Strongly nonlinear elliptic boundary value problems in Musielak–Orlicz spaces

Abstract

In this paper we prove an existence result of solutions for some strongly nonlinear elliptic problems with lower order term and $L^1$-data in Musielak–Orlicz–Sobolev spaces.

Appendix

Let $\varphi$ and $\psi$ be two complementary Musielak–Orlicz functions.

We are interested here in the Dirichlet problem for the operator

$$\begin{aligned} A(u) \equiv \sum _{|\alpha |\le m} (-1)^{|\alpha |} D^{\alpha } A_{\alpha }(x, u, \nabla u,\ldots , \nabla ^m u) \end{aligned}$$

(6.1)

on $\Omega$.

The basic conditions imposed on the coefficients $A_{\alpha }$ of (6.1) are the followings:

$(A_1)$ :

Each $A_{\alpha }(x, \xi )$ is a real valued function defined on $\Omega \times \mathbb {R}^{N}$ is measurable in x for fixed $\xi$ and continuous in $\xi$ for fixed x.

$(A_2)$ :

There exist two Musielak–Orlicz functions $\gamma$ and $\varphi$ with $\gamma \prec \prec \varphi$, functions $a_{\alpha }$ in $E_{\psi }(\Omega )$, constants $c_1$ and $c_2$ such that for all x in $\Omega$ and $\xi$ in $\mathbb {R}^{N}$,if

$$\begin{aligned} |\alpha |= & {} m : |A_{\alpha }(x, \xi )| \le a_{\alpha }(x) + c_1 \sum _{|\beta | = m} \psi _x^{-1}(\varphi (x,c_2 \xi _\beta )) \\&+\, c_1 \sum _{|\beta | < m} \overline{\gamma _x}^{-1}(\varphi (x,c_2 \xi _\beta )), \end{aligned}$$

if

$$\begin{aligned} |\alpha |< & {} m : |A_{\alpha }(x, \xi )| \le a_{\alpha }(x) + c_1 \sum _{|\beta | = m} \psi _x^{-1}(\gamma (x,c_2 \xi _\beta )) \\&+\, c_1 \sum _{|\beta | < m} \psi _x^{-1}(\varphi (x,c_2 \xi _\beta )). \end{aligned}$$

$(A_3)$ :

For each $x\in \Omega , \ \eta \in \mathbb {R}^{N}, \ \xi ,$ and $\xi '$ in $\mathbb {R}^{N}$ with $\xi \ne \xi '$,

$$\begin{aligned} \sum _{|\alpha | = m} (A_{\alpha }(x, \xi , \eta ) - A_{\alpha }(x, \xi ', \eta )) \left( \xi _{\alpha } - \xi _{\alpha }'\right) > 0. \end{aligned}$$

$(A_4)$ :

There exist functions $b_\alpha (x)$ in $E_{\psi }(\Omega )$, b(x) in $L^1(\Omega )$, positive constants $d_1$ and $d_2$ such that, for some fixed element v in $W_0^mE_\varphi (\Omega )$,

$$\begin{aligned} \sum _{|\alpha |\le m} A_\alpha (x, \xi ) ( \xi _\alpha - D^\alpha v) \ge d_1 \sum _{|\alpha |\le m} \varphi (x,d_2\xi _\alpha )- \sum _{|\alpha |\le m} b_\alpha (x) \xi _\alpha - b(x) \end{aligned}$$

for all x in $\Omega$ and $\xi$ in $\mathbb {R}^{N}$.

Associated to the differential operator 6.1 we define a mapping T from

$$\begin{aligned} D(T)=\left\{ u\in W_0^mL_\varphi (\Omega ); A_\alpha (\xi (u)) \in L_{\psi }(\Omega ) \text{ for } \text{ all } |\alpha | \le m\right\} \subset W_0^mL_\varphi (\Omega )\end{aligned}$$

into $W^{-m}L_{\psi }(\Omega )$ by the formula

$$\begin{aligned} < v, Tu > = \int _\Omega \sum _{|\alpha | \le m} A_\alpha (\xi (u)) D^\alpha v dx \end{aligned}$$

for $v\in W_0^mL_\varphi (\Omega ).$

For the perturbing function $g : \Omega \times \mathbb {R} \rightarrow \mathbb {R},$, we assume the usual conditions. $(G_1)$ g(x, u) is a Carathéodory function and satisfies the sign condition

$$\begin{aligned} g(x,u) u \ge 0 \end{aligned}$$

for a.e. x in $\Omega$ and all u in $\mathbb {R}$.

$(G_2)$ For each $r\ge 0$, there exists $h_r \in L^1(\Omega )$ such that

$$\begin{aligned} |g(x,u) | \le h_r(x) \end{aligned}$$

for a.e. x in $\Omega$ and all u in $\mathbb {R}$ with $|u|\le r$.

Theorem 6.1

[9] Let $\Omega$ be a bounded Lipschitz domain in $\mathbb {R}^{N}$. Assume that the coefficients of (6.1) satisfy $(A_1),\ldots , (A_4)$ with $m=1$. Let T the corresponding mapping in the complementary system $(W_0^1L_\varphi (\Omega ), W_0^1E_\varphi (\Omega ), W^{-1}L_{\psi }(\Omega ), W^{-1}E_{\psi }(\Omega ))$. Let g(x, u) satisfy $(G_1)$ and $(G_2)$. Then there exists $u \in D(T)$ such that $g(x, u) \in L^1(\Omega ), \ g(x, u) u \in L^1(\Omega )$ and

$$\begin{aligned} \langle u-v, T u \rangle + \int _\Omega g(x,u) (u-v) dx = \langle u-v, f \rangle \end{aligned}$$

(6.2)

for all $v \in W^1_0L_\varphi (\Omega )\cap L^\infty (\Omega ).$