Infinitely many solutions for semilinear Δλ-Laplace equations with sign-changing potential and nonlinearity

In this paper, we prove the existence of infinitely many solutions for the following class of boundary value elliptic problems where Ω is a bounded domain in RN (N ≥ 2), Δλ is a strongly degenerate elliptic operator, V (x) is allowing to be sign-changing and f is a function with a more general super-quadratic growth, which is weaker than the Ambrosetti-Rabinowitz type condition.

Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators

Let w be either in the Muckenhoupt class of A2(ℝn) weights or in the class of QC(ℝn) weights, and let be the degenerate elliptic operator on the Euclidean space ℝn, n ≥ 2. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space associated with , and when with , the authors prove that the associated Riesz transform is bounded from to the weighted classical Hardy space .

Regularity theory for tangent-point energies: The non-degenerate sub-critical case

In this article we introduce and investigate a new two-parameter family of knot energies ${\operatorname{TP}^{(p,\,q)}}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will first characterize the curves of finite energy ${\operatorname{TP}^{(p,\,q)}}$ in the sub-critical range p ∈ (q+2,2q+1) and see that those are all injective and regular curves in the Sobolev–Slobodeckiĭ space ${W^{\scriptstyle (p-1)/q,q}(\mathbb {R}/\mathbb {Z},\mathbb {R}^n)}$. We derive a formula for the first variation that turns out to be a non-degenerate elliptic operator for the special case q = 2: a fact that seems not to be the case for the original tangent-point energies. This observation allows us to prove that stationary points of $\operatorname{TP}^{(p,2)}$ + λ length, p ∈ (4,5), λ > 0, are smooth – so especially all local minimizers are smooth.