Existence of weak solutions of parabolic systems with p, q-growth

Existence of weak solutions for a class of (p,q)$(p,q)$-Laplacian systems

Existence and Uniqueness of Weak Solutions to Parabolic IVP’s

= −divax, t∇u + b⃗x, t ⋅ ∇u + cx, tux, t with aij, bj, c ∈ LUT and for a0 > 0, z⃗ ⋅ aijx, t z⃗ ≥ a0 | z⃗| ∀z⃗ ∀x, t ∈ UT. Then L is said to be uniformly elliptic on UT. We recall that the assumptions on the coefficients aij, bj, c, imply the existence of positive constants, a1,a0 and μ0 such that for all u,v ∈ H1U, and μ > μ0, |Bu,v, t| ≤ a1‖u‖1‖v‖1 a.e. t ∈ 0,T Bu,u, t +...

Existence of non-trivial solutions for fractional Schrödinger-Poisson systems with subcritical growth

In this paper, we are concerned with the following fractional Schrödinger-Poisson system:    (−∆s)u + u + λφu = µf(u) +|u|p−2|u|, x ∈R3 (−∆t)φ = u2, x ∈R3 where λ,µ are two parameters, s,t ∈ (0,1] ,2t + 4s > 3 ,1 < p ≤ 2∗ s and f : R → R is continuous function. Using some critical point theorems and truncation technique, we obtain the existence and multiplicity of non-trivial solutions with ...

Regularity of Weak Solutions for Nonlinear Parabolic Problem with p(x)-Growth

In this paper, we study the nonlinear parabolic problem with p(x)growth conditions in the space W L(Q), and give a regularity theorem of weak solutions for the following equation ∂u ∂t + A(u) = 0 where A(u) = −diva(x, t, u,∇u) + a0(x, t, u,∇u), a(x, t, u,∇u) and a0(x, t, u,∇u) satisfy p(x)-growth conditions with respect to u and ∇u.

Local Boundedness of Weak Solutions for Nonlinear Parabolic Problem with p(x)-Growth

The study of variational problems with nonstandard growth conditions is an interesting topic in recent years. p x -growth problems can be regarded as a kind of nonstandard growth problems and they appear in nonlinear elastic, electrorheological fluids and other physics phenomena. Many results have been obtained on this kind of problems, for example 1–9 . Let Q be Ω × 0, T , where T > 0 is given...