Nonhomogeneous Dirichlet problems for the p-Laplacian

Abstract

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems $-\Delta _p \,u=f_\lambda (x,u)$ in $\Omega , u = \varphi \ \text{ on } ~\partial \Omega$, where $\lambda > 0$ is a parameter. The family we consider includes in particular the Pohozaev type equation $-\Delta _p \,u = \lambda u^{p^{*}-1}$. The main new feature is the consideration of the p-Laplacian $-\Delta _p$ together with a nonzero boundary condition $\varphi$. In order to deal with these nonhomogeneous problems, it has been important to extend to this new context several basic results such as the Brezis-Nirenberg theorem on local minimization in $W^{1,p}$ and $C^1$, a $C^{1,\alpha }$ estimate for a family of equations with critical growth, and a variational approach to the method of upper–lower solutions. These extensions have an independent interest for applications in other situations.

Appendix

1.1 Upper–lower solution

Let g(x, s) be a Carathéodory function on $\Omega \times \mathbb {R}$ with the property that for any $s_0 > 0$, there exists a constant A such that $|g (x,s)| \le A$ for a.e. $x \in \Omega$ and all $s \in [-s_0, s_0]$. Let $u_0 \in W^{1,p} (\Omega ) \cap L^\infty (\Omega )$ and $u_1 \in W^{1-1/p,p}(\partial \Omega ) \cap L^\infty (\partial \Omega )$.

A function $u \in W^{1,p} (\Omega ) \cap L^\infty (\Omega )$ is called a lower solution of the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p (u + u_0)= g (x,u) \ \text{ in } \ \Omega ,\\ u= u_1 \ \text{ on } \ \partial \Omega \end{array}\right. } \end{aligned}$$

(4.1)

if $u \le u_1 \ \text{ on } \ \partial \Omega$ and

$$\begin{aligned} \int _\Omega |\nabla (u + u_0) |^{p-2} \nabla (u + u_0) \nabla \psi \ \le \ \int _\Omega \ g (x,u) \psi \end{aligned}$$

for all $\psi \in W^{1,p}_0 (\Omega )$ with $\psi \ge 0$ in $\Omega$. An upper solution is defined similarly.

Proposition 4.1

Assume $\underline{u}$ and $\bar{u}$ are respectively lower and upper solutions for (4.1), with $\underline{u} \le \bar{u}$ in $\Omega .$ Consider the functional

$$\begin{aligned} \Phi (u) := \frac{1}{p} \int _\Omega |\nabla (u + u_0)|^p - G (x,u) \end{aligned}$$

where $G (x,s):= \int ^s_0 g (x,t) dt$, and the interval

$$\begin{aligned} M:= \{u \in W^{1,p} (\Omega ) : \underline{u} \le u \le \bar{u} \ \text{ in } \ \Omega \ \text{ and } \ u = u_1 \ \text{ on } \ \partial \Omega \}. \end{aligned}$$

Then $\Phi$ achieves its infimum on M at some u, and any such minimizer u is a solution of (4.1).

Proof

It is adapted from [26] and [13] which deal with the homogeneous case $u_0 = u_1 = 0$, for $p=2$ and $1<p<\infty$ respectively. By coercivity and weak lower semicontinuity, one easily sees that the infimum of $\Phi$ on M is achieved at some u. The proof that any such minimizer u solves (4.1) follows exactly the same argument as in Proposition 3.1 of [13], using in particular the monotonicity of $-\Delta _p$. One point worth mentioning here is that, with the notations from p. 729 in [13], the function $\varphi ^\epsilon := (u + \epsilon \varphi - \bar{u})^+$ vanishes on $\partial \Omega$ and so is an admissible testing function in the definition of an upper solution. $\square$

Note that the general restriction $p < N$ plays no role in Proposition 4.1.

1.2 Regularity and a priori estimate

The following result provides some regularity for the solutions of (1.1) in the subcritical and critical case. It also gives an a priori estimate which will be of importance in the study of local minimization in $C^1$ and $W^{1,p}$ in the next subsection.

Proposition 4.2

Let a sequence $u_k \in W^{1,p}(\Omega )$ satisfy

$$\begin{aligned} -\Delta _p u_k = h_k (x,u_k) \ \text{ in } \ \Omega \end{aligned}$$

(4.2)

where the Carathéodory functions $h_k (x,s)$ verify the uniform growth condition

$$\begin{aligned} |h_k (x,s)| \le C_1 + C_2 |s|^{p^*-1} \end{aligned}$$

for a.e. $x \in \Omega$, all $s \in \mathbb {R}$ and all k. Assume that $u_k$ remains bounded in $W^{1,p} (\Omega )$ and that $u_k | _{\partial \Omega }$ belongs to $L^\infty (\partial \Omega )$ and remains bounded in $L^\infty (\partial \Omega )$. Moreover assume that $\int _E |u_k|^{p^*} \rightarrow 0$ as $|E| \rightarrow 0$, uniformly with respect to k. Then $u_k$ belongs to $L^\infty (\Omega )$ and remains bounded in $L^\infty (\Omega )$. If in addition $u_k|_{\partial \Omega }$ belongs to $C^{1,\alpha } (\partial \Omega )$ and remains bounded in $C^{1,\alpha } (\partial \Omega )$, then $u_k$ belongs to $C^{1,\beta } (\bar{\Omega })$ for some $\beta = \beta (\alpha , N, p)$ and remains bounded in $C^{1,\beta } (\bar{\Omega })$.

For a single $u_k$ which vanishes on $\partial \Omega$, the regularity result provided by Proposition 4.2 was obtained in [16, 18]. The a priori estimate for $u_k$ vanishing on $\partial \Omega$ was obtained in [13]. Comments on the necessity of the requirement relative to the uniform equiintegrability of $u_k$ in $L^{p^*} (\Omega )$ can be found in [13, p. 731].

Remark 4.3

In the subcritical case where $h_k$ satisfies $|h_k (x,s)| \le C_1 + C_2 |s|^{q-1}$ for some $q < p^*$, no requirement of uniform equiintegrability is needed. The conclusion of Proposition 4.2 then follows directly by using successively [3] and [20].

Remark 4.4

The same conclusions as above hold for $u_k$ if (4.2) is replaced by $-\Delta _p (u_k + v_k) = h_k (x , u_k)$ provided $v_k$ remains bounded in $W^{1,p} (\Omega )$ as well as in $L^\infty (\Omega )$ (or $C^{1,\alpha } (\bar{\Omega })$ for the last part of the conclusion). Just apply Proposition 4.2 to $w_k:= u_k + v_k$.

Proof of Proposition 4.2

We break it in 3 steps: (i) there exists $r > p^*$ such that $u_k$ remains bounded in $L^r (\Omega )$, (ii) $u_k$ remains bounded in $L^\infty (\Omega )$, (iii) under the additional assumption on the $C^{1,\alpha }$ behavior of $u_k$ on $\partial \Omega$, there exists $\beta \in ] 0,1[$ such that $u_k$ remains bounded in $C^{1,\beta } (\bar{\Omega })$.

Proof of step (i). It is obtained by adapting arguments from [16] and [13]. Take $\beta > 1$ with $\beta p < p^*$ and let $l > 0$. Define for $s \ge 0$, as in Moser iteration technique,

$$\begin{aligned} F (s):= {\left\{ \begin{array}{ll} s^\beta \ \text{ if } \ s \le l, \\ \beta l^{\beta - 1} (s -l) + l^\beta \ \text{ if } \ s > l, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} G (s):= {\left\{ \begin{array}{ll} s^{(\beta -1) p + 1} \ \text{ if } \ s \le l, \\ \beta ((\beta - 1) p+1) l^{(\beta -1) p} (s -l) + l^{(\beta -1) p+1} \ \text{ if } \ s > l. \end{array}\right. } \end{aligned}$$

For any non negative $\eta \in C^\infty _c (\mathbb {R}^N)$, consider the function $\eta ^p G ((u_k - M)^+)$ where the constant M satisfies $\Vert u_k\Vert _{L^\infty (\partial \Omega )} \le M$ for all k. Since this function vanishes on $\partial \Omega$, it belongs to $W^{1,p}_0 (\Omega )$ and so is an admissible testing function in (4.2). At this point one can follow word by word the calculations on [16, p. 951–952] to reach

$$\begin{aligned}&\left( \int _\Omega [F((u_k - M)^+]^{p^*} \eta ^{p^*}\right) ^{p/p^*} \le C_1 \int _\Omega |\nabla \eta |^p [F ((u_k - M)^+)]^p \nonumber \\&\quad +\, C_2 \int _\Omega \eta ^p [(u_k - M)^+]^{p^*-p} [F ((u_k - M)^+)]^p + C_3 \end{aligned}$$

(4.3)

for some constants $C_1, C_2, C_3$ independent of $k, \eta , l$. We now use the assumption of uniform equiintegrability to find $R > 0$ such that

$$\begin{aligned} \Vert (u_k - M)^+\Vert ^{p^* - p}_{L p^* (B \cap \Omega )} \le 1/2 C_2 \end{aligned}$$

for any ball B of radius $\le R$ in $\mathbb {R}^N$ and for all k. Restricting $\eta$ to have support in such a ball and using Hölder inequality, one obtains

$$\begin{aligned} C_2 \int _\Omega \eta ^p [(u_k - M)^+]^{p^* - p} \,[F((u_k - M)^+)]^p \le \frac{1}{2} \left( \int _\Omega \eta ^{p^*} [F((u_k - M)^+)]^{p^*}\right) ^{p/p^*}. \end{aligned}$$

Consequently (4.3) implies

$$\begin{aligned} \left( \int _\Omega \eta ^{p^*}[ F((u_k - M)^+)]^{p^*}\right) ^{p/p^*} \le C (\eta ) \int _\Omega [F((u_k - M)^+)]^p + C_3 \end{aligned}$$

with $C (\eta )$ a constant independent of k, l. Letting $l \rightarrow \infty$ yields

$$\begin{aligned} \left( \int _\Omega \eta ^{p^*}((u_k - M)^+)^{\beta p^*}\right) ^{p/p^*}\le C (\eta ) \int _\Omega ((u_k - M)^+)^{\beta p} + C_3. \end{aligned}$$

Covering $\bar{\Omega }$ with a finite number of balls of radius $\le R$, one deduces that $(u_k - M)^+$ remains bounded in $L^{\beta p^*} (\Omega )$. And a similar argument applied to $(u_k + M)^-$ yields the conclusion of step (i).

Proof of step (ii). Theorem 7.1 from [19] can be applied to (4.2) to derive that $u_k$ remains bounded in $L^\infty (\Omega )$. In fact a particular case of this result of [19] suffices here, which is recalled as Lemma 4.5 below. Here are some details on the application of this Lemma 4.5 to (4.2): (4.4) clearly holds with $\varphi _1 \equiv 0$; taking $r > p^*$ as given by step (i), one can verify (4.5) with $\alpha _2 = p^* - 1$ and $\varphi _2$ a suitable constant by picking $r_2$ sufficiently large.

Proof of step (iii). Once the $L^\infty$ estimate of step (ii) is obtained, the global regularity result of [20] can be applied and yields the conclusion of step (iii). $\square$

Lemma 4.5

(cf. [19]). Let $u \in W^{1,p} (\Omega ) \cap L^r (\Omega )$ with $r \ge p^*$ satisfy $u /_{\partial \Omega } \in L^\infty (\partial \Omega )$ and

$$\begin{aligned} \int _\Omega a (x,u, \nabla u) \nabla v = \int _\Omega b (x,u) v \end{aligned}$$

for v of the form $(u - C)^+$ or $(u + C)^-$, C any constant with $C > \Vert u\Vert _{L^\infty (\partial \Omega )}$. Here the function $a (x,s,\eta )$ and b(x, s) are assumed to verify for $x \in \Omega , s \in \mathbb {R}$ and $\eta \in \mathbb {R}^N$,

$$\begin{aligned} \langle a (x,s, \eta ), \eta \rangle\ge & {} \nu |\eta |^P - (1 + |s|^{\alpha _1}) \varphi _1 (x), \end{aligned}$$

(4.4)

$$\begin{aligned} (\text{ sign } \ s) b (x,s)\le & {} (1 + |s|^{\alpha _2}) \varphi _2 (x), \end{aligned}$$

(4.5)

with $\nu$ a positive constant, $0 \le \varphi _i \in L^{r_i} (\Omega ), \ r_i > N/p, \ 0 \le \alpha _1 < p \frac{N + r}{N-1} - \frac{r}{r_1}$ and $0 \le \alpha _2 < p \frac{N + r}{N} - 1 - \frac{r}{r_2}$. Then $u \in L^\infty (\Omega )$ and $||u||_{L^\infty (\Omega )}$ can be estimated in terms of $\Vert u\Vert _{L^r(\Omega )},$ $\Vert u\Vert _{L^\infty (\partial \Omega )}, \ \nu , \alpha _i, \Vert \varphi _i\Vert _{L^{r_i} \Omega }$ and $\Omega .$

1.3 Local minimization in $C^1$ and $W^{1,p}$

In our study of multiplicity, we use the following extension of a well-known result from [8].

Proposition 4.6

Let $\Phi (u)$ be a functional of the form $\Phi (u):= \frac{1}{p} \int _\Omega |\nabla u|^P - \int _\Omega G (x, u)$ where $G (x,s) : = \int ^s_0 g (x,t \ dt)$ and g satisfies the growth condition

$$\begin{aligned} |g (x,s)| \le d_1 + d_2 |s|^{p^* - 1} \end{aligned}$$

for some constants $d_1, d_2$. Let

$$\begin{aligned} \mathcal {M} := \{u \in W^{1,p} (\Omega ): u|_{\partial \Omega } = \varphi \} \end{aligned}$$

where $\varphi \in C^{1,\alpha } (\partial \Omega )$. Let $u_0$ be a local minimum of $\Phi$ on $\mathcal {M}$ for the $C^1 (\bar{\Omega })$ topology, i.e. $\Phi (u_0) \le \Phi (u_0 + w)$ for some $\epsilon > 0$ and all $w \in C^1_0 (\bar{\Omega })$ with $\Vert w\Vert _{C^1_0 (\bar{\Omega )}} \le \epsilon$. Then $u_0 \in C^{1,\beta } (\bar{\Omega })$ for some $\beta \in ]0,1[$ and $u_0$ is a local minimizer of $\Phi$ on $\mathcal {M}$ for the $W^{1,p}(\Omega )$ topology, i.e. $\Phi (u_0) \le \Phi (u_0 + w)$ for some $\epsilon _1 > 0$ and all $w \in W^{1,p}_0 (\Omega )$ with $\Vert w\Vert _{W^{1,p}_0 (\Omega )} \le \epsilon _1$.

Proposition 4.6 was proved in [8] for $p = 2$ and $\varphi \equiv 0$, and in [13] for $1< p < \infty$ and $\varphi \equiv 0$ (see also [10, 17], ... for related works). The proof below is adapted from [13].

Remark 4.7

The same conclusion holds if one replaces $\Phi (u)$ by

$$\begin{aligned} \tilde{\Phi } (u) : = \frac{1}{p} \int _\Omega |\nabla (u + u_1)|^p \ \ -\int _\Omega G (x,u) \end{aligned}$$

where $u_1$ is a fixed function in $C^{1,\alpha } (\bar{\Omega })$. It suffices to apply Proposition 4.6 to $v : = u + u_1$.

Proof of Proposition 4.6

Since $u_0$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p u_0 = g (x, u_0) \ \text{ in } \ \Omega , \\ u_0 = \varphi \ \text{ on } \ \partial \Omega , \end{array}\right. } \end{aligned}$$

Proposition 4.2 implies $u_0 \in C^{1, \beta } (\bar{\Omega }).$ Assume by contradiction that $u_0$ is not a local minimum of $\Phi$ on $\mathcal {M}$ for the $W^{1,p} (\Omega )$ topology. This means that for any $\epsilon > 0$, there exists $v_\epsilon \in \mathcal {M}$ with $\Vert v_\epsilon - u_0\Vert _{W^{1,p}} \le \epsilon$ and $\Phi (v_\epsilon ) < \Phi (u_0)$. For later use of Lagrange multiplier rule, it will be convenient to use only a consequence of that, namely $\Vert v_\epsilon -u_0\Vert _{L^{p^*}} \le \epsilon$ and $\Phi (v_\epsilon ) < \Phi (u_0)$.

The argument exactly follows the same lines as in [13] and we only sketch it. One introduces the same truncated functional $\Phi _j$ as in [13] and finds $j_\epsilon$ such that $\Phi _{j_\epsilon } (v_\epsilon ) < \Phi (u_0)$. One also observes that $\Phi _{j_\epsilon }$ achieves it infimum on $\{v \in \mathcal {M} : \Vert v - u_0\Vert _{L^{p^*}} \le \epsilon \}$ at some $u_\epsilon$. The main point of the proof then consists in proving that $u_\epsilon$ remains bounded in some $C^{1,\beta } (\bar{\Omega })$ as $\epsilon \rightarrow 0$. To prove this latter fact, one writes the Euler equation satisfied by $u_\epsilon :$

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} -\Delta _p u_\epsilon = g_{j_\epsilon } (x , u_\epsilon ) + \mu _\epsilon |u_\epsilon - u_0|^{p^* - 2} (u_\epsilon - u_0)\ \text{ in }\,\,\, {\Omega },\\ u_\epsilon = \varphi \ \text{ on } \ \partial \Omega . \end{array} \end{array}\right. } \end{aligned}$$

(4.6)

where $g_{j_\epsilon }$ comes from the truncation of g and where $\mu _\epsilon$ is a Lagrange multiplier associated to the constraint $\Vert v - u_0\Vert _{L^{P^*}} \le \epsilon$. Since $u_0 - u_\epsilon$ belongs to $W^{1,p}_0 (\Omega )$, it is an admissible testing function in (4.6), and one can deduce $\mu _\epsilon \le 0$. The rest of the argument is then identical to that in [13], using the estimate of Proposition 4.2. The only point worth mentioning is that one should take M in formula (3.17) from [13] larger than $\Vert \varphi \Vert _{L^\infty (\partial \Omega )}$ in order to guarantee that $(u_\epsilon - M)^+$ and $(u_\epsilon + M)^-$ are admissible testing function in (4.6). $\square$