Semilinear degenerate elliptic boundary value problems via the Semenov approximation

Abstract

This paper is devoted to the study of static bifurcation theory for a class of degenerate boundary value problems for semilinear elliptic differential operators of second-order which includes as particular cases the Dirichlet and Robin problems. The purpose of this paper is to generalize some results of Szulkin (Nonlinear Anal TMA 6:95–116, 1982) to the degenerate case. We study the behavior of nontrivial solution branches for bounded nonlinear terms, and prove that the branches turn back towards the first eigenvalue of the linearized degenerate boundary value problem. Our proof is based on global static bifurcation theory for positive mappings in ordered Banach spaces, with the special emphasis on the Semenov approximation in Chemistry. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the $L^{p}$ theory of pseudo-differential operators.

Appendix: The maximum principle in \(L^{p}\) Sobolev spaces

In this appendix we formulate various maximum principles for second-order, elliptic differential operators with discontinuous coefficients such as the weak and strong maximum principles (Theorems 10.1 and 10.2) and the Hopf boundary point lemma (Lemma 10.1) in the framework of $L^{p}$ Sobolev spaces. The results here are adapted from Bony [20], Troianiello [72, Chapter 3] and also Taira [64, Chapter 8].

Let D be a bounded domain in Euclidean space $\mathbf {R}^{N}$, $N \ge 3$, with boundary $\partial D$ of class $C^{1,1}$. We consider a second-order, uniformly elliptic differential operator A with real discontinuous coefficients in non-divergence form

$$\begin{aligned} Au := - \sum _{i,j=1}^{N} a^{ij}(x) \dfrac{\partial ^{2}u}{\partial x_{i} \partial x_{j}} + \sum _{i=1}^{N} b^{i}(x) \dfrac{\partial u}{\partial x_{i}} + c(x)u. \end{aligned}$$

More precisely, we assume that the coefficients $a^{ij}(x)$, $b^{i}(x)$ and c(x) of the differential operator A satisfy the following three conditions:

• (1) $a^{ij}(x) \in L^{\infty }(D)$, $a^{ij}(x) = a^{ji}(x)$ for almost all $x \in D$ and there exist a constant $\lambda > 0$ such that

  $$\begin{aligned} \dfrac{1}{\lambda } \vert \xi \vert ^{2} \le \sum _{i,j=1}^{N} a^{ij}(x)\xi _{i}\xi _{j} \le \lambda \left| \xi \right| ^{2} \quad \text{ for } \text{ almost } \text{ all } x \in D \text{ and } \text{ all } \xi \in \mathbf {R}^{N}. \end{aligned}$$

• (2) $b^{i}(x) \in L^{\infty }(D)$ for all $1 \le i \le N$.

• (3) $c(x) \in L^{\infty }(D)$ and $c(x) \ge 0$ for almost all $x \in D$.

First, we state a variant of the weak maximum principle in the framework of $L^{p}$ Sobolev spaces, due to Bony [20] [72, Chapter 3, Lemma 3.25]:

Theorem 10.1

(the weak maximum principle) If a function $u \in W^{2,p}(D)$, $N< p < \infty$, satisfies the condition

$$\begin{aligned} Au(x) \le 0 \quad \text{ for } \text{ almost } \text{ all } x \in D, \end{aligned}$$

then we have the inequality

$$\begin{aligned} \max _{\overline{D}}u \le \max _{\partial D} u^{+}, \end{aligned}$$

where

$$\begin{aligned} u^{+}(x) = \max \left\{ u(x),0\right\} \quad \text{ for } x \in \overline{D}. \end{aligned}$$

A detailed proof of Theorem 10.1 is given in Taira [67, Theorem 8.1].

Secondly, the Hopf boundary point lemma reads as follows [72, Chapter 3, Lemma 3.26, 68, Lemma 6.1]:

Lemma 10.1

(the boundary point lemma) Assume that a function $u \in W^{2,p}(D)$, $N< p < \infty$, satisfies the condition

$$\begin{aligned} Au(x) \le 0 \quad \text{ for } \text{ almost } \text{ all } x \in D. \end{aligned}$$

If u(x) attains a non-negative, strict local maximum at a point $x_{0}^{\prime }$ of $\partial D$, then we have the inequality

$$\begin{aligned} \dfrac{\partial u}{\partial \varvec{\nu }}(x_{0}^{\prime }) > 0 \end{aligned}$$

(see Fig. 1).

Finally, we can obtain the following strong maximum principle for the operator A [20, Théorème 2, 68, Theorem 6.2, 72, Chapter 3, Theorem 3.27]:

Theorem 10.2

(the strong maximum principle) Assume that a function $u \in W^{2,p}(D)$, $N< p < \infty$,satisfies the condition

$$\begin{aligned} Au(x) \le 0 \quad \text{ for } \text{ almost } \text{ all } x \in D. \end{aligned}$$

If u(x) attains a non-negative maximum at an interior point $x_{0}$ of D, then it is a (non-negative) constant function.