**3. The Harnack inequality**

For any measurable set *E* ⊂*R<sup>l</sup>* and any integrable function *f*, we denote the mean value

$$(f)\_E = \frac{1}{mes(E)} \int\_E f(y) dy$$

where *mes E*ð Þ is the Lebesgue measure of the set *<sup>E</sup>*<sup>⊂</sup> *Rl* .

Let us assume that Ω ⊂ *Rl* is a bounded open set. We are going to consider the Harnack inequality for a weak solution to the following differential problem with the bounded condition

$$\begin{aligned} \Lambda u &= 0 \quad \text{in } \Omega, \\ u &= f \quad \text{in } \, R^l \backslash \Omega, \end{aligned} \tag{14}$$

where <sup>Λ</sup> � �ð Þ <sup>Δ</sup> *<sup>s</sup>* � *<sup>b</sup>* � <sup>∇</sup> acts on *<sup>L</sup><sup>p</sup>*, 1<sup>≤</sup> *<sup>p</sup>*<sup>&</sup>lt; <sup>∞</sup> under the condition that its kernel satisfies the following inequality

$$\frac{\mu}{|\boldsymbol{\omega} - \boldsymbol{\uprho}|^{l+sp}} \le K(\boldsymbol{\upomega}, \ \boldsymbol{\uprho}) \le \frac{\lambda}{|\boldsymbol{\upomega} - \boldsymbol{\uprho}|^{l+sp}} \tag{15}$$

for almost all *x*, *y*∈ *Rl* , j*x* � *y*j≤1 and some *μ*, *λ* such that 0<*μ*≤ *λ*< ∞.

The general information can be found in [3, 18]. By the standard method, the following statements (1–3) can be proven.

**Statement 1.** *Assuming that u* ∈*W<sup>p</sup> <sup>s</sup> is a weak solution to* (14) *and nonnegative in a ball with the center in point x*<sup>0</sup> *with radius ρ. Then, the following inequality*

$$\begin{aligned} &\sigma^{\frac{pr}{p-1}} \left\langle |\max\{u,\mathbf{0}\}|^{p-1} \frac{\mathbf{1}}{|\cdot-\mathbf{x}\_{0}|^{l+sp}} \right\rangle\_{\mathbb{R}^{l}\backslash\mathcal{B}(\mathbf{x}\_{0},r)}^{\frac{1}{p-1}} \\ &\leq C \sup\_{\mathcal{B}(\mathbf{x}\_{0},r)} u + C \left(\frac{r}{\rho}\right)^{\frac{p}{p-1}} \left\langle |\max\{-u,\mathbf{0}\}|^{p-1} \frac{\mathbf{1}}{|\cdot-\mathbf{x}\_{0}|^{l+sp}} \right\rangle\_{\mathbb{R}^{l}\backslash\mathcal{B}(\mathbf{x}\_{0},\rho)}^{\frac{1}{p-1}} \end{aligned}$$

holds for 0 <*r*<*ρ*.

**Statement 2.** *Assuming that u* ∈*W<sup>p</sup> <sup>s</sup> is a weak solution to* (14) *and nonnegative in a ball with the center in point x*<sup>0</sup> *with radius ρ. Then, for all r*, *ρ such that* 0< *r*< *ρ, the following inequality*

$$\sqrt[s]{\left(f^{\delta}\right)\_{B(\mathbf{x}\_0,r)}} \le C \inf\_{B(\mathbf{x}\_0,r)} u +$$

$$+ C \left(\frac{r}{\rho}\right)^{\frac{p}{p-1}} \left\langle \left| \max\{-u, \mathbf{0}\} \right|^{p-1} \frac{1}{\left| -\infty\_0 \right|^{l+sp}} \right\rangle\_{\mathbb{R}^l \backslash B(\mathbf{x}\_0, \rho)}^{\frac{1}{p-1}}$$

holds for any *δ*∈ð Þ 0, 1 .

**Statement 3.** *Assuming that u*∈ *W<sup>p</sup> <sup>s</sup> is a weak solution to* (14) *and nonnegative in a ball with the center in point x*<sup>0</sup> *with radius ρ. Then, for all r*, *ρ such that* 0< *r*< *ρ, the following inequality*

$$\begin{aligned} \sup\_{B(\mathbf{x}\_0, r)} u &\le C \varepsilon^{-\frac{p-1}{p^2}} \sqrt[p]{((\max\{u, \mathbf{0}\})^p)\_{B(\mathbf{x}\_0, \rho)}} + \\ &+ C \varepsilon r^{\frac{p}{p-1}} \left\langle |\max\{u, \mathbf{0}\}|^{p-1} \frac{\mathbf{1}}{\left| \cdot - \mathbf{x}\_0 \right|^{l+sp}} \right\rangle\_{R^l \backslash B(\mathbf{x}\_0, r)}^{\frac{1}{p-1}} \end{aligned}$$

holds for *δ*∈ð Þ 0, 1 .

Now, we can prove the next theorem.

Theorem 2. Let *u*∈*W<sup>p</sup> <sup>s</sup> Rl* � � be a weak solution to the problem

$$\begin{cases} (-\Delta)\_p^\circ - b \cdot \nabla \rfloor u = 0 \quad \text{in } \Omega, \\ u = f \quad \text{in } R^l \backslash \Omega, \end{cases} \tag{16}$$

where 1≤*<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup> and *<sup>s</sup>*∈ð Þ 0, 1 . Assuming *<sup>u</sup>* <sup>∈</sup>*W<sup>p</sup> <sup>s</sup> <sup>R</sup><sup>l</sup>* � � is nonnegative in a ball with the center in point *x*<sup>0</sup> with the radius *ρ*, then

$$\sup\_{B(x\_0, r)} u \le C \inf\_{B(x\_0, r)} u + C \left( \frac{r}{\rho} \right)^{\frac{p}{p-1}} \left\langle |\max \{ -u, 0 \}|^{p-1} \frac{1}{|\cdot - x\_0|^{l+sp}} \right\rangle\_{R^l(B(x\_0, \rho))}^{\frac{1}{p-1}}$$

for *r*, *ρ* such that 0< *r*< *ρ*.

**Proof.** From statement 3, we can write the estimation

$$\begin{aligned} \sup\_{\mathcal{B}\left(\boldsymbol{x}\_{0},\frac{\boldsymbol{r}}{2}\right)} & u \leq C e^{-\frac{p-1}{p^{2}}} \Big( ( (\max\{\boldsymbol{u},\boldsymbol{0}\})^{p} )\_{\boldsymbol{B}\left(\boldsymbol{x}\_{0},\boldsymbol{0}\right)} \Big)^{\frac{1}{p}} + \\ + \bar{C} e r^{\frac{p}{p-1}} \Big< \Big| \max\{\boldsymbol{u},\boldsymbol{0}\} |^{p-1} \frac{1}{\left| \cdot - \boldsymbol{x}\_{0} \right|^{l+sp}} \Big>\_{\mathcal{R}^{l}\left(\boldsymbol{\mathcal{B}}\left(\boldsymbol{x}\_{0},\frac{\boldsymbol{r}}{2}\right)\right)} \end{aligned}$$

and using statement 1, we have the following inequality

$$\begin{aligned} &r^{\frac{p}{p-1}} \left\langle |\max\{u,\mathbf{0}\}|^{p-1} \frac{\mathbf{1}}{|\cdot-\mathbf{x}\_{0}|^{l+sp}} \right\rangle\_{\mathbb{R}^{l}\backslash\mathcal{B}\left(\mathbf{x}\_{0},\mathbf{f}\right)}^{\frac{1}{p-1}} \\ &\leq \mathbf{C} \sup\_{\mathcal{B}(\mathbf{x}\_{0},r)} u + \tilde{\tilde{\mathbf{C}}} r^{\frac{p}{p-1}} \rho^{-\frac{p}{p-1}} \left\langle |\max\{-u,\mathbf{0}\}|^{p-1} \frac{\mathbf{1}}{|\cdot-\mathbf{x}\_{0}|^{l+sp}} \right\rangle\_{\mathbb{R}^{l}\backslash\mathcal{B}(\mathbf{x}\_{0},\mathbf{o})}^{\frac{1}{p-1}}.\end{aligned}$$

So, we obtain the estimation

$$\begin{split} \sup\_{\mathcal{B}\left(\mathbf{x}\_{0},\frac{\mathbf{r}}{2}\right)} & u \leq C\varepsilon \sup\_{\mathcal{B}\left(\mathbf{x}\_{0},r\right)} u + C\varepsilon^{-\frac{p-1}{p^{2}}} \Big( (\max\{u,\mathbf{0}\})^{p} \big)\_{\mathcal{B}\left(\mathbf{x}\_{0},\rho\right)} \Big)^{\frac{1}{p}} \\ &+ C\varepsilon r^{\frac{p}{p-1}} \Big\langle \max\{u,\mathbf{0}\} \big|\mathcal{P}^{p-1} \frac{\mathbf{1}}{\left|\cdot - \mathbf{x}\_{0}\right|^{l+sp}} \Big\rangle\_{\mathcal{R}^{l}\left(\mathbf{x}\_{0},r\right)}^{\frac{1}{p}}. \end{split}$$

*Methods of the Perturbation Theory for Fundamental Solutions to the Generalization… DOI: http://dx.doi.org/10.5772/intechopen.109366*

Choosing <sup>1</sup> <sup>2</sup> < *η*<~*η*< 1 applying the standard argument and the Young inequality, we obtain

$$\begin{aligned} \sup\_{B(\mathbf{x}\_0, r\bar{\eta})} u &\leq \frac{1}{2} \sup\_{B(\mathbf{x}\_0, r\bar{\eta})} u + \tilde{c} \sqrt[s]{\left(f^{\delta}\right)\_{B(\mathbf{x}\_0, r)}} + \\ &+ C \left(\frac{r}{\rho}\right)^{\frac{pr}{p-1}} \left\langle \left| \max\{-u, \mathbf{0} \} \right|^{p-1} \frac{\mathbf{1}}{\left| -\mathbf{x}\_0 \right|^{l+sp}} \right\rangle\_{\mathbb{R}^l \backslash \mathcal{B}(\mathbf{x}\_0, \rho)}^{\frac{1}{p-1}} \end{aligned}$$

,

now, iterating this argument and applying statement 2, we have proven Theorem 2.

Using Theorem 2, we write

$$\begin{aligned} \mathop{\mathrm{osc}}\limits\_{B\left(\mathbf{x}\_{0},\epsilon^{i}\_{\!\!T}\right)} & u \equiv \sup\_{B\left(\mathbf{x}\_{0},\epsilon^{i}\_{\!\!T}\right)} u - \inf\_{B\left(\mathbf{x}\_{0},\epsilon^{i}\_{\!\!T}\right)} u \leq \\ & \leq C\left(\varepsilon^{i}\frac{r}{\rho}\right)^{\frac{p}{p-1}}\sqrt[p]{(|\!\!u\!\!u\!\!)^{p}\_{B\left(\mathbf{x}\_{0},r\!\!D}\right)} + \\ & + C\left(\varepsilon^{i}\frac{r}{\rho}\right)^{\frac{p}{p-1}}\left\langle |\max\{u,\mathbf{0}\}|^{p-1}\frac{1}{|\!\!\!-\infty\_{0}|^{\left[l+sp\right]}}\right\rangle\_{R^{l}\left(\mathbf{x}\_{0},\mathbf{\tilde{z}}\right)}^{\frac{1}{p}} \end{aligned}$$

for all *i* ∈ *N*, thus Theorem 2 is a consequence of Theorem 1.
