On the antimaximum principle for the p-Laplacian and its sublinear perturbations

Abstract

We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $-\Delta _p u = \lambda \,m(x)|u|^{p-2}u+ \eta \,a(x)|u|^{q-2}u+ f(x)$ in a bounded domain $\Omega \subset \mathbb {R}^N$, where $q<p$. Under certain regularity and qualitative assumptions on the weights m, a and the source function f, we identify ranges of the parameters $\lambda$ and $\eta$ for which solutions satisfy maximum and antimaximum principles in weak and strong forms. Some of our results, especially on the validity of the antimaximum principle under low regularity assumptions, are new for the unperturbed problem with $\eta =0$, and among them there are results providing new information even in the linear case $p=2$. In particular, we show that for any $p>1$ solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the p-Laplacian provided $m,f \in L^\gamma (\Omega )$ with $\gamma >N$. For completeness, we also investigate the existence of solutions.

Appendices

Regularity

We start with an $L^\infty (\Omega )$-bound for solutions of the problem ($\mathcal {P}$). In view of the Morrey lemma, it will be sufficient to investigate only the case $N \ge p$. The proof uses the classical bootstrap argument and we present it sketchily for completeness.

Proposition A.1

Let $N \ge p>1$. Assume that $m, a, f \in L^\gamma (\Omega )$ for some $\gamma >N/p$, and $M_1 > 0$ is any constant such that $\Vert m\Vert _\gamma , \Vert a\Vert _\gamma , \Vert f\Vert _\gamma \le M_1$. Let r be such that $p\gamma ^\prime< r < p^*$, where $\gamma ^\prime = \gamma /(\gamma -1)$ and $p^*$ is defined in (3.1). Assume that $|\lambda |, |\eta | \le M_2$ for some $M_2 > 0$. Then there exists $C=C(|\Omega |,M_1,M_2,p,q,\gamma ,r)>0$ such that any solution u of ($\mathcal {P}$) satisfies

$$\begin{aligned} \Vert u\Vert _\infty \le C \left( \,1+\Vert u\Vert _{r}\right) . \end{aligned}$$

(A.1)

Proof

Let $C_*>0$ be the best constant of the embedding $W_0^{1,p}(\Omega ) \hookrightarrow L^r(\Omega )$, and set $M_0 =\max \{1,|\Omega |\}$. Let u be any solution of ($\mathcal {P}$) and denote, for brevity, $v = u_+$. For any $l>0$ and $M>0$, we denote $v_M = \min \{v,M\}$ and take $v_M^{l+1} \in W_0^{1,p}(\Omega )$ as a test function for ($\mathcal {P}$). Concerning the left-hand side of ($\mathcal {P}$), we have

$$\begin{aligned} \int _\Omega&|\nabla u|^{p-2}\nabla u\nabla (v_M^{l+1})\,dx =(l+1)\int _\Omega v_M^l|\nabla v_M|^p\,dx =(l+1)\left( \frac{p}{p+l}\right) ^p\,\int _\Omega |\nabla (v_M^{1+l/p})|^p\,dx\nonumber \\&\ge (l+1)\left( \frac{p}{C_*(p+l)}\right) ^p \Vert v_M^{1+l/p}\Vert _{r}^p \ge \frac{1}{C_*^{p}(p+l)^p} \Vert v_M\Vert _{(p+l)r/p}^{p+l}. \end{aligned}$$

(A.2)

On the other hand, temporarily assuming that $v\in L^{(p+l)\gamma ^\prime }(\Omega )$, we apply the Hölder inequality to estimate the right-hand side of ($\mathcal {P}$) from above by the following expressions:

$$\begin{aligned}&|\lambda |\Vert m\Vert _{\gamma }\Vert v\Vert _{(p+l)\gamma ^\prime }^{p+l} +|\eta |\Vert a\Vert _{\gamma }\Vert v\Vert _{(q+l)\gamma ^\prime }^{q+l} +\Vert f\Vert _{\gamma } \Vert v\Vert _{(1+l)\gamma ^\prime }^{1+l}\nonumber \\&\quad \le M_1 \Big (M_2 + M_2 M_0^\frac{(p-q)}{(p+l)\gamma ^\prime } + M_0^\frac{(p-1)}{(p+l)\gamma ^\prime } \Big ) \max \{1,\Vert v\Vert _{(p+l)\gamma ^\prime }^{p+l}\}\nonumber \\&\quad \le M_1 \Big (M_2 + M_2 M_0^\frac{(p-q)}{p\gamma ^\prime } + M_0^\frac{(p-1)}{p\gamma ^\prime } \Big ) \max \{1,\Vert v\Vert _{(p+l)\gamma ^\prime }^{p+l}\}. \end{aligned}$$

(A.3)

Consequently, if $v\in L^{(p+l)\gamma ^\prime }(\Omega )$ for some $l>0$, then we let $M\rightarrow \infty$ and deduce from (A.2) and (A.3) that $v\in L^{(p+l)r/p}(\Omega )$ and

$$\begin{aligned} \Vert v\Vert _{(p+l)r/p} \le \left( C(p+l)^p\right) ^{1/(p+l)}\, \max \{1,\Vert v\Vert _{(p+l)\gamma ^\prime }\}, \end{aligned}$$

(A.4)

where $C\ge 1$ is a constant independent of $l>0$ and v. Now we define a sequence $\{l_m\}$ as follows:

$$\begin{aligned} (p+l_0)\gamma ^\prime =r\quad \textrm{and}\quad l_{m+1}=\frac{p+l_m}{P}-p, \quad \textrm{where}\quad P:=\frac{\,\gamma ^\prime p\,}{r}<1. \end{aligned}$$

In particular, we do have $v\in L^{(p+l_0)\gamma ^\prime }(\Omega )$. Denoting $d_m = (C(p+l_m)^p)^{1/(p+l_m)}$, we infer from (A.4) that

$$\begin{aligned} \Vert v\Vert _{(p+l_{m+1})\gamma ^\prime }\le d_m \max \{1,\Vert v\Vert _{(p+l_m)\gamma ^\prime }\} \le \max \{1,\Vert v\Vert _{(p+l_0)\gamma ^\prime }\}\prod _{k=0}^m d_k \end{aligned}$$

(A.5)

for every m. Let us show that $\prod _{k=0}^\infty d_k$ is finite. We have

$$\begin{aligned} \log \prod _{k=0}^\infty d_k= \log C\,\sum _{k=0}^\infty \frac{1}{p+l_k}+ p\,\sum _{k=0}^\infty \frac{\log (p+l_k)}{p+l_k}. \end{aligned}$$

(A.6)

Noting that $P=(p+l_k)/(p+l_{k+1})<1$, we deduce that $l_k \rightarrow \infty$ and get

$$\begin{aligned} \frac{\log (p+l_{k+1})}{p+l_{k+1}}\, \frac{p+l_k}{\log (p+l_k)} =P\,\left( 1-\frac{\log P}{\log (p+l_k)} \right) \rightarrow P<1 \quad \text {as}~k\rightarrow \infty . \end{aligned}$$

Hence, the series in (A.6) are convergent, which completes the proof of the boundedness of v by letting $m\rightarrow \infty$ in (A.5). Repeating the same procedure with $v=-u_-$, we obtain the boundedness of $u_-$ and hence of u in the form (A.1). $\square$

After we established the boundedness of solutions of ($\mathcal {P}$), we can consider the whole right-hand side of ($\mathcal {P}$) as a function which maps $\Omega$ to $\mathbb {R}$ and investigate the regularity of solutions of the corresponding Poisson problem in order to get the regularity of solutions of ($\mathcal {P}$). Namely, we consider the problem

$$\begin{aligned} \left\{ \begin{aligned} -\Delta _p u&= g(x){} & {} \text {in}\ \Omega , \\ u&=0{} & {} \text {on}\ \partial \Omega . \end{aligned} \right. \end{aligned}$$

(A.7)

The following result on the local Hölder regularity of the (unique) solution of (A.7) is well known (see, e.g., [48, Theorem 7.3.1]) and we omit the proof.

Proposition A.2

Let $\Vert g\Vert _\gamma \le M$ for some $M >0$, where $\gamma >N/p$ if $N\ge p$ and $\gamma =1$ if $N<p$. Then the solution $u \in W_0^{1,p}(\Omega )$ of (A.7) satisfies $u \in C(\Omega ) \cap L^\infty (\Omega )$. Moreover, there exists $\beta = \beta (p,N,\gamma ) \in (0,1)$ such that for any compact subset $K \subset \Omega$ there exists $C = C(\Omega ,K,M,p,\gamma )>0$ such that $\Vert u\Vert _{C^{0,\beta }(K)} \le C$.

Let us discuss a higher regularity of solutions of (A.7) under a higher integrability assumption on the source function g. The proof of the following result is inspired by [46, Proposition 2.1] (see also [2, Proposition 2.1] and compare with [19, Corollary]). We expand the approach of [46, Proposition 2.1] in order to provide more explicit dependence of the regularity of solutions of (A.7) on g, which is necessary for the proofs of our main results formulated under the assumptions ($\mathcal {O}$), $(\mathcal {M})$, $(\mathcal {A})$, $(\mathcal {F})$.

Proposition A.3

Let $\Omega$ satisfy ($\mathcal {O}$). Let $\Vert g\Vert _\gamma \le M$ for some $M >0$ and $\gamma >N$. Then there exist $\beta = \beta (M,p,N,\gamma ) \in (0,1)$ and $C = C(\Omega ,M,p,\gamma )>0$ such that the solution $u \in W_0^{1,p}(\Omega )$ of (A.7) satisfies $u \in C^{1,\beta }(\overline{\Omega })$ and $\Vert u\Vert _{C^{1,\beta }(\overline{\Omega })} \le C$.

Proof

Throughout the proof, we denote by $C>0$ a universal constant, for convenience. First, we consider the following problem for the linear Laplace operator:

$$\begin{aligned} \left\{ \begin{aligned} -\Delta v&= g(x){} & {} \text {in}\ \Omega , \\ v&=0{} & {} \text {on}\ \partial \Omega , \end{aligned} \right. \end{aligned}$$

(A.8)

where the function g is the same as in (A.7). Since $\Omega$ is of class $C^{1,1}$ and $g \in L^\gamma (\Omega )$ with $\gamma >N$, the problem (A.8) has a unique solution $v \in W^{2,\gamma }(\Omega )$, see, e.g., [27, Theorem 9.15] (in fact, here $\gamma >1$ is enough). Moreover, this solution has the following property, see, e.g., [27, Lemma 9.17]:

$$\begin{aligned} \Vert v\Vert _{W^{2,\gamma }(\Omega )} \le C \Vert g\Vert _\gamma , \end{aligned}$$

where C does not depend on v and g. Thanks to the regularity of $\Omega$ and the assumption $\gamma >N$, the embedding $W^{2,\gamma }(\Omega ) \hookrightarrow C^{1,\kappa }(\overline{\Omega })$ is continuous with $\kappa = 1-\frac{N}{\gamma } \in (0,1)$, see, e.g., [27, Theorem 7.26] (in fact, here $C^{0,1}$-regularity of $\Omega$ is enough). Consequently,

$$\begin{aligned} \Vert v\Vert _{C^{1,\kappa }(\overline{\Omega })} \le C \Vert v\Vert _{W^{2,\gamma }(\Omega )} \le C \Vert g\Vert _\gamma \le C M, \end{aligned}$$

where C is independent of v. Recall, for convenience, that

$$\begin{aligned} \Vert v\Vert _{C^{1,\kappa }(\overline{\Omega })}:= \sup _{x \in \Omega } |v(x)| + \max _{i=1,\dots ,N}\sup _{x \in \Omega } |v'_{x_i}(x)| + \max _{i=1,\dots ,N}\sup _{x,y \in \Omega ,~x \ne y} \frac{|v'_{x_i}(x)-v'_{x_i}(y)|}{|x-y|^\kappa }. \end{aligned}$$

Denoting $V(x) = \nabla v(x)$, we have

$$\begin{aligned} V \in C^{0,\kappa }(\overline{\Omega }; \mathbb {R}^N). \end{aligned}$$

(A.9)

Subtracting (A.8) from (A.7), we see that the solution u of (A.7) weakly solves the problem

$$\begin{aligned} \left\{ \begin{aligned} -\text {div}\left( |\nabla u|^{p-2} \nabla u - V(x)\right)&= 0{} & {} \text {in}\ \Omega , \\ u&=0{} & {} \text {on}\ \partial \Omega . \end{aligned} \right. \end{aligned}$$

(A.10)

Let us show that the regularity result [39, Theorem 1] is applicable to (A.10). Denote $A(x,z) = |z|^{p-2} z - V(x)$ and $a^{ij}(z) = \frac{\partial A^i(x,z)}{\partial z_j}$, $z \in \mathbb {R}^N$. The matrix $(a^{ij}(z))$ is a symmetric $N \times N$-matrix corresponding to the linearization of the p-Laplacian and we have

$$\begin{aligned} (a^{ij}(z)) = |z|^{p-2} \left( I + (p-2) \frac{z \otimes z}{|z|^2}\right) , \quad z \in \mathbb {R}^N {\setminus } \{0\}, \end{aligned}$$

where $z \otimes z:= (z_i z_j)$ is a matrix. We set $(a^{ij}(0))$ to be a zero matrix. It is not hard to see that

$$\begin{aligned} \min \{1,p-1\}|z|^{p-2} |\xi |^2 \leqslant \sum _{i,j=1}^N a^{ij}(z)\xi _i \xi _j \leqslant \max \{1,p-1\} |z|^{p-2} |\xi |^2 \end{aligned}$$

for any $z,\xi \in \mathbb {R}^N$, see, e.g., [53, Section 5.1]. Thanks to (A.9), we have the following estimate for all $x,y \in \overline{\Omega }$ and $z \in \mathbb {R}^N$:

$$\begin{aligned} |A(x,z)-A(y,z)| = |V(x)-V(y)| \le C |x-y|^{\kappa }, \end{aligned}$$

where C depends on M but does not depend on v, x, y, z. We also mention that since $\Omega$ satisfies ($\mathcal {O}$), $\Omega$ automatically belongs to the class $C^{1,\kappa }$.

Finally, we recall that the solution u of (A.7) is bounded. More precisely, in the case $N \ge p$, Proposition A.1 gives the bound

$$\begin{aligned} \Vert u\Vert _\infty \le C\,(\,1+\Vert u\Vert _{r}), \end{aligned}$$

(A.11)

where $p\gamma ^\prime< r < p^*$ and C does not depend on u, and a similar bound holds in the case $N<p$ due to the Morrey lemma. Notice that since $\gamma >N$, we have $\gamma ' < p^*$. Since u satisfies $\int _\Omega |\nabla u|^p \, dx = \int _\Omega g u \,dx$, we use the Hölder inequality and the continuity of the embedding $W_0^{1,p}(\Omega ) \hookrightarrow L^{\gamma '}(\Omega )$ to deduce that

$$\begin{aligned} \Vert \nabla u\Vert _p^p \le \Vert g\Vert _\gamma \Vert u\Vert _{\gamma '} \le C \Vert g\Vert _\gamma \Vert \nabla u\Vert _{p}, \end{aligned}$$

which yields

$$\begin{aligned} \Vert u\Vert _{r} \le C \Vert \nabla u\Vert _p \le C \Vert g\Vert _\gamma ^\frac{1}{p-1} \le C M^\frac{1}{p-1}, \end{aligned}$$

where C does not depend on M and u. Combining this estimate with (A.11), we finally arrive at the bound $\Vert u\Vert _\infty \le C$, where C is independent of u.

Thus, all the requirements of [39, Theorem 1] are satisfied, which guarantees that $u \in C^{1,\beta }(\overline{\Omega })$, where $\beta = \beta (M,p,N,\gamma ) \in (0,1)$ and $\Vert u\Vert _{C^{1,\beta }(\overline{\Omega })} \le C(\Omega ,M,p,\gamma )$. $\square$

Weak form of the Picone inequality

In the proofs of Theorem 2.12 and Proposition 4.2 (and hence of Theorem 2.9 (ii)), we need to employ a version of the standard Picone inequality [3, Theorem 1.1] applicable to purely Sobolev functions, i.e., when no a priori information on the a.e.-differentiability is available. We start with the following auxiliary results.

Lemma B.1

Let $\varphi \in W^{1,p}(\Omega ) \cap L^\infty (\Omega )$, $u \in W^{1,p}(\Omega )$, and $\varepsilon >0$. Then $|\varphi |^p/(|u|+\varepsilon )^{p-1} \in W^{1,p}(\Omega ) \cap L^\infty (\Omega )$ and its weak gradient is expressed as follows:

$$\begin{aligned} \nabla \left( \frac{|\varphi |^p}{(|u|+\varepsilon )^{p-1}}\right) = p \frac{|\varphi |^{p-2}\varphi }{(|u|+\varepsilon )^{p-1}} \nabla \varphi - (p-1) \frac{|\varphi |^{p}}{(|u|+\varepsilon )^{p}} \left( \nabla u_+ + \nabla u_- \right) . \end{aligned}$$

(B.1)

If, in addition, $\varphi \in W^{1,p}_0(\Omega )$, then $|\varphi |^p/(|u|+\varepsilon )^{p-1} \in W^{1,p}_0(\Omega )$.

Proof

The proof is based on classical arguments, so we will be sketchy. First, we observe that $|\varphi |^p \in W^{1,p}(\Omega ) \cap L^\infty (\Omega )$. Indeed, since $\varphi \in L^\infty (\Omega )$, we can find a function $G \in C^1(\mathbb {R})$ such that $G(s)=|s|^p$ for $s \in [-\Vert \varphi \Vert _\infty ,\Vert \varphi \Vert _\infty ]$ and $|G'(s)| \le M$ for all $s \in \mathbb {R}$ and some uniform constant $M>0$. Then [30, Theorem 1.18] ensures that $G(\varphi ) \equiv |\varphi |^p \in W^{1,p}(\Omega )$ and its weak gradient is calculated according to the classical rules. Clearly, we also have $|\varphi |^p \in L^\infty (\Omega )$.

It can be shown in a similar way that $1/(|u|+\varepsilon )^{p-1} \in W^{1,p}(\Omega ) \cap L^\infty (\Omega )$. Indeed, we can find a function $H \in C^1(\mathbb {R})$ such that $H(s) = 1/s^{p-1}$ for $s \in [\varepsilon ,\infty )$ and $|H'(s)| \le M$ for all $s \in \mathbb {R}$ and some uniform constant $M>0$. Since $\Omega$ is bounded, we have $|u|+\varepsilon \in W^{1,p}(\Omega )$. Hence, we deduce from [30, Theorem 1.18] that $H(|u|+\varepsilon ) \equiv 1/(|u|+\varepsilon )^{p-1} \in W^{1,p}(\Omega )$, and its weak gradient can be expanded by the classical rules. Since $1/(|u|+\varepsilon )^{p-1} \le 1/\varepsilon ^{p-1}$, we conclude that $1/(|u|+\varepsilon )^{p-1} \in L^\infty (\Omega )$.

Applying now [30, Theorem 1.24 (i)], we see that $|\varphi |^p/(|u|+\varepsilon )^{p-1} \in W^{1,p}(\Omega ) \cap L^\infty (\Omega )$ and its weak gradient is given by the expression (B.1). If we additionally assume that $\varphi \in W^{1,p}_0(\Omega )$, then, by a simple amendment of the proof of [30, Theorem 1.18], we have $|\varphi |^p \in W_0^{1,p}(\Omega )$, and hence $|\varphi |^p/(|u|+\varepsilon )^{p-1} \in W_0^{1,p}(\Omega )$ by [30, Theorem 1.24 (ii)]. $\square$

Under stronger requirements on the functions $\varphi$ and u, we can omit $\varepsilon$ in Lemma B.1.

Lemma B.2

Let $\varphi \in W^{1,p}(\Omega ) \cap L^\infty (\Omega )$ and $u \in W^{1,p}(\Omega )$ be such that $K:=\textrm{supp}\, \varphi \subset \Omega$ and $\mathrm {ess\,inf}_{K}\,|u| > 0$. Then $|\varphi |^p/|u|^{p-1} \in W_0^{1,p}(\Omega ) \cap L^\infty (\Omega )$ and its weak gradient is expressed as in (B.1) with $\varepsilon =0$.

Proof

Arguments are similar to those from the proof of Lemma B.1 and hence we omit the details. $\square$

In view of the expression (B.1), one can argue exactly as in the proof of [3, Theorem 1.1] to obtain the following weak version of the Picone inequality, see also [1, Section 2] and [53, Section 3.2] for related results.

Lemma B.3

Let $\varphi \in W^{1,p}(\Omega ) \cap L^\infty (\Omega )$ and $u \in W^{1,p}(\Omega )$ be such that $u \ge 0$ a.e. in $\Omega$. Let $\varepsilon >0$. Then the following inequality holds:

$$\begin{aligned} \int _\Omega |\nabla u|^{p-2} \nabla u\nabla \left( \frac{|\varphi |^p}{(u+\varepsilon )^{p-1}}\right) dx \le \int _\Omega |\nabla \varphi |^{p} \,dx. \end{aligned}$$

(B.2)

If, in addition, $K:=\textrm{supp}\, \varphi \subset \Omega$ and $\mathrm {ess\,inf}_{K}\,u > 0$, then (B.2) holds with $\varepsilon =0$.