In this paper we study a class of nonlocal elliptic problems driven by an unbalanced growth operator of double phase type. For the original problem, whose reaction term contains the nonlocal factor ( ∫ _Ω F(x,u) dx) ^r-1, we prove the existence of two nontrivial constant sign weak solutions, one nonnegative and one nonpositive. This part is obtained by using suitable truncations, the mountain pass theorem, and a Cerami compactness argument. In the second part of the paper, we consider an auxiliary nodal problem naturally associated with the positive and negative parts of the solution, where the corresponding nonlocal terms are separated. For this auxiliary problem we prove the existence of a sign-changing weak solution by minimizing the energy functional on a Nehari-type constraint set. The proof combines variational methods with the quantitative deformation lemma and the Poincaré-Miranda theorem. Finally, we provide a concrete example showing the applicability of the assumptions.
We consider a nonlinear Dirichlet problem driven by a differential operator with (p, q)-growth and a reaction which exhibits the combined effects of a singular term and of a parametric (p-1) -superlinear perturbation. The exponent η of the singular term satisfies η > 1 (strong singularity). Using an auxiliary problem to bypass the singularity and variational tools together with truncations and critical groups, we show that for all small values of the parameter, the problem has at least two positive solutions.
Given two variable exponent Muckenhoupt weights w is an element of A(p)(& centerdot;) and w(1) is an element of A(p1)(& centerdot;) , we prove that for all small enough theta > 0, there holds that w(0 )is an element of A(p0(& centerdot;)), where the weight is determined by w = w(0)(1-theta )w(1)(theta) and exponent of the weight class by 1/p(& centerdot;) = (1 - theta)/p(0)(& centerdot;) + theta/p(1)(& centerdot;). The proof is based on a recent reverse Holder's inequality for variable exponent Muckenhoupt weights by Cruz-Uribe and Penrod. We upgrade these factorizations to the restricted range context by using a recent transformation formula due to Nieraeth. Then, following an extrapolation of compactness scheme by Hytonen and Lappas, we provide an alternative proof of the recent extrapolation of compactness results of Lorist and Nieraeth in the context of weighted variable exponent Lebesgue spaces.
In this paper, we study the normalized solutions and regularity of solutions for a class of the Biharmonic equations with p-Laplacian as follows {[ Δ ^2u+Δ _p u=μ |u|^q-2u+λ u, in ℝ^N,; ∫ _ℝ^N|u|^2dx=c, ]. where μ >0 , λ <0 is a Lagrange multiplier, 2
In this paper, we prove that the Artinian Gorenstein 𝕂-algebra A_F_s of codimension n, socle degree d and Macaulay dual generator F_s := ℓ_1^d + … + ℓ_s^d ∈𝕂[X_1, …, X_n] where ℓ _1, ⋯ , ℓ_s are general linear forms satisfies the strong Lefschetz property (SLP). This result allows us to study whether the Waring rank of F_s is exactly s. Furthermore, we show that A_F_s is the doubling of a suitable 0-dimensional scheme Z_F_s in ℙ^n-1, the so-called tight annihilating scheme of A_F_s, and we compute the minimal free resolution of A_F_s in terms of the minimal free R-resolution of I(Z_F_s). Finally, we determine the linear general Jordan type of A_F_s.
