Bernstein Functions and the Dirichlet Problem

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

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Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

Advances in Calculus of Variations ◽

10.1515/acv-2020-0055 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Xavier Cabré ◽

Pietro Miraglio ◽

Manel Sanchón

Keyword(s):

Dirichlet Problem ◽

Optimal Condition ◽

Bounded Domain ◽

Laplace Operator ◽

Nonlinear Equations ◽

Open Problem ◽

Optimal Range ◽

Optimal Regularity ◽

Smooth Bounded Domain ◽

Stable Solutions

AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}. Instead, when {n<p+\frac{4p}{p-1}}, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior {C^{\alpha}} bound for stable solutions which holds for every nonnegative {f\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds. When {p\in(1,2)}, we obtain the same result under the nonsharp assumption {n<5p}. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when {p=2} in the optimal range {n<10}.

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Measure Data Problems for a Class of Elliptic Equations with Mixed Absorption-Reaction

Advanced Nonlinear Studies ◽

10.1515/ans-2021-2124 ◽

2021 ◽

Vol 21 (2) ◽

pp. 261-280

Author(s):

Marie-Françoise Bidaut-Véron ◽

Marta Garcia-Huidobro ◽

Laurent Véron

Keyword(s):

Dirichlet Problem ◽

Compact Subset ◽

Elliptic Equations ◽

Radon Measure ◽

Measure Data ◽

Nonnegative Solutions

Abstract In the present paper, we study the existence of nonnegative solutions to the Dirichlet problem ℒ p , q M ⁢ u := - Δ ⁢ u + u p - M ⁢ | ∇ ⁡ u | q = μ {{\mathcal{L}}^{{M}}_{p,q}u:=-\Delta u+u^{p}-M|\nabla u|^{q}=\mu} in a domain Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} where μ is a nonnegative Radon measure, when p > 1 {p>1} , q > 1 {q>1} and M ≥ 0 {M\geq 0} . We also give conditions under which nonnegative solutions of ℒ p , q M ⁢ u = 0 {{\mathcal{L}}^{{M}}_{p,q}u=0} in Ω ∖ K {\Omega\setminus K} , where K is a compact subset of Ω, can be extended as a solution of the same equation in Ω.

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