The Brezis–Nirenberg problem for systems involving divergence-form operators

Abstract

We study a system of nonlinear elliptic partial differential equations involving divergence-form operators. The problem under consideration is a natural generalization of the classical Brezis–Nirenberg problem. We find conditions on the domain, the coupling coefficients and the coefficients of the differential operator under which positive solutions are guaranteed to exist and conditions on these objects under which no positive solution exists.

Appendix

Here we record the details of some computations whose inclusion in the main body of the text would have distracted from the story.

Proof of Lemma 4.1

For any $\xi , \zeta \in {\mathbb {R}}^m$ we have

$$\begin{aligned} H(\xi ) - H(\xi - \zeta ) = \sum _{i = 1}^m (A_i(\xi , \zeta ) + B_i(\xi , \zeta )), \end{aligned}$$

(8.1)

where

$$\begin{aligned} A_i(\xi , \zeta ) = (|\xi _i|^{q_i} - |\xi _i - \zeta _i|^{q_i} - |\zeta _i|^{q_i})P_i(\xi , \zeta ), \qquad B_i(\xi , \zeta ) = |\zeta _i|^{q_i} P_i(\xi , \zeta ), \end{aligned}$$

and

$$\begin{aligned} P_i(\xi , \zeta ) = \prod _{j = 1}^{i - 1}|\xi _j - \zeta _j|^{q_j}\cdot \prod _{\ell = i + 1}^m|\xi _\ell |^{q_\ell }. \end{aligned}$$

In our notation for $P_i(\xi , \zeta )$ empty products (for example products of the form $\prod _{j = 1}^0 c_j$ or of the form $\prod _{j = m + 1}^m c_j$) are understood to equal 1. For any $1< r< \infty$ and any $\epsilon >0$, the inequality

$$\begin{aligned} ||a + b|^r - |a|^r - |b|^r| \le \epsilon |a|^r + C_\epsilon |b|^r \end{aligned}$$

holds for all $a, b\in {\mathbb {R}}$. For each $i = 1, \ldots , m$, applying this inequality to to the first factor of $A_i(\xi , \zeta )$ (with $a = \xi _i-\zeta _i$, $b= \zeta _i$ and $r = q_i$) gives

$$\begin{aligned} \left| |\xi _i|^{q_i} - |\xi _i - \zeta _i|^{q_i} - |\zeta _i|^{q_i}\right| \le \epsilon |\xi _i - \zeta _i|^{q_i} + C_\epsilon |\zeta _i|^{q_i}. \end{aligned}$$

Therefore, using (8.1) we obtain

$$\begin{aligned} \begin{aligned}&|H(\xi ) - H(\xi - \zeta ) - H(\zeta )|\\&\quad \le \sum _{i = 1}^m\left( \epsilon |\xi _i - \zeta _i|^{q_i} + C_\epsilon |\zeta _i|^{q_i}\right) P_i(\xi , \zeta ) + |\zeta _1|^{q_1}\bigg |P_1(\xi , \zeta )\\ {}&\qquad - \prod _{j = 2}^m|\zeta _j|^{q_j}\bigg | + \sum _{i = 2}^m B_i(\xi , \zeta ) \end{aligned} \end{aligned}$$

(8.2)

for all $\epsilon >0$ and all $(\xi , \zeta )\in {\mathbb {R}}^m\times \mathbb R^m$. For $\epsilon >0$, and with ${\varvec{u}}^k$ as in the hypotheses of the lemma, setting

$$\begin{aligned} f^k_\epsilon = \left( \left| H(\varvec{u}^k) - H(\varvec{u}^k- \varvec{u}) - H(\varvec{u})\right| - \epsilon \sum _{i = 1}^m|u^k_i - u_i|^{q_i} P_i(\varvec{u}^k, \varvec{u})\right) ^+ \end{aligned}$$

we have $f_\epsilon ^k \rightarrow 0$ a.e. in $\Omega$. Moreover, in view of (8.2) we have $0\le f_\epsilon ^k \le g_\epsilon ^k$, where

$$\begin{aligned} g_\epsilon ^k = C_\epsilon \sum _{i = 1}^m|u_i|^{q_i}P_i(\varvec{u}^k, \varvec{u}) + |u_1|^{q_1}\bigg |P_1(\varvec{u}^k, \varvec{u}) - \prod _{j = 2}^m|u_j|^{q_j}\bigg | + \sum _{i = 2}^m B_i(\varvec{u}^k, \varvec{u}). \end{aligned}$$

For every $i = 1, \ldots , m$, $P_i(\varvec{u}^k, \varvec{u})$ is bounded in $L^{p/(p-q_i)}(\Omega )$ and

$$\begin{aligned} P_i(\varvec{u}^k,\varvec{u}) \rightarrow {\left\{ \begin{array}{ll} \prod _{\ell = 2}^m|u_\ell |^{q_\ell } &{} \text { if }i = 1\\ 0 &{} \text { if } i\in \{2, \ldots , m\} \end{array}\right. } \qquad \text { a.e. }x\in \Omega . \end{aligned}$$

Therefore, $P_1(\varvec{u}^k, \varvec{u})\rightharpoonup \prod _{\ell = 2}^m|u_\ell |^{q_\ell }$ weakly in $L^{p/(p - q_1)}(\Omega )$ and, for $i \in \{2, \ldots , m\}$, $P_i(\varvec{u}^k,\varvec{u})\rightharpoonup 0$ weakly in $L^{p/(p - q_i)}(\Omega )$. Consequently,

$$\begin{aligned} \sum _{i = 1}^m\int _\Omega |u_i|^{q_i}P_i(\varvec{u}^k, \varvec{u})\; \textrm{d}x \rightarrow \int _\Omega H(\varvec{u})\; \textrm{d}x \qquad \text { as }k\rightarrow \infty . \end{aligned}$$

By a similar argument, we find both that $\big |P_1(\varvec{u}^k, \varvec{u}) - \prod _{j = 2}^m|u_j|^{q_j}\big |\rightharpoonup 0$ weakly in $L^{p/(p - q_1)}(\Omega )$ and that $B_i(\varvec{u}^k, \varvec{u})\rightharpoonup 0$ weakly in $L^{p/(p - q_i)}(\Omega )$ for $i\in \{2, \ldots , m\}$, so we deduce that $\int _\Omega g_\epsilon ^k\; \textrm{d}x \rightarrow C_\epsilon \int _\Omega H(\varvec{u})\; \textrm{d}x$ as $k\rightarrow \infty$. The (generalized) Dominated Convergence Theorem now guarantees that $\int _\Omega f_\epsilon ^k \; \textrm{d}x\rightarrow 0$. Finally,

$$\begin{aligned} \begin{aligned}&\int _\Omega |H(\varvec{u}^k)- H(\varvec{u}^k - \varvec{u}) - H(\varvec{u})|\; \textrm{d}x\\&\quad \le \int _\Omega f_\epsilon ^k \; \textrm{d}x + \epsilon \sum _{i = 1}^m\int _\Omega |u_i^k - u_i|^{q_i} P_i(\varvec{u}^k,\varvec{u})\; \textrm{d}x \\&\quad \le C\epsilon + \circ (1), \end{aligned} \end{aligned}$$

where $\circ (1)\rightarrow 0$ as $k\rightarrow \infty$ and C depends on m, p, $\Vert \varvec{u}\Vert _{L^p(\Omega ; {\mathbb {R}}^m)}$, and an upper bound for $\{\Vert \varvec{u}^k\Vert _{L^p(\Omega ; {\mathbb {R}}^m)}\}_{k = 1}^\infty$, but is independent of k. Since $\epsilon >0$ is arbitrary, the lemma is established.$\square$