Existence and uniqueness for a class of nonlinear elliptic equations with measure data

Abstract

We study existence and uniqueness of Radon measure-valued solutions for a class of nonlinear elliptic equations in inhomogeneous media. Solutions are constructed by a regularization procedure which relies on a standard approximation of the measure data and satisfy both a persistence property and a compatibility condition prescribing the structure of their concentrated and diffuse parts with respect to a suitable capacity.

Appendix

Let $\Omega \subset \mathbb {R}^N$ be a bounded open set with smooth boundary $\partial \Omega$. In this section, besides $(H_1)$, we shall always assume that

(A):

(i) $\phi :{\overline{\Omega }}\times \mathbb {R}\rightarrow \mathbb {R}$ is continuous, $\phi (x,0)=0$ for every $x\in {\overline{\Omega }}$;

(ii) for every $x\in {\overline{\Omega }}$ the map $\mathbb {R}\ni z\mapsto \phi (x,z)$ is strictly increasing, and

$$\begin{aligned} \lim _{z\rightarrow \pm \infty }\phi (x,z)=\pm \infty \ \,\text{ uniformly } \text{ for } x\in {\overline{\Omega }}\,. \end{aligned}$$

(A.1)

Let us consider the elliptic equation

$$\begin{aligned} \left\{ \begin{array}{rl} -\text{ div }\big (\mathbf{a}(\nabla \phi (x,u))\big )+u=f&{}\text{ in }\ \,\Omega \,,\\ u=0&{}\text{ on }\ \,\partial \Omega \,. \end{array}\right. \end{aligned}$$

(A.2)

Definition A.1

For every $f\in L^{\infty }(\Omega )$, by a solution of (A.2), we mean any $u\in L^{\infty }(\Omega )$ such that $\phi (\cdot ,u)\in W^{1,p}_0(\Omega )$ and

$$\begin{aligned} \int _{\Omega }{} \mathbf{a}(\nabla \phi (x,u))\cdot \nabla \rho \,dx+\int _{\Omega }u\rho \,dx=\int _{\Omega }f\rho \,dx \end{aligned}$$

(A.3)

for all $\rho \in W^{1,p}_0(\Omega )$.

Observe that the first term in the left-hand side of (A.3) makes sense by assumption $(H_1)$-(i) and the condition $\phi (\cdot ,u)\in W^{1,p}_0(\Omega )$ (see Remark 3.1).

Uniqueness of solutions of (A.2) is a consequence of the following comparison principle.

Proposition A.1

Let $u_1,\,u_2$ be two solutions of (A.2) with data $f_1,\,f_2\in L^{\infty }(\Omega )$, $f_1\le f_2$ a.e. in $\Omega$, respectively. Then $u_1\le u_2$ a.e. in $\Omega$.

Proof

Choosing $\rho =[\phi (x,u_1)-\phi (x,u_2)]^+$ in the weak formulation of $u_i$ ($i=1,2$) and subtracting, we obtain

$$\begin{aligned} \int _{\Omega }\big [\mathbf{a}(\nabla \phi (x,u_1))-\mathbf{a}(\nabla \phi (x,u_2))\big ]\cdot \nabla \rho \,dx+\int _{\Omega }[u_1-u_2]\,\rho \,dx=\int _{\Omega }[f_1-f_2]\,\rho \,dx\,. \end{aligned}$$

(A.4)

By $(H_1)$-(ii) and the assumption $f_1\le f_2$ a.e. in $\Omega$, we get

$$\begin{aligned} \int _{\Omega }[u_1-u_2][\phi (x,u_1)-\phi (x,u_2)]^+\,dx\le 0 \,. \end{aligned}$$

In view of the strictly increasing character of the map $z\mapsto \phi (\cdot ,z)$ (see (A)-(ii)), the previous inequality implies that $u_1\le u_2$ a.e. in $\Omega$. $\square$

Remark A.1

Let u be the solution of (A.2) with data $f\ge 0$ a.e. in $\Omega$ ($f\le 0$, respectively). Since $\mathbf{a}(0)=0$ and $\phi (x,0)=0$ for all $x\in \Omega$, it is immediately seen that $v\equiv 0$ is the solution of (A.2) with data $f=0$. Thus, from Proposition A.1, it follows that $u\ge 0$ a.e. in $\Omega$ ($u\le 0$, respectively).

Proposition A.2

For every $f\in L^{\infty }(\Omega )$, the solution u of (A.2) satisfies

$$\begin{aligned} \Vert u\Vert _{L^1(\Omega )}\le \Vert f\Vert _{L^1(\Omega )}\,. \end{aligned}$$

(A.5)

Proof

For every $K>0$, choosing $\rho =T_K(\phi (\cdot ,u))$ as test function in (A.3) we get

$$\begin{aligned} \alpha _0\int _{\Omega }|\nabla T_K(\phi (x,u))|^p\,dx+\int _{\Omega }u\,T_K(\phi (x,u))\,dx \le \int _{\Omega } f\,T_K(\phi (x,u)\,dx\le K\Vert f\Vert _{L^1(\Omega )} \end{aligned}$$

(here, we have also used $(H_1)$-(i)), whence

$$\begin{aligned} \int _{\Omega }u\,\frac{T_K(\phi (x,u))}{K}\,dx \le \Vert f\Vert _{L^1(\Omega )}\,. \end{aligned}$$

Since $\phi (x,z)>0$ for $z>0$ and $\phi (x,z)<0$ for $z<0$ (recall that $\phi (x,0)=0$ and the map $z\mapsto \phi (x,z)$ is strictly increasing for all $x\in \Omega$), letting $K\rightarrow 0^+$ in the above equality gives

$$\begin{aligned} \int _{\Omega } |u|\,dx=\lim _{K\rightarrow 0^+}\int _{\Omega }u\,\frac{T_K(\phi (x,u))}{K}\,dx \le \Vert f\Vert _{L^1(\Omega )}\,. \end{aligned}$$

$\square$

Let us address the existence part.

Proposition A.3

For every $f\in L^{\infty }(\Omega )$ , there exists a solution of (A.2) in the sense of Definition A.1.

Proof

Let us consider any sequence $\{\phi _{j}\}\subseteq C^{\infty }({\overline{\Omega }}\times \mathbb {R};\mathbb {R})\cap \mathrm{Lip}(\Omega \times \mathbb {R};\mathbb {R})$ such that

$$\begin{aligned}&\phi _{j}\rightarrow \phi \quad \text{ in }\ \,C_{\mathrm{loc}}({\overline{\Omega }}\times \mathbb {R})\ \ \text{ as } j\rightarrow \infty \,, \end{aligned}$$

(A.6)

$$\begin{aligned}&\phi _{j}(x,0)=0\quad \text{ for } \text{ every } x\in {\overline{\Omega }}\,, \end{aligned}$$

(A.7)

$$\begin{aligned}&\partial _{z}\phi _{j}(x,z)\ge \frac{1}{j}\quad \text{ for } \text{ every } (x,z)\in {\overline{\Omega }}\times \mathbb {R}\,. \end{aligned}$$

(A.8)

Applying the results in [24] with

$$\begin{aligned} \mathbf{{\tilde{a}}}(x,z,\xi )=\mathbf{a}_j(\nabla _x\phi _j(x,z)+\partial _z\phi _j(x,z)\xi ) \end{aligned}$$

(here $\mathbf{a}_j(\xi )=\mathbf{a}(\xi )+\frac{1}{j}|\xi |^{p-2}\xi$ for $\xi \ne 0$ and $\mathbf{a}_j(0)=0$), it can be easily checked that for every $f\in L^{\infty }(\Omega )$, there exists $u_j\in W^{1,p}_0(\Omega )$ which solves in the weak sense the elliptic equation

$$\begin{aligned} \left\{ \begin{array}{rl} -\text{ div }\big (\mathbf{a}_j(\nabla \phi _j(x,u_j))\big )+u_j=f&{}\text{ in }\ \,\Omega \,,\\ u_j=0&{}\text{ on }\ \,\partial \Omega \,. \end{array}\right. \end{aligned}$$

(A.9)

The rest of the proof is devoted to take the limit as $j\rightarrow \infty$ in (A.9).

A priori estimates. Choosing $\rho =\phi _j(\cdot ,u_j)$ as test function in (A.9), it easily follows from assumption $(H_1)$–(i) that the sequence $\{\phi _j(\cdot ,u_j)\}$ is bounded in $W^{1,p}_0(\Omega )$. Moreover, by $(H_1)$-(i) we obtain

$$\begin{aligned} \sup _{j\in \mathbb {N}}\int _{\Omega }|\mathbf{a}_j(\nabla \phi _j(x,u_j))|^{\frac{p}{p-1}}dx \le \sup _{j\in \mathbb {N}}\int _{\Omega } (\beta _0+1)\,|\nabla \phi _j(x,u_j)|^pdx <\infty \,. \end{aligned}$$

(A.10)

Let us also prove that

$$\begin{aligned} \{u_j\} \hbox { is bounded in } L^{\infty }(\Omega ). \end{aligned}$$

(A.11)

To this aim, let $K_0:=\Vert f\Vert _{L^{\infty }(\Omega )}$ and set

$$\begin{aligned} M:=\max _{(x,z)\in {\overline{\Omega }}\times [-K_0,K_0]} |\phi (x,z)|\,. \end{aligned}$$

Observe that $M>0$, and by (A.6) there exists $j_M\in \mathbb {N}$ such that

$$\begin{aligned} \max _{(x,z)\in {\overline{\Omega }}\times [-K_0,K_0]} |\phi _j(x,z)|\le 2M\quad \text{ for } \text{ all } j\ge j_M\,. \end{aligned}$$

(A.12)

Moreover, there holds

$$\begin{aligned}&\int _{\Omega } \mathbf{a}_j(\nabla \phi _j(x,u_j))\cdot \nabla [\phi _j(x,u_j)-2M]^+\,dx + \int _{\Omega } u_j[\phi _j(x,u_j)-2M]^+\,dx \\&\quad = \int _{\Omega } f\,[\phi _j(x,u_j)-2M]^+\,dx \le K_0\int _{\Omega } [\phi _j(x,u_j)-2M]^+\,dx\,, \end{aligned}$$

whence (see $(H_1)$-(i))

$$\begin{aligned} \alpha _0\int _{\Omega } |\nabla [\phi _j(x,u_j)-2M]^+|^p\,dx + \int _{\Omega } [u_j-K_0][\phi _j(x,u_j)-2M]^+\,dx\le 0\,. \end{aligned}$$

In view of (A.12) and since $\phi _j(x,u_j)-2M<0$ if $u_j<0$, from the above inequality, we infer that for all $j\ge j_M$,

$$\begin{aligned} \int _{\{u_j\ge K_0\}} [u_j-K_0][\phi _j(x,u_j)-2M]^+\,dx = \int _{\Omega } [u_j-K_0][\phi _j(x,u_j)-2M]^+\,dx\le 0\,. \end{aligned}$$

Hence, $\phi _j(x,u_j(x))\le 2M$ for every such j and for a.e. x such that $u_j(x)\ge K_0$. Now let $K>K_0$ be chosen so that $\phi (x,K)\ge 4M$ for all $x\in {\overline{\Omega }}$ (see (A.1)) and fix any $j_K\in \mathbb {N}$ such that $|\phi _j(x,K)-\phi (x,K)|<M$ for all $x\in {\overline{\Omega }}$ and $j\ge j_K$ (see (A.6)). If $u_j(x)\ge K$ for some x and $j\ge \max \{j_M,j_K\}$, we would have

$$\begin{aligned} 3M\le \phi (x,K)-M\le \phi _j(x,K)\le \phi _j(x,u_j)\le 2M\,, \end{aligned}$$

a contradiction. This proves that $u_j\le K$ a.e. in $\Omega$ for all j large enough. Arguing analogously, it can be easily seen that there exists $H>0$ such that $u_j\ge -H$ a.e. in $\Omega$, and (A.11) follows at once.

Letting $j\rightarrow \infty$. By the previous step, there exist $u\in L^{\infty }(\Omega )$ and $v\in W^{1,p}_0(\Omega )\cap L^{\infty }(\Omega )$ such that possibly up to a subsequence (not relabeled) there holds

$$\begin{aligned}&u_j{\mathop {\rightharpoonup }\limits ^{*}}u\quad \text{ in }\ \,L^{\infty }(\Omega )\,, \end{aligned}$$

(A.13)

$$\begin{aligned}&\phi _j(\cdot ,u_j)\rightharpoonup v\quad \text{ in }\ \,W^{1,p}_0(\Omega )\,, \end{aligned}$$

(A.14)

$$\begin{aligned}&\phi _j(x,u_j(x))\rightarrow v(x)\quad \text{ for } {\textit{ a.e. }} x\in \Omega \,. \end{aligned}$$

(A.15)

Moreover, since (A.6) and (A.11) ensure that $[\phi _j(\cdot ,u_j)-\phi (\cdot ,u_j)]\rightarrow 0$ a.e. in $\Omega$, we have

$$\begin{aligned} \phi (x,u_j(x))\rightarrow v(x)\quad \text{ for } {\textit{ a.e. }} x\in \Omega \,. \end{aligned}$$

(A.16)

From the above convergence and the strictly increasing character of the map $z\mapsto \phi (\cdot ,z)$, for a.e. $x\in \Omega$ we get the convergence of the sequence $\{u_j(x)\}$. Thus (see (A.13))

$$\begin{aligned} u_j\rightarrow u\quad {\textit{ a.e }}. \hbox { in } \Omega \,, \end{aligned}$$

(A.17)

and the equality $v(x)=\phi (x,u(x))$ holds true for a.e. $x\in \Omega$. In view of the above remarks, we also get

$$\begin{aligned} u_j\rightarrow u\quad \text{ in }\ \,L^q(\Omega )\qquad (1\le q<\infty )\,, \end{aligned}$$

(A.18)

and the convergences in (A.14)–(A.16) can be rephrased as

$$\begin{aligned}&\phi _j(\cdot ,u_j)\rightharpoonup \phi (\cdot ,u)\quad \text{ in }\ \,W^{1,p}_0(\Omega )\,, \end{aligned}$$

(A.19)

$$\begin{aligned}&\phi (x,u_j(x)),\,\phi _j(x,u_j(x))\rightarrow \phi (x,u(x))\quad \text{ for } {\textit{ a.e. }} x\in \Omega \,. \end{aligned}$$

(A.20)

By (A.18), letting $j\rightarrow \infty$ in the weak formulation of problems (A.9), i.e.,

$$\begin{aligned} \int _{\Omega }\mathbf{a}_j(\nabla \phi _j(x,u_j))\cdot \nabla \rho \,dx+\int _{\Omega }u_j\rho \,dx=\int _{\Omega }f\rho \,dx\,, \end{aligned}$$

(A.21)

will give equality (A.3) for every $\rho \in W^{1,p}_0(\Omega )$, if we prove that

$$\begin{aligned} \mathbf{a}_j(\nabla \phi _j(\cdot ,u_j))\rightharpoonup \mathbf{a}(\nabla \phi (\cdot ,u))\quad \text{ in }\ \,L^{p'}(\Omega ) \end{aligned}$$

(A.22)

($p'=p/(p-1)$). To this aim, we observe that by (A.10) there exists $\mathbf{z}\in L^{p'}(\Omega ;\mathbb {R}^N)$ such that (possibly up to a subsequence not relabeled)

$$\begin{aligned} \mathbf{a}_j(\nabla \phi _j(\cdot ,u_j))\rightharpoonup \mathbf{z}\quad \text{ in }\ \,L^{p'}(\Omega )\,. \end{aligned}$$

(A.23)

Therefore, letting $j\rightarrow \infty$ in (A.21), for all $\rho \in W^{1,p}_0(\Omega )$ we obtain

$$\begin{aligned} \int _{\Omega }\mathbf{z}\cdot \nabla \rho \,dx+\int _{\Omega }u\rho \,dx=\int _{\Omega }f\rho \,dx\,. \end{aligned}$$

(A.24)

On the other hands, for every nonnegative $\psi \in C^1_c(\Omega )$ and $l\in \mathbb {R}^N$, choosing $\rho =[\phi _j(\cdot ,u_j)-l\cdot x]\,\psi$ as test function in (A.21), and letting $j\rightarrow \infty$ gives

$$\begin{aligned}&\lim _{j\rightarrow \infty }\int _{\Omega }{} \mathbf{a}_j(\nabla \phi _j(x,u_j))\cdot \nabla \big [(\phi _j(x,u_j)-l\cdot x)\psi \big ]\,dx \nonumber \\&\quad =\int _{\Omega }f[\phi (x,u)-l\cdot x]\psi \,dx-\int _{\Omega }u[\phi (x,u)-l\cdot x]\,\psi \,dx \nonumber \\&\quad =\int _{\Omega }{} \mathbf{z}\cdot \nabla \big [(\phi (x,u)-l\cdot x)\psi \big ]\,dx \end{aligned}$$

(A.25)

(see (A.18), (A.19)–(A.20) and (A.24)), whereas from assumption $(H_1)$-(ii), we have

$$\begin{aligned}&\lim _{j\rightarrow \infty }\int _{\Omega }{} \mathbf{a}_j(\nabla \phi _j(x,u_j))\cdot \nabla \big [[\phi _j(x,u_j)-l\cdot x]\psi \big ]\,dx \\&\quad =\lim _{j\rightarrow \infty }\left\{ \int _{\Omega }\mathbf{a}_j(\nabla \phi _j(x,u_j))\cdot \nabla [\phi _j(x,u_j)-l\cdot x]\,\psi \,dx \right. \\&\left. \qquad + \int _{\Omega }{} \mathbf{a}_j(\nabla \phi _j(x,u_j))\cdot \nabla \psi \,[\phi _j(x,u_j)-l\cdot x]\,dx\right\} \\&\quad =\limsup _{j\rightarrow \infty }\left\{ \int _{\Omega }[\mathbf{a}_j(\nabla \phi _j(x,u_j))-\mathbf{a}_j(l)]\cdot \nabla [\phi _j(x,u_j)-l\cdot x]\,\psi \,dx \right. \\&\left. \qquad +\int _{\Omega }{} \mathbf{a}_j(l)\cdot \nabla [\phi _j(x,u_j)-l\cdot x]\,\psi \,dx\right\} \\&\qquad + \int _{\Omega }{} \mathbf{z}\cdot \nabla \psi \,[\phi (x,u)-l\cdot x]\,dx\ge \int _{\Omega }{} \mathbf{a}(l)\cdot \nabla [\phi (x,u)-l\cdot x]\,\psi \,dx \\&\qquad +\int _{\Omega }{} \mathbf{z}\cdot \nabla \psi \,[\phi (x,u)-l\cdot x]\,dx \end{aligned}$$

(see also (A.19)–(A.20) and (A.23)), namely

$$\begin{aligned}&\lim _{j\rightarrow \infty }\int _{\Omega }{} \mathbf{a}_j(\nabla \phi _j(x,u_j))\cdot \nabla \big [[\phi _j(x,u_j)-l\cdot x]\psi \big ]\,dx \nonumber \\&\quad \ge \int _{\Omega }{} \mathbf{a}(l)\cdot \nabla [\phi (x,u)-l\cdot x]\,\psi \,dx+\int _{\Omega }{} \mathbf{z}\cdot \nabla \psi \,[\phi (x,u)-l\cdot x]\,dx\,. \end{aligned}$$

(A.26)

Combining (A.25) and (A.26) plainly gives

$$\begin{aligned} \int _{\Omega }\left[ \mathbf{a}(l)-\mathbf{z}\right] \cdot [\nabla \phi (x,u)-l]\,\psi \,dx\le 0 \end{aligned}$$

for all $l\in \mathbb {R}^N$ and $\psi$ as above. Arguing as in step (i) of the proof of Theorem 5.6, by the above inequality, we get $\mathbf{z}=\mathbf{a}(\nabla \phi (\cdot ,u))$ a.e. in $\Omega$, whence (A.22) immediately follows (see also (A.23)). $\square$