Some Problems To P(x)-Laplacian Equations

The study of problems involving p(x)-Laplace operator has received considerable attentionin recent years. In this paper, we collect some papers about the existence of solutions to p(x)-Laplace problems, which contain existence of solutions to Dirichlet problems on a bounded domain, on unbounded domains, and Neumann boundary value problems and problems in R^N with periodic perturbations.Firstly, the paper introduces the existence of solutions to p(x)-Laplace problems on a bounded domain.The problems we consider are as follows:whereΩis a bounded domain in Rⁿ, 0 < a₀≤a(x)∈L^∞(Ω), 0 < b₀≤b(x)∈L^∞(Ω), p(x) is Lipschitz continuous on (?). and satisfiesOur object is to obtain advisable conditions on f for (1.1) to admit nontrivial and nonnegativesolutions in the following prototype casesFor the p(x)-Laplacian Problems (1.1), we define two functionals K(u) and J(u) as follows:whereF(x, t) =(?) We consider the existence of solutions to Problem (1.1) when f satisfies the following assumptions.(H1) f∈C((?)×R), f(x,t) > 0,(x,t)∈Ω₀×(0, +∞)for some nonempty open setΩ₀∈Ωand f(x, t) = 0 for all x∈Ωand t≤0;(H2) |f(x,t)|≤c₁ + c₂|t|^α(x), orα+ 1∈C(?) with (?) = (?) - p(x) +1 > 0 and a = (?) -α(x) - 1 > 0. Here c₁, c₂ are positive constants;(H3) |f(x, t)|≤(?),β+1∈P(Ω), 0≤β(x) with b = (?)-β(x) - 1 > 0. Here (?), (?) are positive constants;(H4) There exist constants M > 0 andμ> p(x) with (?)-p(x) > 0 such thatμF(x, t)≤tf(x, t) for x∈Ωand |t| > M, and f(x, t) = o(?) as t→0.Under assumptions (H1), (H2), (H3) and (H4), we can show the existence of solutions to Problem (1.1) in bounded domains by Theorems 1.1, 1.2, 1.3 and 1.4.Theorem 1.1 Suppose that f satisfies (H1) and (H2) or (H3). Then K(u) is differentiateon W^1,px withTheorem 1.2 Suppose that f satisfies (H1) and (H2) or (H3). Then K'(u) is a continuousand compact mapping from W₀^1,p(x) to W^-1,p'(x)For the prototype case ( 1.3 ), we suppose that f satisfies the following additional condition.(H4) There exist constants M > 0 andμ> p(x) with (?){μ- p(x)} > 0 such thatμF(x, t)≤tf(x, t), for x∈Ω, |t|≥M, and f(x, t) = o(|t|^p(x)-1) as t→0.Theorem 1.3 Under conditions (H1 ),(H2) and (H4), the p(x)-Laplacian Problem (1.1) has a nontrivial and nonnegative solution u∈W₀^1,p(x)(Ω).Secondly, the paper introduces existence of solutions to p(x)-Laplace problems in unboundeddomains. The problems we consider are of the following typewhereΩ(?) R^N is an unbounded domain with smooth boundary (?)Ω. a(x), b(x) will be assumed throughout this paper satisfying 0 < a₁≤a(x)≤a₂,01≤b(x)≤b₂, (?)x∈Q. Associated with (1.5) is the energy functional I defined byWe will also assume that f :Ω×R→R is a Caratheodory function and satisfies(f1) f(x, t)≤C(|t|^p(x)-1 + |t|^α(x)-1) whereα(x)∈(?), andα(x) < p^*(x),α^-> p⁺;(f2) (?)θ> p⁺ such thatand F(x,t)=∫₀¹f(x,s)ds;(f3) f(x, t) = o(|t|^p(x)-1), as |t|→0, uniformly x∈Ω;(f4) (?) is strictly increasing in t≥0, (?)x∈Ω.Since we seek only positive solutions of Problem (1.5), it is convenient to define f(x, t)≡0 for t≤0, x∈(?).Now we give the dedinitin of the weak solutions to Problem (1.5).Definition1.1(1) We call that u∈X is a weak solution of Problem (1.5), ifUnder the conditions (f1), (f2), (f3) and (f4), we can show the existence of positive solutions to Problem (1.5) by Theorems 1.5, 1.6, and 1.7.Theorem1.5 Suppose (f₁) - (f₄) hold and for some large k∈N,α_N(Ω) <(?), then there exists a ground state solution u of Problem (1.5) with I(u) =α_N(Ω). Theorem1.6 Suppose (f₁) - (f₄) hold and there existsα(PS)_c sequence withα_N(Ω)? c <α_N(?) for some large k∈N, then Problem (1.5) admits a higher energy solution v with c≥I(v)≥a_N(Ω).Theorem1.7 Suppose (f₁) - (f₄) hold and there exists a (PS)_c sequence with c> 0 and c (?) for some m∈N, then there exists a positive solution of Problem (1.5).Thirdly, we introduce the nonlinear Neumann boundary value problem of the following form:Using the variational method, under appropriate conditions on f and g, we obtain a number of results on the existence and multiplicity of solutions to (1.6), whereΩ(?) R^N is a bounded domain with smooth boundary (?)Ω,(?)is the outer unit normal derivative, p(x)∈C(?), p(x) > 1, (?)x∈(?), andλ,μ∈R. We assume thatλ² +μ²â‰ 0.Next, we need the following assumptions on f and g to get the existence results of Problem (1.6).(f₀) f :Ω×R→R satisfies the Caratheodory condition and there exist two constantsC₁≥0, C₂ > 0 such thatwhereα(x)∈C₊(Ω), andα(x) < p^*(x), (?)x∈(?);(f₁) (?)M₁ > 0,θ₁ > P+, such that 0 <θ₁, F(x, t)≤f(x, t), |t|≥M₁ (?)X∈Ω;(f₂) f(x,t)=o(|t|^p+-1), t→0;(f₃) f(x, -t) = -f(x, t),x∈Ω,tΩR;(g₀) f : (?)Ω×R→R satisfies the Caratheodory condition and there exist two constants C'₁≥0, C'₂≥0 such thatwhereβ(x)∈C₊((?)Ω.),β(x) < p^*(x), (?)x∈(?)Ω(g₁) (?)M₂ > 0,θ₂ > P⁺, such that (g₂) f(x, t) = o(|t|^p+-1 ),t→0 for x∈(?)Ωuniformly;(g₃) g(x, -t) = -g(x, t), x∈(?)Ω,t∈R.We can see the existence results of Problem (1.6) form Theorems 1.8, and 1.9.Theorem1.8 If (f₀), (g₀) hold andα⁺,β⁺ < p^-, then (1.6) admits a weak solution.Theorem1.9 If (f₀), (f₁), (f₂), (g₀), (g₁), (g₂) hold andα⁺,β⁺ < p^-,λ,μ≥0, then (1.6) has a nontrivial weak solution.At last, we consider the p(x)-Laplcian in R^N with periodic perturbations.Now, we consider the p(x)-Laplcian in R^N with periodic data formIn this section, we denote by R the space of all real numbers, R+ = [0, +∞].Let {e₁,e₂,...e_N} be the standard basis of R^N. Let T_i > 0, i = 1,2...,N. Denote T = (T₁,T₂,..., T_N). A function p : R^N→R is called T-periodic, ifFor Problem (1.7), we need the following assumptions:(p₁) The function p : R^N→R is Lipschitz continuous and(p₂) p is T-periodic;(V₁) V∈C⁰(R^N, R⁺), 0 < V_< V₊ <∞;(V₂) V is T-periodic;(f₁) f∈C⁰(R^N×R⁺,R)andwhere C₁ is a positive constant, q∈C⁰(R^N, R) and p≤q≤p^*,p^* is defined by (f₂) There is a positive constantβ> p⁺, such thatwhere F(x, t) =∫₀¹f(x, s) ds;(f₃) f(x, t) = o(|t|^p+-1 ),as t→0(f₄) f(·,t) 0 is T-periodic for every t∈R;(f₅) For each x∈R, (?) is an increasing function of r on R\{0}Under the assumptions mentioned above, we can show Theorems 2.0 and 2.1 on the existence of weak solutions to Problem (1.7).Theorem2.0 Suppose (p₁), (p₂), (V₁), (V₂) hold, then(1) Problem(1.7) has a nontrivial solution.(2) Problem(1.7) has a positive solution and a negative solution.Theorem2.1 Suppose that in addition to the assumptions of Theorem 2.0, (f₅) holds.Then(1) Problem(1.7) has a ground state solution u_*, that is u_* is a nontrivial solution of (1.7) and(2) Problem(1 .7) has a positive solution v and a negative solution w such that...