Optimal regularity of stable solutions to nonlinear equations involving the <i>p</i>-Laplacian

Abstract

Abstract We consider the equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {-\Delta_{p}u=f(u)} in a smooth bounded domain of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {\mathbb{R}^{n}} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\Delta_{p}} is the p -Laplace operator. Explicit examples of unbounded stable energy solutions are known if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:mrow> </m:math> {n\geq p+\frac{4p}{p-1}} . Instead, when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>&lt;</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:mrow> </m:math> {n&lt;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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mi>α</m:mi> </m:msup> </m:math> {C^{\alpha}} bound for stable solutions which holds for every nonnegative <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:mrow> </m:math> {f\in C^{1}} whenever <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {p\geq 2} and the optimal condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>&lt;</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:mrow> </m:math> {n&lt;p+\frac{4p}{p-1}} <

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