Regularizing effect of the interplay between coefficients in some nonlinear Dirichlet problems with distributional data

Abstract

We prove that the solution u of Dirichlet problem (1.1) has exponential summability under the only assumption that there exists $R>0$ such that $|F(x)|^{2} \le R\,a(x)$; furthermore, we prove the boundedness of u under the slightly stronger assumption that there exists $R>0$ such that $|F(x)|^{p} \le R\,a(x)$, $p>2$.

Appendix: Existence for bounded data F

We prove here the existence of a solution of (1.1) if |F| is a function in $L^{m}(\Omega )$, with $m> N (> 2)$. First of all, let $a_{n}(x) = \min (a(x),n)$. Then, by a straightforward application of the results of [8] (note that the datum $-{\text {div}}(F)$ belongs to the dual of $W_0^{1,2}(\Omega )$), there exists a solution $u_{n}$ in $W_0^{1,2}(\Omega )$ of

$$\begin{aligned} -{\text {div}}(M(x,\nabla u_{n})) + a_{n}(x)\,u_{n}= -{\text {div}}(F). \end{aligned}$$

Choosing $u_{n}$ as test function, and using (1.2), as well as Young's inequality, we have that

$$\begin{aligned} \alpha \int _{\Omega }|\nabla u_{n}|^{2} + \int _{\Omega }a_{n}(x)u_{n}^{2} = \int _{\Omega }F(x)\,\nabla u_{n}\le \frac{1}{2\alpha } \int _{\Omega }|F(x)|^{2} + \frac{\alpha }{2} \int _{\Omega }|\nabla u_{n}|^{2}, \end{aligned}$$

from which it follows that the sequence $\{u_{n}\}$ is bounded in $W_0^{1,2}(\Omega )$. Furthermore, using Stampacchia’s result (see [12]), and the fact that by (1.3) $a_{n}\ge 0$, one can prove that the sequence $\{u_{n}\}$ is also bounded in $L^{\infty }(\Omega )$. Thus, up to subsequences, $u_{n}$ converges to some function u in $W_0^{1,2}(\Omega )\cap L^{\infty }(\Omega )$, weakly in $W_0^{1,2}(\Omega )$, $*$-weakly in $L^{\infty }(\Omega )$ and almost everywhere in $\Omega$. Since $0 \le a_{n}(x) \le a(x) \in L^{1}(\Omega )$, the boundedness of $\{u_{n}\}$ in $L^{\infty }(\Omega )$, and its almost everywhere convergence to u allow us to apply Lebesgue theorem to prove that

$$\begin{aligned} a_{n}(x)\,u_{n}\quad \text{ strongly } \text{ converges } \text{ to } a(x)\,u \hbox { in } L^{1}(\Omega ). \end{aligned}$$

In order to pass to the limit in the approximate equations, we will use Minty’s trick: let $\varphi$ in $W_0^{1,2}(\Omega )\cap L^{\infty }(\Omega )$, and choose $u_{n}- \varphi$ as test function in the equation for $u_{n}$. We obtain

$$\begin{aligned} \int _{\Omega }M(x,\nabla u_{n}) \nabla (u_{n}- \varphi ) + \int _{\Omega }a_{n}(x)u_{n}\,(u_{n}- \varphi ) = \int _{\Omega }F \nabla (u_{n}- \varphi ). \end{aligned}$$

Adding and subtracting the term

$$\begin{aligned} \int _{\Omega }M(x,\nabla \varphi ) \nabla (u_{n}- \varphi ), \end{aligned}$$

and using the fact that $M(x,\cdot )$ is monotone, we arrive, after passing to the limit, to

$$\begin{aligned} \int _{\Omega }M(x,\nabla \varphi ) \nabla (u - \varphi ) + \int _{\Omega }a(x) u \,(u - \varphi ) \le \int _{\Omega }F \nabla (u - \varphi ). \end{aligned}$$

Choosing $\varphi = u - t\,\psi$, with $t \ne 0$ and $\psi$ in $W_0^{1,2}(\Omega )\cap L^{\infty }(\Omega )$, we thus have that

$$\begin{aligned} t\,\int _{\Omega }M(x,\nabla (u - t\psi )) \nabla \psi + t\,\int _{\Omega }a(x) u \,\psi \le t\,\int _{\Omega }F \nabla \psi . \end{aligned}$$

Dividing by $t > 0$ and letting t tend to zero, we arrive at

$$\begin{aligned} \int _{\Omega }M(x,\nabla u) \nabla \psi + \int _{\Omega }a(x) u \,\psi \le \int _{\Omega }F \nabla \psi , \end{aligned}$$

while the reverse inequality can be obtained dividing by $t < 0$, and then letting t tend to zero. Thus, we have proved that

$$\begin{aligned} \int _{\Omega }M(x,\nabla u) \nabla \varphi + \int _{\Omega }a(x)\,u\,\varphi = \int _{\Omega }F(x) \nabla \varphi , \qquad \forall \varphi \in W_0^{1,2}(\Omega )\cap L^{\infty }(\Omega ), \end{aligned}$$

that is, problem (1.1) has a solution u belonging to $W_0^{1,2}(\Omega )\cap L^{\infty }(\Omega )$.