ϕ ≤ f (u; ϕ) for all ϕ ∈ X}, which is a non-empty, 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 ∈ ∂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 (u; z) coincides with j (u), z f (u; z) + j (u), z X X and (f + j)0 (u, z) = for all u, z ∈ X (2) The set-valued mapping u → ∂f (u) is upper semi-continuous in the sense that for each u0 ∈ X, ε > 0, v ∈ X, there is a δ > 0, such that for each w ∈ ∂f (u) with w − u0 < δ, there is w0 ∈ ∂f (u0 ) | w − w0 , v | < ε (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 x∗ ∈ ∂f (w) such that w f (u) − f (v) = x∗ , u − v w X (4) The function m(u) = w∈∂f (u) w X∗ exists, and is lower semi continuous; i.e., lim inf m(u) ≥ m(u0 ) u→u0 In the following we need the nonsmooth version of Palais–Smale condition Definition 2.1 We say that ϕ satisfies the (PS)c -condition if any sequence {un } ⊂ X such that ϕ(un ) → c and m(un ) → 0, as n → +∞, has a strongly convergent subse9 which shows that the condition (B) with θ = is satisfied In case (3), noting that (B2 ) and (A) imply (B1 ), by the conclusion (1) and (2) we know ϕ satisfies (PS) condition if µ ≤ Below assume µ > The conditions (B2 ) and (A) imply that, for x ∈ Ω and u ∈ X, G2 (x, u) ≤ θ u, ξ2 + c3 , and F2 (x, u) ≤ θµa2 (x) u, ξ2 + c3 µa2 (x), so we have F (x, u) − θλa1 (x) ξ1 , u − θµa2 (x) ξ2 , u = (F1 (x, u) − θλa1 (x) ξ1 , u ) + (F2 (x, u) − θµa2 (x) ξ2 , u ) ≤ c1 a1 (x) + c2 a1 (x) |u|q1 (x) + c3 µa2 (x), which shows (B) holds The proof is complete As X is a separable and reflexive Banach space, there exist (see [34, Section 17]) {en }∞ ⊂ X and {fn }∞ ⊂ X ∗ such that n=1 n=1 fn (em ) = δn,m =     if n = m    if n = m, ∗ X = span{en : n = 1, 2, , }, X ∗ = spanW {fn : n = 1, 2, , } For k = 1, 2, , denote Xk = span {ek } , Yk = ⊕k Xj , j=1 Zk = ⊕∞ Xj j=k (3.3) Proposition 3.5 [35] Assume that Ψ : X → R is weakly-strongly continuous and Ψ (0) = Let γ > be given Set βk = βk (γ) = sup u∈Zk , u ≤γ 14 |Ψ (u)| Then βk → 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 > and e1 ∈ X with e1 > r, max{ϕ(0), ϕ(e1 )} ≤ 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 c = inf max ϕ(γ(t) γ∈Γ t∈[0,1] where Γ = {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = e1 } Proposition 3.7 (Nonsmooth Fountain theorem, see [36]) Assume (F1 ) X is a Banach space, ϕ : X → R be an invariant locally Lipschitz functional, the subspaces Xk , Yk and Zk are defined by (3.3) If, for every k ∈ N, there exist ρk > rk > such that (F2 ) ak := inf ϕ(u) → ∞, u∈Zk u =rk k → ∞, (F3 ) bk := max ϕ(u) ≤ 0, u∈Yk u =ρk (F4 ) 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 (F1 ) is satisfied 15 and there is a k0 > such that, for each k ≥ k0 , there exists ρk > γk > such that (D1 ) ak := inf ϕ(u) ≥ 0, u∈Zk u =ρk (D2 ) bk := max ϕ(u) < 0, u∈Yk u =rk (D3 ) dk := inf ϕ(u) → 0, u∈Zk u ≤ρk k → ∞, (D4 ) ϕ satisfies the nonsmooth (PS)∗ condition for every c ∈ [dk0 , 0), c then ϕ has a sequence of negative critical values converging to Remark 3.2 We say ϕ that satisfies the nonsmooth (PS)∗ condition at level c ∈ R c (with respect to (Yn )) if any sequence {un } ⊂ X such that nj → ∞, unj ∈ Ynj , ϕ(unj ) → c, m Ynj (un ) → contains a subsequence converging to a critical point of ϕ 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: + (O1 ) ∃δ1 > 0, c3 > and q3 ∈ C(Ω) with q3 (x) < p∗1 (x) (x) for x ∈ Ω and q3 < p− , such a that G1 (x, t) ≥ c3 tq3 (x) , ∀x ∈ Ω, ∀t ∈ (0, δ1 ] − (O2 ) ∃δ2 > 0, c4 > and q4 ∈ C(Ω) with q4 (x) < p∗2 (x) (x) for x ∈ Ω and q4 > p+ , such a that |G2 (x, t)| ≤ c4 |t|q4 (x) (S) For i = 1, 2, Gi (x, −t) = Gi (x, t), , ∀x ∈ Ω, ∀x ∈ Ω, 16 ∀ |t| ≤ δ2 ∀t ∈ R Remark 4.1 (1) It follows from (A), (A2 ) and (O2 ) that |G2 (x, t)| ≤ c4 |t|q4 (x) + c5 |t|q2 (x) , ∀x ∈ Ω, ∀t ∈ R (2)It follows from (A) and (B2 ) that (see [33, p 298]) G(x, t) ≥ c6 |t|1/θ − c7 , ∀x ∈ Ω, ∀t ∈ R The following is the main result of this article Theorem 4.1 Assume (P), (A), (A1 ) hold (1) If (B1 ) 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 (B1 ), (A2 ), (O1 ), (O2 ) hold, then for every λ > and µ ≤ 0, problem (1.1) has a nontrivial solution v1 such that v1 is a minimizer of ϕ and ϕ(v1 ) < (3) If (A2 ), (B2 ), (O2 ) hold, then for every µ > 0, there exists λ0 (µ) > such that when |λ| ≤ λ0 (µ), problem (1.1) has a nontrivial solution u1 such that ϕ(u1 ) > (4) If (A2 ), (B2 ), (O1 ), (O2 ) holds, then for every µ > 0, there exists λ0 (µ) > such that when < λ ≤ λ0 (µ), problem (1.1) has two nontrivial solutions u1 and v1 such that ϕ(u1 ) > and ϕ(v1 ) < (5) If (A2 ), (B2 ), (O1 ), (O2 ) and (S) holds, then for every µ > and λ ∈ R, problem (1.1) has a sequence of solutions {±uk } such that ϕ(±uk ) → ∞ as k → ∞ (6) If (A2 ), (B2 ), (O1 ), (O2 ) and (S) holds, then for every λ > and µ ∈ R, problem (1.1) has a sequence of solutions {±vk } such that ϕ(±vk ) < and ϕ(±vk ) → as k → ∞ 17 Proof We will use c, c and ci as a generic positive constant By Corollary 3.1, under the assumptions of Theorem 4.1, ϕ satisfies nonsmooth (PS) condition We write Ψ1 (u) = λ a1 (x)G1 (x, u) dx, Ψ2 (u) = µ Ω a2 (x)G2 (x, u) dx, Ω then Ψ = Ψ1 + Ψ2 , ϕ(u) = J(u) − Ψ(u) = J(u) − Ψ1 (u) − Ψ2 (u) Firstly, we use Ψi to denote its extension to Lqi (x) (Ω), where i = 1, From (A) and Theorem 1.3.10 of [33] (or Chang [22]), we see that Ψi (u) is locally Lipschitz on Lqi (x) (Ω) and ∂ Ψi (u) ⊆ {ξi (x) ∈ Lqi (Ω) : ξi (u) ∈ ∂Gi (x, u)} for a.e x ∈ Ω and i = 1, In view of Proposition 2.4 and Theorem 2.2 of [22], we have that Ψi = Ψi ∂Ψ1 (u) ⊆ λ Ω a1 (x)∂G1 (x, u) dx, ∂Ψ2 (u) ⊆ µ Ω X is also locally Lipschitz, and a2 (x)∂G1 (x, u) dx (see [38]), where Ψi X stands for the restriction of Ψi to X for i = 1, Therefore, ϕ is a locally Lipschitz functional on X (1) Let λ ∈ R and µ ≤ By (A), a1 (x) |u|q1 (x) dx + c2 ≤ c1 (|u| |Ψ1 (u)| ≤ c1 q1 (x),a1 (x) + q1 + c3 ≤ c4 u + q1 + c3 Ω By (B1 ), Ψ2 (u) ≤ −µc0 Ω a2 (x) dx = c5 Hence ϕ(u) ≥ p+ u p− − c4 u + q1 − c6 By + (A1 ), q1 < p− , so ϕ is coercive, that is, ϕ(u) → ∞ as u → ∞ Thus ϕ has a minimizer which is a solution of (1.1) (2) Let λ > 0, µ ≤ and the assumptions of (2) hold By the above conclu∞ sion (1), ϕ has a minimizer v1 Take v0 ∈ C0 (Ω) such that ≤ v0 (x) ≤ min{δ1 , δ2 }, Ω a1 (x)v0 (x)q3 (x) dx = d1 > and Ω a2 (x)v0 (x)q4 (x) dx = d2 > By (O1 ) and (O2 ) we 18 have, for t ∈ (0, 1) small enough, |t v0 |p(x) dx − λ p(x) ϕ(tv0 ) = Ω a1 (x)G1 (x, tv0 (x)) dx − µ Ω Ω | v0 |p(x) dx − λ p(x) p− ≤ t Ω a2 (x)G2 (x, tv0 (x)) dx a1 (x)c3 (tv0 (x))q3 (x) dx Ω a2 (x)c4 (tv0 (x))q4 (x) dx − µ Ω − + | v0 |p(x) dx − tq3 λc3 d1 − tq4 µc4 d2 p(x) p− ≤ t Ω + − Since q3 < p− < q4 , we can find t0 ∈ (0, 1) such that ϕ(t0 v0 ) < 0, and this shows ϕ(v1 ) = inf u∈X ϕ(u) < So v1 = because ϕ(0) = The conclusion (2) is proved (3) Let µ > and the assumptions of (3) hold By Remark 4.1.(1), for sufficiently small u , a2 (x) c4 |u|q4 (x) + c5 |u|q2 (x) dx Ψ2 (u) ≤ µ Ω − q4 ≤ µc4 |u|(q4 (x),a2 (x)) ≤ µc8 u − q4 + u − q2 + µc5 |u|(q2 (x),a2 (x)) − q2 − − Since p+ < q2 and p+ < q4 , there exists γ > and α > such that J(u) − Ψ2 (u) ≥ α for u ∈ Sγ We can find λ0 (µ) > such that when |λ| ≤ λ0 (µ), Ψ1 (u) ≤ α/2 for u ∈ Sγ So when |λ| ≤ λ0 (µ), ϕ(u) ≥ α/2 > for u ∈ Sγ By Remark 4.1.(2), noting + that 1/θ > p+ > q1 , we can find a u0 ∈ X such that u0 > γ and ϕ(u0 ) < By Proposition 3.6 problem (1.1) has a nontrivial solution u1 such that ϕ(u1 ) > (4) Let µ > and the assumptions of (4) hold By the conclusion (3), we know that, there exists λ0 (µ) > such that when < λ ≤ λ0 (µ), problem (1.1) has a nontrivial solution u1 such that ϕ(u1 ) > Let γ and α be as in the proof of (3), that is, ϕ(u) ≥ α/2 > for u ∈ Sγ By (O1 ), (O2 ) and the proof of (2), there exists w ∈ X such that 19 w < γ and ϕ(w) < It is clear that there is v1 ∈ Bγ , a minimizer of ϕ on Bγ Thus v1 is a nontrivial solution of (1.1) and ϕ(v1 ) < (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 Z2 to prove (5) (see Proposition 3.7) Denote Ψ(u) = F (x, u) dx = λ a1 (x)G1 (x, u) dx + µ Ω Ω a2 (x)G2 (x, u) dx Ω Let βk (γ) be as in Proposition 3.5 By Proposition 3.5, for each positive integer n, there exists a positive integer k0 (n) such that βk (n) ≤ for all k ≥ k0 (n) We may assume k0 (n) < k0 (n + 1) for each n We define {γk : k = 1, 2, , } by    n if k (n) ≤ k < k (n + 1)  0 γk =   if ≤ k < k (1)  Note that γk → ∞ as k → ∞ Then for u ∈ Zk with u = γk we have 1 − | u|p(x) dx − Ψ(u) ≥ + (γk )p − p(x) p ϕ(u) = Ω and consequently inf u∈Zk , u =γk ϕ (u) → ∞ as k → ∞, i.e., the condition (F2 ) of Proposition 3.7 is satisfied By (A), (A1 ), (B2 ) and Remark 4.1.(2), we have ϕ(u) ≤ u p− p+ + + c1 |λ| (|u|(q1 (x),a1 (x)) )q1 − c6 µ |u|(1/θ,a2 (x)) 1/θ + c9 + Noting that 1/θ > p+ > q1 and all norms on a finite dimensional vector space are equivalent each other, we can see that, for each Yk , ϕ(u) → −∞ as u ∈ Yk and u → ∞ Thus for each k there exists ρk > γk such that ϕ(u) < for u ∈ Yk ∩ Sρk , so the condition 20 (F3 ) of Proposition 3.7 is satisfied As was noted earlier, ϕ satisfies nonsmooth (PS) condition By Proposition 3.7 the conclusion (5) is true (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 Z2 For Ψ(u) = Ω F (x, u) dx = Ψ1 (u) + Ψ2 (u) let βk (1) be as in Proposition 3.5, that is βk (1) = sup |Ψ (u)| u∈Zk , u ≤1 By Proposition 3.5, there exists a positive integer k0 such that βk (1) ≤ 2p+ for all k ≥ k0 Setting ρk = 1, then for k ≥ k0 and u ∈ Zk ∩ S1 , we have ϕ(u) ≥ 1 − + = + > 0, + p 2p 2p which shows that the condition (D1 ) of Proposition 3.8 is satisfied 1,p(x) Since X = W0 ∞ is the closure of C0 (Ω) in W 1,p(x) (Ω) , we may choose {Yk : k = 1, 2, , }, a sequence of finite dimensional vector subspaces of X defined by (3.5), such ∞ that Yk ⊂ C0 (Ω) for all k For each Yk , because all norms on Yk are equivalent each other, there is ε ∈ (0, 1) such that for every u ∈ Yk ∩ Bε , |u|∞ ≤ min{δ1 , δ2 }, |u|(q3 (x),a1 (x)) ≤ and |u|(q4 (x),a2 (x)) ≤ By (O1 ) and (O2 ), for u ∈ Yk ∩ Bε we have ϕ(u) ≤ u p− p− a1 (x) |u|q3 (x) dx + |µ| c4 − λc3 Ω ≤ − u p p− Ω − λc3 |u|(q3 (x),a1 (x)) + q3 + |µ| c4 |u|(q4 (x),a2 (x)) + − Because q3 < p− < q4 , there exists γk ∈ (0, ε) such that bk := a2 (x) |u|q4 (x) dx max u∈Yk , u =γk ϕ (u) < 0, thus the condition (D2 ) of Proposition 3.8 is satisfied 21 − q4 Because Yk ∩ Zk = ∅ and γk < ρk , we have dk := inf u∈Zk , u ≤ρk ϕ (u) ≤ bk := max u∈Yk , u =rk ϕ (u) < On the other hand, for any u ∈ Zk with u ≤ = ρk , we have ϕ(u) = J(u) − Ψ(u) ≥ −Ψ(u) ≥ −βk (1) Noting that βk → as k → ∞, we obtain dk → 0, i.e., (D3 ) of Proposition 3.8 is satisfied Finally let us prove that ϕ satisfies nonsmooth (PS)∗ condition for every c ∈ R c Suppose {unj } ⊂ X, nj → ∞, unj ∈ Ynj , ϕ unj → c and m|Ynj unj → Similar to the process of verifying the (PS) condition in the proof of Proposition 3.3, we can get unj → u in X Let us prove ∈ ∂ϕ(u) below Notice that ≤ m(u) = m(u) − m(unj ) + m(unj ) = m(u) − m(unj ) + m|Ynj (unj ) Using Proposition 2.8.4, Going to limit in the right side of above equation, we have m(u) ≤ 0, so m(u) ≡ 0, i.e., ∈ ∂ϕ(u), this shows that ϕ satisfies the nonsmooth (PS)∗ condition c for every c ∈ R So all conditions of Proposition 3.8 are satisfied and the conclusion (6) follows from Proposition 3.8 The proof of Theorem 4.1 is complete 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 a1 (x) = |x| , a2 (x) = s1 (x) |x|s2 (x) , all conditions of Theorem 4.1 are satisfied provided (P), (A∗ ), (A1 ), and (A2 ) hold 22 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 Acknowledgement The authors are very grateful to the anonymous 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