Anisotropic obstacle Neumann problems in weighted Sobolev spaces with Hardy potential and variable exponent

Abstract

In this paper, we focus on a class of anisotropic obstacle problems governed by a Leray-Lions operator, involving non-linear elliptic equations with a Hardy potential exhibiting variable growth. Additionally, these problems are equipped by homogeneous Neumann boundary conditions. Using truncation techniques and the monotonicity method, we establish the existence of entropy solutions for the studied problem within the framework of anisotropic weighted Sobolev spaces with a variable exponent.

Appendix: Proof of Lemma 5

This section demonstrates that the operator ${\mathcal {G}}_m$ is both coercive and pseudo-monotone.

First of all, ${\mathcal {B}}_m$ is bounded. In fact, thanks to the Hölder’s inequality and by using the (5) we have

$$\begin{aligned}&\vert \langle {\mathcal {B}}_{m}w,v \rangle \vert \le \sum _{i=1}^{N}\int _{\Omega } \big \vert b_{i}(T_{m}(w))\partial _i v \big \vert dz+\varrho \int _{\Omega } \frac{\vert T_{m}(w)\vert ^{q_{0}(z)-1}}{\vert z \vert ^{q_{0}(z)}+\frac{1}{m}}\vert v \vert dz \nonumber \\&\quad \le \sup _{\vert \sigma \vert \le m}\vert b_{i}(\sigma )\vert \sum _{i=1}^{N} \int _{\Omega } \big \vert \partial _i v \big \vert \nu _i^{\frac{-1}{p_i(z)}} \nu _i^{\frac{1}{p_i(z)}}dz +\varrho \int _{\Omega } \frac{\vert T_{m}(w)\vert ^{q_{0}(z)-1}}{\vert z \vert ^{q_{0}(z)}+\frac{1}{m}}\vert v \vert dz\nonumber \\&\quad \le C' \Vert v\Vert +\varrho \left( \int _{\Omega }\left( \frac{\vert T_{m}(w)\vert ^{q_0(z)-1}}{\vert z \vert ^{q_{0}(z)}+\frac{1}{m}}\right) ^{q'_{0}(z)}dz\right) ^{\frac{1}{(q'_{0})^-}}\Vert v\Vert _{q_{0}(z)}\nonumber \\&\quad \le C \Vert v\Vert . \end{aligned}$$

(37)

where C and $C'$ are constants that depend on m.

It can be easily deduced that ${\mathcal {G}}_m$ is bounded by means the outcome obtained from Eq. (6), applying Hölder’s inequality, and combining it with Eq. (37). Thereafter, to establish the coercivity, let $w_0$ belongs to ${\mathcal {K}}_\Delta$. Then, for any w in ${\mathcal {K}}_\Delta$, we obtain

$$\begin{aligned}&\Big \vert \langle {\mathcal {A}}_{m}w,w_0\rangle \Big \vert =\Big \vert \sum _{i=1}^{N} \int _{\Omega }\Phi _{i}(w, T_{m}(w), \nabla w) \partial _{i}w_0dz+\frac{1}{m}\int _{\Omega }\vert w\vert ^{q_0(x)-2}w w_0dz \Big \vert \nonumber \\&\quad \le \sum _{i=1}^{N}\left( \int _{\Omega }\vert \Phi _{i}(z, T_{m}(w), \nabla w)\vert ^{p'_{i}(z)}\nu ^{1-p'_{i}(z)}dz\right) ^{\frac{1}{(p'_{i})^-}}\Vert \nu ^{\frac{1}{p_{i}(z)}} \partial _{i}w_0\Vert _{L^{p_i(z)}(\Omega )}\nonumber \\&\qquad +\frac{1}{m}\left( \int _{\Omega }\vert w\vert ^{(q_0(z)-1)q'_{0}(z)}dz \right) ^{\frac{1}{(q^-_{0})'}}\Vert w_0\Vert _{q_0(z)}\nonumber \\&\quad \le \beta \sum _{i=1}^{N}\left( \int _{\Omega }\left( R_{i}^{p'_{i}(z)}+\vert T_{m}(w)\vert ^{p_{i}(z)}+\sum _{i=1}^{N}\nu \vert \partial _{i}w \vert ^{p_{i}(z)}\right) \right) ^{\frac{1}{(p'_{i})^-}}\Vert \partial _{i}w_0\Vert _{L^{p_{i}(z)}(\Omega ,\nu _i)}\nonumber \\&\qquad +\frac{1}{m}\left( \int _{\Omega }\vert w\vert ^{q_0(z)}dz \right) ^{\frac{1}{(q_{0})^-}}\Vert w_0\Vert _{q_0(z)}\le C'\Vert w_0 \Vert . \end{aligned}$$

(38)

According to (6), we obtain

$$\begin{aligned} \Big \vert \langle {\mathcal {A}}_{m}w,w\rangle \Big \vert&\ge \alpha \sum _{i=1}^{N}\int _{\Omega }\big \vert \partial _i w\big \vert ^{p_i(z)} \nu _i(z)dz+\frac{1}{m}\int _{\Omega }\vert w\vert ^{q_0(x)}dz \Big \vert \nonumber \\&\ge \underline{\alpha }\Vert w\Vert , \end{aligned}$$

(39)

with $\displaystyle \underline{\alpha }=\min (\alpha ,\frac{1}{m})$.

Combining (38) and (39), we have

$$\begin{aligned} \langle {\mathcal {A}}_{m}w,w-w_0 \rangle&=\langle {\mathcal {A}}_{m}w, w \rangle -\langle {\mathcal {A}}_{m}w,w_0 \rangle \\&\ge \underline{\alpha }\sum _{i=0}^{N}\int _{\Omega }\nu \vert \partial _{i}w\vert ^{p_{i}(z)}dz-C_{4}\Vert w_0 \Vert \\&\ge \underline{\alpha }\Vert w \Vert ^{\underline{p}}-C_{4}\Vert w_0 \Vert , \end{aligned}$$

it follows that

$$\begin{aligned} \frac{\langle {\mathcal {A}}_{m}w,w-w_0\rangle }{\Vert w \Vert }\ge \frac{\underline{\alpha }}{\Vert w \Vert }\Vert w \Vert ^{\underline{p}}-\frac{C'}{\Vert w \Vert } \Vert w_0 \Vert \longrightarrow +\infty \text { as } \Vert w \Vert \rightarrow \infty . \end{aligned}$$

Witch implies that

$$\begin{aligned} \frac{\langle {\mathcal {G}}_{m}w,w-w_0\rangle }{\Vert w \Vert }=\frac{\langle {\mathcal {A}}_{m}w,w-w_0\rangle }{\Vert w \Vert }+\frac{\langle {\mathcal {B}}_{m}w,w-w_0\rangle }{\Vert w \Vert }\longrightarrow +\infty \text { as } \Vert w \Vert \rightarrow \infty . \end{aligned}$$

We still need to prove that ${\mathcal {G}}{m}$ is pseudo-monotone. Let $(w_{n})_{n\in {\mathbb {N}}}$ be a sequence in $W^{1,\textbf{p}(z)}(\Omega ,\mathbf {\nu })$ satisfying the following condition

$$\begin{aligned} \left\{ \begin{array}{l} w_{n} \rightharpoonup w \qquad \qquad \text { in } \qquad W^{1,\textbf{p}(z)}(\Omega ,\mathbf {\nu }),\\ {\mathcal {G}}_{m}w_{m}\rightharpoonup \chi _{m} \qquad \textrm{in} \qquad (W^{1,\textbf{p}(z))}(\Omega ,\mathbf {\nu }))^*,\\ \limsup \limits _{n \rightarrow \infty }\langle {\mathcal {G}}_{m}w_{n},w_{n}\rangle \le \langle \chi _{m},w\rangle . \end{array}\right. \end{aligned}$$

(40)

We will show that $\chi _{m}={\mathcal {G}}_{m}w$ and $\langle {\mathcal {G}}_{m}w_{n}, w_{n}\rangle \rightarrow \langle \chi _{m}, w\rangle$ as $n\rightarrow +\infty .$

With the help of the compact embedding $W^{1,\textbf{p}(z)}(\Omega ,\mathbf {\nu }) \hookrightarrow L^{\underline{p}}(\Omega ),$ we obtain $w_{n}$ converges to w in $L^{\underline{p}}(\Omega )$ for a subsequence noted again $(w_{n})_{n\in {\mathbb {N}}}.$

Since $(w_n)_{n\in {\mathbb {N}}}$ is a bounded sequence in $W^{1,\textbf{p}(z)}(\Omega ,\mathbf {\nu })$, by (7) it is obvious that the sequence $(\Phi _{i}(z,T_{m}(w_n),\nabla w_n))_{n \in {\mathbb {N}}}$ is bounded in $L^{p'_{i}(z)}(\Omega ,\nu _i^*)$, which implies the existence of a measurable function $\Theta _{i}^{m}\in L^{p'_{i}(z)}(\Omega ,\nu _i^*)$ such that

$$\begin{aligned} \Phi _{i}(z,T_{m}(w),\nabla w)\rightharpoonup \Theta _{i}^{m} \qquad in \quad L^{p'_{i}(z)}(\Omega ,\nu _i^{*})\ as \ n \rightarrow \infty . \end{aligned}$$

(41)

We apply the Lebesgue dominated convergence theorem to obtain

$$\begin{aligned} \frac{ \vert T_{m}(w_n)\vert ^{q_{0}(z)-2}T_{m}(w_n)}{\vert z \vert ^{q_{0}(z)}+\frac{1}{m}}\rightarrow \frac{\vert T_{m}(w)\vert ^{q_{0}(z)-2} T_{m}(w)}{\vert z \vert ^{q_{0}(z)}+\frac{1}{m}} \quad in \quad L^{q'_{0}(z)}(\Omega ). \end{aligned}$$

(42)

Also, we have

$$\begin{aligned} \frac{1}{m} \vert w_{n} \vert ^{q_0(z)-2}w_{n}\rightharpoonup \frac{1}{m}\vert w \vert ^{q_0(z)-2}w \ in \ L^{q'_{0}(z)}(\Omega ). \end{aligned}$$

(43)

What is more, as $\big (b_{i}^m(w_n)\big )_{n\in {\mathbb {N}}}$ is bounded in $L^{p'_i(z)}(\Omega ,\nu _i^*)$ and $b_{i}^m(w_n)\longrightarrow b_{i}^m(w)$ a.e. in $\Omega$, we get

$$\begin{aligned} b_{i}^m(w_n) \rightarrow b_{i}^m(w) \text { strongly in } L^{p'_i(z)}(\Omega ,\nu _i^*) \text { as } n\rightarrow \infty . \end{aligned}$$

(44)

For all $\nu \in W^{1,\textbf{p}(z)}(\Omega ,\mathbf {\nu })$, we obtain

$$\begin{aligned} \langle \chi _{m},\varphi \rangle&=\lim _{n \rightarrow \infty } \langle B_{m}w_n,\varphi \rangle \lim _{n \rightarrow \infty }\sum _{i=1}^{N}\int _{\Omega }\Phi _{i}(z,T_{m}(w_n),\nabla w_n)\partial _{i} \varphi dx\nonumber \\&\quad +\lim _{n \rightarrow \infty }\frac{1}{m}\int _{\Omega }\vert w_n \vert ^{q_{0}(z)-2}w_n \varphi dz-\lim _{n\rightarrow \infty }\varrho \int _{\Omega }\frac{\vert T_{m}(w_{n})\vert ^{q_0(z)-2}T_{m}(w_{n})}{\vert z\vert ^{q_{0}(z)}+\frac{1}{m}}\varphi dz\nonumber \\&\quad +\lim _{n \rightarrow \infty }\sum _{i=1}^{N}\int _{\Omega }b_{i}^m(w_{n})\partial _{i}\varphi dz\nonumber \\ {}&=\sum _{i=1}^{N}\int _{\Omega }\Theta _{i}^{m} \partial _{i}\varphi dz+\frac{1}{m}\int _{\Omega }\vert w\vert ^{q_0(z)-2}w \varphi dz\nonumber \\&\quad - \varrho \int _{\Omega }\frac{\vert T_{m}(w)\vert ^{q_{0}-2}T_{m}(w)}{\vert z \vert ^{q_{0}}+\frac{1}{m}}\varphi dz+\lim _{k\rightarrow \infty }\sum _{i=1}^{N}\int _{\Omega }b_{i}^m(w)\partial _{i}\varphi dz. \end{aligned}$$

(45)

From (40) and (45), we have

$$\begin{aligned}&\limsup \limits _{ n \rightarrow \infty }\langle {\mathcal {G}}_{m}(w_{n}),w_{n} \rangle \\ {}&\quad =\limsup \limits _{n \rightarrow \infty }\left\{ \sum _{i=1}^{N}\int _{\Omega } \Phi _{i}(z,T_{m}(w_{n}),\nabla w_{n})\partial _{i}w_{n}dz \right. \\&\qquad \left. +\frac{1}{m}\int _{\Omega }\vert w_n\vert ^{q_{0}(z)}dz-\varrho \int _{\Omega }\frac{\vert T_{m}(w)\vert ^{q_0(z)-2}T_{m}(w)}{\vert z \vert ^{q_{0}(z)}+\frac{1}{m}}w_n dz +\sum _{i=1}^{N}\int _{\Omega }b_{i}^m(w_{n})\partial _{i}w_n dz \right\} \\&\quad \le \sum _{i=1}^{N}\int _{\Omega }\Theta _{i}^{m}\partial _{i}wdz+\frac{1}{m}\int _{\Omega }\vert w \vert ^{q_{0}(z)}dz-\varrho \int _{\Omega }\frac{\vert T_{m}(w)\vert ^{q_{0}(z)-2}T_{m}(w)}{\vert z \vert ^{q_{0}(z)}+\frac{1}{m}}wdz \\&\qquad +\sum _{i=1}^{N}\int _{\Omega }b_{i}^m(w)\partial _{i}w dz. \end{aligned}$$

By using (42) and (44), we obtain

$$\begin{aligned} \int _{\Omega }\frac{\vert T_{m}(w_n)\vert ^{q_0(z)-2}T_{m}(w_n)}{\vert z\vert ^{q_0(z)}+\frac{1}{m}}w_n dz \rightarrow \int _{\Omega }\frac{\vert T_{m}(w)\vert ^{q_0(z)-2}T_{m}(w)}{\vert z\vert ^{q_0(z)}+\frac{1}{m}}wdz, \end{aligned}$$

(46)

and

$$\begin{aligned} \sum _{i=1}^{N}\int _{\Omega }b_{i}^m(w_{n})\partial _{i} w_n dz \rightarrow \sum _{i=1}^{N}\int _{\Omega }b_{i}^m(w)\partial _{i} w dz. \end{aligned}$$

(47)

Which implies that

$$\begin{aligned}{} & {} \limsup \limits _{n \rightarrow \infty }\left( \sum _{i=1}^{N}\int _{\Omega }\Phi _{i}(z,T_{m}(w_{n}),\nabla w_{n})\partial _{i}w_{n}dz+\frac{1}{m}\int _{\Omega }\vert w_{n}\vert ^{q_0(z)}dz\right) \nonumber \\{} & {} \quad \le \sum _{i=1}^{N}\int _{\Omega }\Theta _{i}\partial _{i}wdz+\frac{1}{m}\int _{\Omega }\vert u\vert ^{q_0(z)}dz. \end{aligned}$$

(48)

On the other side, taking into account (8), we get

$$\begin{aligned}{} & {} \sum _{i=1}^{N}\int _{\Omega }(\Phi _{i}(z,T_{m}(w_{n}), \nabla w_{n})-\Phi _{i}(x,T_{m}(w_{n}),\nabla w))(\partial _{i}w_{n}-\partial _{i}w)dz\\{} & {} \quad +\frac{1}{m}\int _{\Omega }(\vert w_{n}\vert ^{q_0(z)-2}w_{n}-\vert w\vert ^{q_0(z)-2}w)(w_{n}-w)dz\ge 0, \end{aligned}$$

hence

$$\begin{aligned}&\sum _{i=1}^{N}\int _{\Omega }\Phi _{i}(z,T_{m}(w),\nabla w)\partial _{i}wdz+\frac{1}{m}\int _{\Omega }\vert w_n\vert ^{q_0(z)}dz\\&\quad \ge \sum _{i=1}^{N}\int _{\Omega }\Phi _{i}(z,T_{m}(w),\nabla w)\partial _{i}wdz+\frac{1}{m}\int _{\Omega }\vert w\vert ^{q_0(z)-2}w_n wdz\\&\qquad +\sum _{i=1}^{N}\int _{\Omega }\Phi _{i}(z,T_{m}(w_{k}),\nabla w)(\partial _{i}w_{n}-\partial _{i}w)dz+\frac{1}{m}\int _{\Omega }\vert w \vert ^{q_0(z)-2}w(w_{n}-w)dz. \end{aligned}$$

The Lebesgue dominated convergence theorem implies $T_{m}(w_n)\longrightarrow T_{m}(w)$ in $L^{p_{i}(z)}(\Omega ,\nu _i)$, hence $\Phi _{i}(z,T_{m}(w_n),\nabla w)$ converges to $\Phi _{i}(z,T_{m}(w),\nabla w)$ in $L^{p'_{i}(z)}(\Omega ,\nu _i^{*})$, by employing (40) we infer

$$\begin{aligned}&\liminf \limits _{n \rightarrow \infty }\left( \sum _{i=1}^{N}\int _{\Omega }\Phi _{i}(z,T_{m}(w),\nabla w)\partial _{i}wdz+\frac{1}{m}\int _{\Omega } \vert w_n \vert ^{q_0(z)}dz\right) \\&\quad \ge \sum _{i=1}^{N}\int _{\Omega }\Theta _{i}\partial _{i}wdz+\frac{1}{m}\int _{\Omega }\vert w \vert ^{q_0(z)}dz. \end{aligned}$$

According to (48), we deduce that

$$\begin{aligned}{} & {} \lim _{n \rightarrow \infty }\left( \sum _{i=1}^{N}\int _{\Omega }\Phi _{i}(z,T_{m}(w_n),\nabla w_n)\partial _{i}w_ndz+\frac{1}{m}\int _{\Omega }\vert w_n \vert ^{q_{0}(z)}dz \right) \nonumber \\{} & {} \quad =\sum _{i=1}^{N}\int _{\Omega }\Theta _{i}\partial _{i}wdz+\frac{1}{m}\int _{\Omega }\vert w \vert ^{q_{0}(z)}dz. \end{aligned}$$

(49)

Hence, from (45)–(47), it follows that

$$\begin{aligned} \langle {\mathcal {G}}_{m} w_{n},w_{n} \rangle \rightarrow \langle \chi _{m},w \rangle \ as \ n \rightarrow \infty . \end{aligned}$$

In the sequel, by means (49) we can establish that

$$\begin{aligned}{} & {} \lim _{n \rightarrow +\infty }\left( \sum _{i=1}^{N}\int _{\Omega }(\Phi _{i}(z,T_{m}(w),\nabla w)-\Phi _{i}(z,T_{m}(w),\nabla w))(\partial _{i}w-\partial _{i}w)dz \right. \\ {}{} & {} \quad \left. +\frac{1}{m}\int _{\Omega }(\vert w_{n}\vert ^{q_{0}(z)-2}w_{n}-\vert w \vert ^{q_{0}(z)-2}w)(w_{n}-w)dz\right) =0. \end{aligned}$$

Once again, by Lemma 3, we obtain

$$\begin{aligned} w_n \rightarrow w \quad \text { in } \; W^{1,\textbf{p}(z)}(\Omega ,\mathbf {\nu }) \qquad \text { and } \quad \partial _{i}w_n\rightarrow \partial _{i} w \quad \text { a.e. in }\; \Omega , \end{aligned}$$

which means that

$$\begin{aligned} \Phi _{i}(z,T_{m}(w),\nabla w) \rightharpoonup \Phi _{i}(z,T_{m}(w),\nabla w) \quad \text { in } \; L^{p'_{i}(z)}(\Omega ,\nu _i^{*}) \quad \text { for } \ i=1, \ldots , N, \end{aligned}$$

and it follows from (41) to (43) that $\chi _{m}={\mathcal {G}}_{m}w$, which conclude the proof of Lemma 5.