Dirichlet problems for the p-Laplacian with a convection term

Abstract

We consider the nonlinear Dirichlet boundary value problem

in a bounded domain $\Omega \subset \mathbb {R}^N$ with smooth boundary $\partial \Omega$, where $\Delta _p u\mathop {=}\limits ^{\mathrm{{def}}}\mathrm {div} (|\nabla u|^{p-2} \nabla u)$ with $1< p < \infty$, $\lambda \in \mathbb {R}$, and $h\in L^\infty (\Omega )$. The term $B(x,\nabla u)$ is a continuous function assumed to be also homogeneous of degree $(p-1)$ and odd with respect to the second variable; $B(x, \varvec{\eta }) = (\mathbf {a}(x)\cdot \varvec{\eta }) |\varvec{\eta }|^{p-2}$ being a canonical example with a given vector field $\mathbf {a}\in [ C(\overline{\Omega }) ]^N$, for $(x, \varvec{\eta })\in \Omega \times \mathbb {R}^N$. For the corresponding eigenvalue problem obtained by setting $h\equiv 0$, we show existence, simplicity, and isolation of the principal eigenvalue $\lambda _1$ ($\lambda _1 > 0$). When $h\not \equiv 0$ and $-\infty< \lambda < \lambda _1$, we prove that there exists a weak solution $u\in W_0^{1,p}(\Omega )$ to problem (P); this solution is unique provided $\lambda < 0$ (without any further assumptions). When $h\ge 0$, $h\not \equiv 0$, and $0\le \lambda < \lambda _1$, we show that the solution is positive and also unique.

Appendix

We are now gathering together some basic features (estimates, existence results), concerning the auxiliary problem (3.2). Most of the results are essentially well-known; nevertheless, a self-contained account is given for the reader’s convenience.

Lemma 4.2

Let $B = B(x,\varvec{\eta })$, $B:\overline{\Omega }\times \mathbb {R}^N\rightarrow \mathbb {R}^N$ be continuous and homogeneous of degree $p-1$ in $\varvec{\eta }$ and set

$$\begin{aligned} b_1 = \max _{x\in \Omega ,\ |\varvec{\eta }|=1}|B(x,\varvec{\eta })|. \end{aligned}$$

Fix $M\ge b_1$. Then, every solution $u\in W^{1,p}_0(\Omega )$ to

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p u + B(x,\nabla u) +\delta \varphi _p(u)=h&{}\quad x\in \Omega \\ \ u = 0&{} \quad x\in \partial \Omega , \end{array}\right. } \end{aligned}$$

(4.3)

where $\delta \in \mathbb {R}$ and $h\in W^{-1,p'}(\Omega )$ (the dual space of $W^{1,p}_0(\Omega )$) satisfies an estimate of the form

$$\begin{aligned} \Vert \nabla u\Vert _p^p\le C(\Vert h\Vert _{-1,p'}^{p'}+\Vert u\Vert _p^p), \end{aligned}$$

(4.4)

where $C = C(\delta ,M)$. Moreover, if $\delta \ge 0$ a constant $C_1$ exists so that every solution to (4.3) satisfies

$$\begin{aligned} \Vert \nabla u\Vert _p^p\le C_1\Vert h\Vert _{-1,p'}^{p'}. \end{aligned}$$

(4.5)

Proof

To achieve (4.4) first use u as a test function in (4.3) to obtain

$$\begin{aligned} \Vert \nabla u\Vert _p^p - M \Vert \nabla u\Vert _p^{p-1}\Vert u\Vert _p + \delta \Vert u\Vert _p^p\le \Vert h\Vert _{-1,p'} \Vert u\Vert _p, \end{aligned}$$

which implies

$$\begin{aligned} (1-\varepsilon )\Vert \nabla u\Vert _p^p - C\Vert u\Vert _p^p + \delta \Vert u\Vert _p^p\le C \Vert h\Vert _{-1,p'}^{p'} + \varepsilon \Vert \nabla u\Vert _p^p, \end{aligned}$$

where C depends on $\varepsilon$ and M. Then it suffices with taking $\varepsilon$ small (observe that the term in $\delta$ can be dropped if $\delta \ge 0$).

When $\delta \ge 0$ we refresh an idea from [1] to get (4.5). In fact, assume that (4.5) does not hold. Then, there exist sequences $u_n\in W^{1,p}_0(\Omega )$, $h_n\in W^{-1,p'}(\Omega )$, $u_n$ solving (4.3) with $h_n$ replacing h, such that

$$\begin{aligned} n\Vert h_n\Vert _{-1,p'}^{p'}< \Vert \nabla u_n\Vert _p^p \le C (\Vert h_n\Vert _{-1,p'}^{p'}+\Vert u_n\Vert _p^p), \end{aligned}$$

for all $n\in \mathbb N$. Setting $t_n = \Vert u_n\Vert _p$, $u_n = t_n \tilde{u}_n$, $h_n = t_n^{p-1} \tilde{h}_n$ it follows from the previous inequalities that $\tilde{u}_n$ remains bounded in $W^{1,p}_0(\Omega )$ while $\tilde{h}_n\rightarrow 0$ in $W^{-1,p'}(\Omega )$. On the other hand, by taking weak limits in $W^{1,p}_0(\Omega )$ (through a subsequence) one finds that $\tilde{u}_n\rightharpoonup \tilde{u}$ where $\Vert \tilde{u}\Vert _p=1$ and $\tilde{u}$ solves (4.3) with $h=0$. However, in view of Theorem 3.2.1 in [29] (see Remark 4.3 below) this means that $\tilde{u} = 0$. So, we get a contradiction and (4.5) holds true. $\square$

Remark 4.3

The argument in the proof of Theorem 3.2.1 in [29] can be improved to cover the case of the operator $-\Delta _p u + B(x,\nabla u) + \delta \varphi _p(u)$ provided that $\delta \ge 0$.

Remark 4.4

For B, h as in Lemma 4.2, let $t\in [0,1]$ be a parameter. It is clear that solutions $u\in W^{1,p}_0(\Omega )$ to the parameterized version of (4.3)

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p u + tB(x,\nabla u) +\delta \varphi _p(u)=h&{}\quad x\in \Omega \\ \ u = 0&{} \quad x\in \partial \Omega , \end{array}\right. } \end{aligned}$$

(4.6)

satisfy estimate (4.5) where $C_1$ can be chosen not depending on t.

We are obtaining next $L^\infty$ estimates for (4.3). When $p>N$, (4.5) implies that the family of all possible solutions to (4.3) ($\delta \ge 0$) corresponding to a uniformly bounded family of data $\{h\}$ in $W^{-1,p'}(\Omega )$, is in turn uniformly bounded in $C^{\beta }(\overline{\Omega })$ with $0 < \beta \le 1-\dfrac{p}{N}$. For $1< p\le N$, a uniform $L^\infty$ bound can be produced when the data h are uniformly bounded in $L^\infty$. It is implicit in the next result that weak solutions to (4.3) are in addition essentially bounded.

Lemma 4.5

Let $B = B(x,\varvec{\eta })$, $b_1$ be as in Lemma 4.2 while $\delta \in \mathbb {R}$. Assume that $1< p\le N$. Then, to each $M\ge \max \{b_1,|\delta |\}$ there corresponds a constant $K = K(M)$ such that every solution $u\in W_0^{1,p}(\Omega )$ to (4.3) with $h\in L^\infty (\Omega )$ and

$$\begin{aligned} \max \{\Vert h\Vert _\infty ,\Vert u\Vert _1\}\le M \end{aligned}$$

satisfies

$$\begin{aligned} \Vert u\Vert _\infty \le K. \end{aligned}$$

(4.7)

Proof

To show that $u^+$ satisfies estimate (4.7) it suffices with proving that there exist positive constants $k_0,\eta$ and C such that

$$\begin{aligned} \int _{A_k}(u-k) \le C k |A_k|^{1+\eta } \end{aligned}$$

(4.8)

is satisfied for all $k\ge k_0$, with $A_k=\{u>k\}$ and $|A_k|$ standing for the measure of $A_k$. Then, the assertion is consequence of Lemma 5.1 of Chapter II in [23] (the same reasoning works for $u^-$).

When $p<N$, by using $v=(u-k)^+$, $k>0$, as a test function in (4.3), we get

$$\begin{aligned} \int _{A_k}|\nabla u|^p\le & {} M\Vert \nabla u\Vert _{p,A_k}^{p-1}\Vert (u-k)^+\Vert _p+ Ck^{p-1}\Vert (u-k)^+\Vert _1+ C\Vert (u-k)^+\Vert _p^p\\\le & {} C|A_k|^{(\frac{1}{p}-\frac{1}{p^*})}\Vert \nabla u\Vert _{p,A_k} ^p+ Ck^{p-1}|A_k|^{ (1-\frac{1}{p^*})}\Vert \nabla u\Vert _{p,A_k} \\\le & {} (C|A_k|^{(\frac{1}{p}-\frac{1}{p^*})}+\varepsilon )\Vert \nabla u\Vert _{p,A_k} ^p + C_\varepsilon k^p|A_k|^{p'(1-\frac{1}{p^*})}, \end{aligned}$$

where $\frac{1}{p^*}=\frac{1}{p}-\frac{1}{N}$, $p'=p/(p-1)$ and $\varepsilon$ can be chosen arbitrarily small. The letter C has been used for different not relevant constants. It has also been employed that $|A_k| = o(1)$ as $k\rightarrow \infty$. In fact, $|A_k|\le k^{-1}\Vert u\Vert _1 \le k^{-1} M$.

Accordingly, the coefficient of $\Vert \nabla u\Vert _{p,A_k} ^p$ in the last inequality can be taken smaller than unity provided that k is larger than some $k_0 = k_0(M)$. Thus,

$$\begin{aligned} \Vert \nabla u\Vert _{p,A_k} ^p \le C k^p |A_k|^{p'(1-\frac{1}{p^*})}. \end{aligned}$$

Finally, by means of Sobolev’s embedding we arrive at

$$\begin{aligned} \int _{A_k}(u-k)\le C k |A_k|^{(1+\frac{p'}{p})\frac{1}{{p^*}'}} = C k |A_k|^{\frac{p'}{{p^*}'}}. \end{aligned}$$

Hence, (4.8) follows from the fact that $p'>{p^*}'$. The case $p=N$ is handled in the same way by replacing $p^*$ with a large $q>p$. $\square$

Remark 4.6

In virtue of [24], Lemma 4.5 entails that weak solutions to both problems (1.4) and (4.3) belong to $C^{1,\beta }(\overline{\Omega })$ for some $0< \beta < 1$.

Lemma 4.7

Assume $B = B(x,\varvec{\eta })$ is as in Lemma rm 4.2 while $\delta \ge 0$. Then, problem (4.3) admits a solution $u\in W_0^{1,p}(\Omega )$ for every $h\in W^{-1,p'}(\Omega )$. Moreover, if $h\in L^\infty (\Omega )$ then (4.3) possesses a solution $u\in C^{1,\beta }(\overline{\Omega })$ for certain $0<\beta < 1$ which is further unique in the class $C(\overline{\Omega })\cap C^1(\Omega )$ provided that $\delta >0$.

Proof

Fix $h\in L^\infty (\Omega )$ and define $H:[0,1]\times C^1(\overline{\Omega })\rightarrow C^1(\overline{\Omega })$ as

$$\begin{aligned} H(t,v): = {\mathcal L}^{-1}(h-tB(x,\nabla v)), \end{aligned}$$

where $u= {\mathcal L}^{-1}(h)$, ${\mathcal L}^{-1}: W^{-1,p'}(\Omega )\rightarrow W_0^{1,p}(\Omega )$, is the solution operator to

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal L} (u) = h&{}\quad x\in \Omega \\ u=0&{}\quad x\in \partial \Omega , \end{array}\right. } \end{aligned}$$

${\mathcal L}(u)=-\Delta _p u +\delta \varphi _p(u)$. Operator H maps bounded sets into relatively compact sets in $C^1(\overline{\Omega })$. In fact, by virtue of (4.5) and Lemma 4.5 ($1<p< N$) such sets are primarily mapped into bounded sets both in $W^{1,p}_0(\Omega )$ and $L^\infty (\Omega )$ (regardless of $t\in [0,1]$). Then, Theorem 1 in [24] implies that such sets are also uniformly bounded in $C^{1,\beta }(\overline{\Omega })$ for some $\beta$ and hence the conclusion follows. The continuity of H is also a consequence of this compactness argument.

We observe now that solving the auxiliary problem (4.6) amounts to solve the fixed point equation

$$\begin{aligned} u=H(t,u)\quad u\in C^1(\overline{\Omega }). \end{aligned}$$

In addition, for fixed $h\in L^\infty (\Omega )$, by Lemmas 4.2, 4.5 and estimates in [24], solutions to (4.6) must lie in an open ball $B = B(0,R)$ in $C^1(\overline{\Omega })$. Thus

$$\begin{aligned} \mathrm{deg}(I-H(t,\cdot ),B,0)=\mathrm{deg}(I-H(0,\cdot ),B,0) = 1, \end{aligned}$$

for all $t\in [0,1]$, where $\mathrm{deg}(\cdot ,\cdot ,\cdot )$ stands for the Leray–Schauder degree. This shows the existence of a solution while uniqueness in the class $C(\overline{\Omega })\cap C^1(\Omega )$ ($\delta >0$) is a consequence of Proposition 2.1.

To solve (4.3) with $h\in W^{1,p'}(\Omega )$ it suffices with approximating h with $h_n\in L^\infty (\Omega )$ and to combine (4.5) with a compactness argument. $\square$

A consequence of Lemma 4.7 is the method of sub and supersolutions which we now state in the framework of problem (4.3). A former general version of next result is presented in [13]. However, being [13] quite prior to the powerful estimates in [24], the regularity degree achieved in the solutions is lower than here.

Lemma 4.8

Assume $h\in L^\infty (\Omega )$, $\delta \in \mathbb {R}$ and let $\underline{u}, \bar{u}\in C^1(\overline{\Omega })$ be a pair of sub an supersolution to (4.3) with $\underline{u}\le \overline{u}$ in $\Omega$. Then there exists a pair $\hat{u},\tilde{u}\in C^{1,\beta }(\overline{\Omega })$ of solutions to (4.3) in the functional interval $[\underline{u},\overline{u}]$ which constitute the minimal and maximal solution, respectively, to (4.3) in that interval.

Proof

Write (4.3) in the equivalent form

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p u + B(x,\nabla u) +(M+\delta )\varphi _p(u)= M \varphi _p(u) + h&{}\quad x\in \Omega \\ \ u = 0&{} \quad x\in \partial \Omega , \end{array}\right. } \end{aligned}$$

(4.9)

with $M+\delta \ge 0$. By introducing the sequence $u_n\in C^1(\overline{\Omega })$ where $u=u_{k+1}$ solves

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p u + B(x,\nabla u) +(M+\delta )\varphi _p(u)= M \varphi _p(u_k) + h&{}\quad x\in \Omega \\ \ u = 0&{} \quad x\in \partial \Omega , \end{array}\right. } \end{aligned}$$

(4.10)

then $\hat{u}$ is obtained as the $C^1(\overline{\Omega })$ limit $\hat{u} = \lim u_n= \sup u_n$ when the initial datum $u_0 = \underline{u}$, meanwhile $\tilde{u} = \lim u_n= \inf u_n$ with $u_0=\overline{u}$. In fact, to prove the convergence, Proposition 2.1 is used to show the monotonicity of $u_n$ in both cases while the previous estimates together with [24] are additionally employed to achieve convergence in $C^1(\overline{\Omega })$. $\square$

We finish the “Appendix” with an statement containing a formulation of Hopf maximum principle for equations in divergence form. It is a generalization to N dimensions of Lemma 7 in [16] where such result is proved for a more restrictive class of operators in dimension $N=2$ (see also Remark 2 in p. 35 of [16]). A later improved version, still for $N=2$, appeared in [21].

Lemma 4.9

Let $\Omega \subset \mathbb {R}^N$ be a $C^2$ bounded domain while $u\in C^1(\overline{\Omega })$ solves

$$\begin{aligned} -\sum _{i,j=1}^N\frac{\partial }{\partial x_i}\left( a_{ij}(x)\frac{\partial u}{\partial x_j}\right) + \sum _{i=1}^N b_i(x)\frac{\partial u}{\partial x_i}\ge 0, \end{aligned}$$

(4.11)

in the weak sense in $\Omega$, where the matrix $A=(a_{ij})$ is uniformly elliptic in $\Omega$, $a_{ij}\in C^\alpha (\overline{\Omega })$, $0< \alpha < 1$, $1\le i,j\le N$, $b_i\in L^\infty (\Omega )$, $1 \le i \le N$. Assume that for a certain $x_0\in \partial \Omega$,

$$\begin{aligned} u(x)>u(x_0)\quad \text {for all}\ \ x\in B\cap \Omega , \end{aligned}$$

where B is a tangent ball to $\partial \Omega$ at $x_0$. Then, the strict inequality

$$\begin{aligned} \frac{\partial u}{\partial v}(x_0)< 0 \end{aligned}$$

holds true for any unitary outer vector v at $x_0$.

The result also holds valid if a zero order term c(x)u, $c\in L^\infty (\Omega )$, is included in the left hand side of (4.11) provided that standard conditions on both the signs of c and u are met. We refer the interested reader to [12] for a proof of Lemma 4.9 and these “extra” features.