Existence and uniqueness results for Navier problems with degenerated operators

1. Introduction

The main purpose of this paper (see Theorem 7) is to establish the existence and uniqueness of solutions for the Navier problem \[ (P)\left\{ \begin{array}{lll} & Lu(x) = f(x) – {\textrm{div}}(G(x)), \ \ {\textrm{in}} \ \ {\Omega}, \\ & u(x) = {\Delta}u(x) = 0, \ \ {\textrm{in}} \ \ {\partial\Omega}, \end{array} \right. \] where $$ Lu(x) = {\Delta}{\big[}{\omega}_1(x)\,{\vert{\Delta}u\vert}^{p-2}{\Delta}u + {\nu}_1(x)\,{\vert{\Delta}u\vert}^{q-2}{\Delta}u{\big]}-{\textrm{div}}{\big[}{\omega}_2(x){\vert {\nabla}u\vert}^{p-2}{\nabla}u + {\nu}_2(x)\, {\vert{\nabla}u\vert}^{s-2}{\nabla}u){\big]},$$ \({\Omega}\,{\subset}\,{\mathbb{R}}^N\) is a bounded open set, \(\displaystyle{\dfrac{f}{{\omega}_2}}\,{\in}L^{p\,’}(\Omega,{\omega}_2)\), \(\displaystyle{\dfrac{G}{{\nu}_2}}\, {\in}\,[L^{s\,’}(\Omega , {\nu}_2)]^N\), \({\omega}_1\), \({\omega}_2\), \({\nu}_1\) and \({\nu}_2\) are four weight functions (i.e., \({\omega}_i\) and \({\nu}_i\), \(i=1,2\) are locally integrable functions on \({\mathbb{R}}^N\) such that \(0< {\omega}_i(x), {\nu}_i(x)< {\infty}\) a.e. \(x{\in}{\mathbb{R}}^N\)), \({\Delta}\) is the Laplacian operator, \(1< q,s< p< {\infty}\), \(1/p + 1/p\,' =1\) and \(1/s+1/s\,'=1\).

For degenerate partial differential equations, i.e., equations with various types of singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces (see [1, 2, 3, 4, 5, 6, 7, 8]). The type of a weight depends on the equation type.

A class of weights, which is particularly well understood, is the class of \(A_p\) weights that was introduced by B.Muckenhoupt in the early 1970’s (see [7]). These classes have found many useful applications in harmonic analysis (see [9] and [10]). Another reason for studying \(A_p\)-weights is the fact that powers of the distance to submanifolds of \({\mathbb{R}}^N\) often belong to \(A_p\) (see [8] and [11]). There are, in fact, many interesting examples of weights (see [7] for p-admissible weights).

In the non-degenerate case (i.e. with \({\omega}(x) \equiv 1\)), for all \(f\, {\in}\,L^p(\Omega)\) the Poisson equation associated with the Dirichlet problem \[ \left\{ \begin{array}{ll} &\, – \, {\Delta}u = f(x), \ {\textrm{in}} \ {\Omega} \\ & u(x) = 0, \ {\textrm{in}} \ {\partial\Omega} \end{array} \right. \] is uniquely solvable in \(W^{2,p}(\Omega)\,{\cap}\, W_0^{1,p}(\Omega)\) (see [12]), and the nonlinear Dirichlet problem \[ \left\{ \begin{array}{ll} &\, – \, {\Delta}_p u = f(x), \ {\textrm{in}} \ {\Omega} \\ & u(x) = 0, \ {\textrm{in}} \ {\partial\Omega} \end{array} \right. \] is uniquely solvable in \(W_0^{1,p}(\Omega)\) (see [13]), where \({\Delta}_p u = {\textrm{div}}({\vert {\nabla}u\vert}^{p-2}{\nabla}u)\) is the p-Laplacian operator. In the degenerate case, the degenerated p-Laplacian has been studied in [11].

The paper is organized as follow. In Section 2 we present the definitions and basic results. In Section 3 we prove our main result about existence and uniqueness of solutions for problem \((P)\).

2. Definitions and basic results

Let \(\Omega\) be an open set in \({\mathbb{R}}^n\). By the symbol \({\mathcal{W}}(\Omega)\) we denote the set of all measurable, a.e. in \(\Omega\) positive and finite functions \({\omega}={\omega}(x)\), \(x\, {\in}\, {\Omega}\). Elements of \({\mathcal{W}}(\Omega)\) will be called weight functions. Every weight \(\omega\) gives rise to a measure on the measurable subsets of \({\mathbb{R}}^N\) through integration. This measure will be denoted by \({\mu}_{\omega}\). Thus, \(\displaystyle {\mu}_{\omega}(E) = \int_E{\omega}(x)\, dx\) for measurable sets \(E\,{\subset}\,{\mathbb{R}}^N\).

Definition 1. Let \(1\,{\leq}\,p< {\infty}\). A weight \(\omega\) is said to be an \(A_p\)-weight, if there is a positive constant \(C = C({p , \omega})\) such that, for every ball \(B\,{\subset}\,{\mathbb{R}}^N\) \begin{eqnarray*} & & {\biggr(}{\frac{1}{{\vert B \vert}}} \int_B {\omega}(x)\,dx{\biggr)}{\biggl(}{\frac{1}{{\vert B \vert}}} \int_B{\omega}^{1/(1-p)}(x)\,dx{\biggr)}^{p-1}\,{\leq}\, C, \ \ {\textrm{if}} \ \ p>1,\\ & & {\biggr(}{\frac{1}{{\vert B \vert}}} \int_B{\omega}(x)\,dx{\biggr)}{\biggl(} { \textrm{ess}}\sup_{x\,{\in}\,B}{\frac{1}{{\omega}(x)}}{\biggr)}\, {\leq}C, \ \ {\textrm{if}} \ \ p=1, \end{eqnarray*} where \(\vert . \vert\) denotes the \(N\)-dimensional Lebesgue measure in \({\mathbb{R}}^N\).

If \(1< q\,{\leq}\,p\), then \(A_q\,{\subset}\,A_p\) (see [5, 6, 8] for more information about \(A_p\)-weights). As an example of an \(A_p\)-weight, the function \({\omega}(x) = {\vert x \vert}^{\alpha}\), \(x\,{\in}\,{\mathbb{R}}^N\), is in \(A_p\) if and only if \(-N< {\alpha}< N(p-1)\) (see [8], Chapter IX, Corollary 4.4). If \({\varphi}\, {\in}\, BMO({\mathbb{R}}^N)\), then \({\omega}(x) = {\textrm{e}}^{{\alpha}\, {\varphi}(x)}\,{\in}\,A_2\) for some \({\alpha}>0\) (see [9]).

Remark 1. If \({\omega}\, {\in}\, A_p\), \(1< p< {\infty}\), then $${\biggl(} {\frac{\vert E \vert}{\vert B \vert}}{\biggr)}^p\, {\leq} \, C \, {\frac{{\mu}_{\omega}(E)}{{\mu}_{\omega}(B)}}$$ for all measurable subsets \(E\) of \(B\) (see 15.5 strong doubling property in [6]). Therefore, \({\mu}_{\omega}(E)=0\) if and only if \({\vert E \vert}=0\); so there is no need to specify the measure when using the ubiquitous expression almost everywhere and almost every, both abbreviated a.e..

Definition 2. Let \(\omega\) be a weight. We shall denote by \(L^p(\Omega ,\omega)\) (\(1\,{\leq}\,p< {\infty}\)) the Banach space of all measurable functions \(f\) defined in \(\Omega\) for which $${\Vert f \Vert}_{L^p(\Omega ,\omega)} = {\bigg(}\int_{\Omega} {\vert f(x) \vert}^p{\omega}(x)\,dx{\bigg)}^{1/p}< {\infty}.$$ We denote \(\displaystyle [L^{p}(\Omega , \omega)]^N = L^{p}(\Omega , \omega)\,{\times}…{\times}\, L^{p}(\Omega , \omega)\).

Remark 2. If \({\omega}\,{\in}\,A_p\), \(1< p< \infty\), then since \({\omega}^{-1/(p-1)}\) is locally integrable, we have \(L^p(\Omega , \omega)\,{\subset}\,L^1_{\textrm{loc}}(\Omega)\) (see [8], Remark 1.2.4). It thus makes sense to talk about weak derivatives of functions in \(L^p(\Omega , \omega)\).

Definition 3. Let \({\Omega}\,{\subset}\,{\mathbb{R}}^N\) be a bounded open set, \(1< p< {\infty}\), \(k\) be a nonnegative integer and \({\omega}\,{\in}\,A_p\). We shall denote by \(W^{k,p}(\Omega , \omega)\), the weighted Sobolev spaces, the set of all functions \(u\,{\in}\,L^p(\Omega , \omega)\) with weak derivatives \(D^{\alpha}u\,{\in}\,L^p(\Omega , \omega)\), \(1\,{\leq}\,{\vert \alpha\vert}\,{\leq}\,k\). The norm in the space \(W^{k,p}(\Omega , \omega)\) is defined by

We also define the space \(W_0^{k,p}(\Omega , \omega)\) as the closure of \(C_0^{\infty}(\Omega)\) with respect to the norm (1). We have that the spaces \(W^{k,p}(\Omega , \omega)\) and \(W_0^{k,p}(\Omega , \omega)\) are Banach spaces (see Proposition 2.1.2 in [8]).

The dual space of \(W_0^{1,p}(\Omega , \omega)\) is the space \([W_0^{1,p}(\Omega , \omega)]^* = W^{-1,p\,’}(\Omega , \omega)\), $$W^{-1,p\,’}(\Omega , \omega) = \{T=f-{\textrm{div}}(G): G=(g_1,…,g_N), {\dfrac{f}{\omega}}, {\dfrac{g_j}{\omega}}\, {\in}\, L^{p\,’}(\Omega , \omega)\}.$$ It is evident that a weight function \(\omega\) which satisfies \(0< C_1\,{\leq}\,{\omega}(x)\,{\leq}\,C_2\), for a.e. \(x\,{\in}\,{\Omega}\), gives nothing new (the space \({\textrm{W}}^{k,p}(\Omega , \omega)\) is then identical with the classical Sobolev space \({\textrm{W}}^{k,p}(\Omega)\)). Consequently, we shall be interested in all above such weight functions \(\omega\) which either vanish somewhere in \({\Omega}\,{\cup}\,{\partial\Omega}\) or increase to infinity (or both). We need the following basics results.

Theorem 4. (The weighted Sobolev inequality) Let \({\Omega}\,{\subset}\,{\mathbb{R}}^N\) be a bounded open set and let \({\omega}\) be an \(A_p\)-weight, \(1< p< {\infty}\). Then there exists positive constants \(C_{\Omega}\) and \(\delta\) such that for all \(\,u{\in}\,W_0^{1,p}(\Omega, \omega)\) and \(\displaystyle 1\,{\leq}\,{\eta}\,{\leq}\, N/(N-1) + {\delta}\)

Proof. Its suffices to prove the inequality for functions \(u\, {\in}\, C_0^{\infty}(\Omega)\) (see Theorem 1.3 in [4]). To extend the estimates (2) to arbitrary \(u\, {\in}\, W_0^{1,p}(\Omega , \omega)\), we let \(\{u_m\}\) be a sequence of \(C_0^{\infty}(\Omega)\) functions tending to \(u\) in \(W_0^{1,p}(\Omega , \omega)\). Applying the estimates (2) to differences \(u_{m_1}-u_{m_2}\), we see that \(\{u_m\}\) will be a Cauchy sequence in \(L^p(\Omega , \omega)\). Consequently the limit function \(u\) will lie in the desired spaces and satisfy (2).

Lemma 5. (a) Let \(1\,< p0\) such that for all \({\xi}, {\eta}\, {\in}\, {\mathbb{R}}^N\), $${\big\vert}{\vert{\xi}\vert}^{p-2}\,{\xi} – {\vert{\eta}\vert}^{p-2}{\eta} {\big\vert}\, {\leq}\, C_p\, {\vert \xi – \eta \vert}(\,{\vert \xi \vert} + {\vert \eta \vert})^{p-2}.$$ (b) Let \(1< p< {\infty}\). There exist two positive constants \({\alpha}_p\) and \({\beta}_p\) such that for every \({\xi},{\eta}\, {\in}\,{\mathbb{R}}^N\) (\(N\,{\geq}\,1)\) $${\alpha}_p(\,{\vert \xi \vert}+{\vert \eta \vert})^{p-2}{\vert \xi – \eta \vert}^2\, {\leq}\,{\langle}\,{\vert \xi \vert}^{p-2}{\xi} – {\vert \eta \vert}^{p-2}{\eta} , {\xi – \eta}{\rangle}\,{\leq}\, {\beta}_p(\,{\vert \xi\vert} + {\vert {\eta}\vert})^{p-2}{\vert \xi – \eta\vert},$$ where \({\langle}. , . {\rangle}\) denotes here the Euclidian scalar product in \({\mathbb{R}}^N\).

Proof. See Proposition 17.2 and Proposition 17.3 in [13].

3. Weak Solutions

Let \({\omega}_1, {\omega}_2\, {\in}\, A_p\) and \({\nu}_1,{\nu}_2\,{\in}\, {\cal W}(\Omega)\), \(1< q,s< p< {\infty}\). We denote by \(X\) the space \(\displaystyle X = W^{2,p}(\Omega , {\omega}_1)\, {\cap}\, W_0^{1,p}(\Omega , {\omega}_2)\) with the norm $${\Vert u \Vert}_X = {\bigg(} \int_{\Omega}{\vert {\nabla}u\vert}^p\, {\omega}_2\, dx + \int_{\Omega}{\vert {\Delta}u\vert}^p\, {\omega}_1\, dx{\bigg)}^{1/p}.$$ In this section we prove the existence and uniqueness of weak solutions \(u\, {\in}\,X\) to the Navier problem \[ (P)\left\{ \begin{array}{lll} & Lu(x) = f(x) – {\textrm{div}}(G(x)),\ \ {\textrm{in}} \ \ {\Omega}, \\ & u(x) = {\Delta}u= 0, \ \ {\textrm{in}} \ \ {\partial\Omega}, \end{array} \right. \] where \(\Omega\) is a bounded open set of \({\mathbb{R}}^N\) (\(N\,{\geq}\,2\)), \(\displaystyle{\dfrac{f}{{\omega}_2}}\,{\in}\,L^{p\,'}(\Omega, {\omega}_2)\) and \(\displaystyle{\dfrac{G}{{\nu}_2}}\,{\in}\, [L^{s\,'}(\Omega , {\nu}_2)]^N\), \(G=(g_1,…,g_N)\).

Definition 6. We say that \(u\, {\in}\,X\) is a weak solution for problem \((P)\) if

for all \({\varphi}\,{\in}\,X\), with \(f/{\omega}_2\, {\in}\, L^{p\,’}(\Omega , {\omega}_2)\) and \(G/{\nu}_2\, {\in}\, [L^{s\,’}(\Omega , {\nu}_2)]^N\), where \({\langle}.,.{\rangle}\) denotes here the Euclidean scalar product in \({\mathbb{R}}^N\).

Remark 3 (a) Since \(1< q,s< p 0\) such that $${\Vert u \Vert}_{L^q(\Omega , {\nu}_1)}\, {\leq}\, M_1{\Vert u \Vert}_{L^p(\Omega, {\omega}_1)} \ {\textrm{and}} \ {\Vert u \Vert}_{L^s(\Omega , {\nu}_2)}\, {\leq}\, M_2{\Vert u \Vert}_{L^p(\Omega, {\omega}_2)}$$ where \(\displaystyle M_1= {\bigg[}\int_{\Omega} {\bigg(}{\dfrac{{\nu}_1}{{\omega}_1}}{\bigg)}{\omega}_1\,dx{\bigg]}^{(p-q)/p\,q}\) and \(\displaystyle M_2= {\bigg[}\int_{\Omega} {\bigg(}{\dfrac{{\nu}_2}{{\omega}_2}}{\bigg)}{\omega}_2\,dx{\bigg]}^{(p-s)/p\,s}\). In fact, since \(1< q,s< p1\) and \(r’ = p/(p-q)\), \begin{eqnarray*} {\Vert u \Vert}_{L^q(\Omega , {\nu}_1)}^q & = & \int_{\Omega}{\vert u \vert}^q\,{\nu}_1\, dx = \int_{\Omega}{\vert u \vert}^q {\dfrac{{\nu}_1}{{\omega}_1}}\, {\omega}_1\, dx\\ &{\leq}& {\bigg(}\int_{\Omega}{\vert u \vert}^{q\, r}\, {\omega}_1\, dx{\bigg)}^{1/r}{\bigg(}\int_{\Omega}{\bigg(}{\dfrac{{\nu}_1}{{\omega}_1}}{\bigg)}^{r\,’}\,{\omega}_1\, dx{\bigg)}^{1/r\,’}\\ & = & {\bigg(}\int_{\Omega}{\vert u \vert}^p\, {\omega}_1\, dx{\bigg)}^{q/p}{\bigg(}\int_{\Omega}{\bigg(}{\dfrac{{\nu}_1}{{\omega}_1}}{\bigg)}^{p/(p-q)}\,{\omega}_1\, dx{\bigg)}^{(p-q)/p}. \end{eqnarray*} Hence, \(\displaystyle {\Vert u \Vert}_{L^q(\Omega , {\nu}_1)}\, {\leq}\, M_1\, {\Vert u \Vert}_{L^p(\Omega , {\omega}_1)}\). Analogously, we obtain \(\displaystyle {\Vert u \Vert}_{L^s(\Omega , {\nu}_2)}\, {\leq}\, M_2\, {\Vert u \Vert}_{L^p(\Omega , {\omega}_2)}\).
(b) Using the estimate in (a) we have \begin{eqnarray*} {\bigg\vert}\int_{\Omega}{\vert {\Delta}u\vert}^{q-2}{\Delta}u\, {\Delta}{\varphi}\, {\nu}_1\, dx{\bigg\vert} & {\leq} & \int_{\Omega}{\vert {\Delta}u\vert}^{q-1}\, {\vert{\Delta}{\varphi}\vert}\, {\nu}_1\, dx\\ & {\leq}&{\bigg(}\int_{\Omega}{\vert{\Delta}u\vert}^{(q-1)\,q\,’}{\nu}_1\, dx{\bigg)}^{1/q\,’}{\bigg(}\int_{\Omega}{\vert{\Delta}{\varphi}\vert}^q\, {\nu}_1\, dx{\bigg)}^{1/q}\\ & = & {\bigg(}\int_{\Omega}{\vert{\Delta}u\vert}^q\, {\nu}_1\, dx{\bigg)}^{(q-1)/q}{\bigg(}\int_{\Omega}{\vert{\Delta}{\varphi}\vert}^q\,{\nu}_1\, dx{\bigg)}^{1/q}\\ & = & {\Vert {\Delta} u \Vert}_{L^q(\Omega , {\nu}_1)}^{q-1}{\Vert {\Delta}{\varphi}\Vert}_{L^q(\Omega , {\nu}_1)}\\ & {\leq} & M_1^{q-1}\, {\Vert {\Delta}u \Vert}_{L^p(\Omega , {\omega}_1)}^{q-1} \, M_1\, {\Vert {\Delta}{\varphi}\Vert}_{L^p(\Omega , {\omega}_1)}\\ & {\leq}& M_1^q\, {\Vert u \Vert}_X\, {\Vert \varphi \Vert}_X, \end{eqnarray*} and, analogously, we also have $${\bigg\vert}\int_{\Omega}{\vert{\nabla}u\vert}^{s-2}{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\, {\nu}_2\, dx{\bigg\vert}\, {\leq}\, M_2^s\,{\Vert u \Vert}_X\, {\Vert \varphi \Vert}_X.$$

Theorem 7. Let \({\omega}_i\, {\in}\, A_p\), \({\nu}_i\, {\in}\,{\cal W}(\Omega)\) (\(i=1,2\)), \(1< q, s < p< {\infty}\). Suppose that
(a) \(\displaystyle {\dfrac{{\nu}_1}{{\omega}_1}}\, {\in}\,L^{p/(p-q)}(\Omega , {\omega}_1)\) and \(\displaystyle {\dfrac{{\nu}_2}{{\omega}_2}}\, {\in}\,L^{p/(p-s)}(\Omega , {\omega}_2)\);
(b) \(f/{\omega}_2\, {\in}\,L^{p\,’}(\Omega , {\omega}_2)\) and \(G/{\nu}_2\, {\in}\, [L^{s\,’}(\Omega , {\nu}_2)]^N\).
Then the problem \((P)\) has a unique solution \(u\, {\in}\,X\) and $${\Vert u \Vert}_X\, {\leq}\,{\bigg[}C_{\Omega} {\bigg\Vert {\dfrac{f}{{\omega}_2}}\bigg\Vert}_{L^{p\,’}(\Omega , {\omega}_2)} + M_2{\bigg\Vert {\dfrac{{\vert G \vert}}{{\nu}_2}}\bigg\Vert}_{L^{s\,’}(\Omega , {\nu}_2)}{\bigg]}^{1/(p-1)},$$ where \(C_{\Omega}\) is the constant in Theorem 3 and \(M_2\) is the constant in 3 (a).

Proof. (I) Existence. By Theorem 4 (with \({\eta}=1\)), we have that \begin{eqnarray*} {\bigg\vert}\int_{\Omega}f\, {\varphi}\,dx{\bigg\vert} & {\leq} & {\bigg(}\int_{\Omega} {\bigg\vert}{\dfrac{f} {{\omega}_2}}{\bigg\vert}^{p\,’}\, {\omega}_2\, dx{\bigg)}^{1/p\,’}{\bigg(}\int_{\Omega}{\vert\,\varphi\vert}^p\, {\omega}_2\, dx{\bigg)}^{1/p} \end{eqnarray*}

and by Remark 3(a) Define the functional \(J:X\, {\rightarrow}\, {\mathbb{R}}\) by \begin{eqnarray*} J(\varphi) & = & {\dfrac{1}{p}}\, \int_{\Omega}{\vert}{\Delta}{\varphi}{\vert}^p\, {\omega}_1\, dx + {\dfrac{1}{q}}\, \int_{\Omega}{\vert}{\Delta}{\varphi}{\vert}^q\, {\nu}_1\, dx\\ && + {\dfrac{1}{p}}\, \int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^p\, {\omega}_2\, dx + {\dfrac{1}{s}}\, \int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^s\, {\nu}_2\, dx – \int_{\Omega}\, f\, {\varphi}\, dx – \int_{\Omega}{\langle}G , {\nabla}{\varphi}{\rangle}\, dx. \end{eqnarray*} Using (4), (5), Remark 3(a) and Young’s inequality (\(a\,b\, {\leq}\, {\dfrac{a^p}{p}} + {\dfrac{b^{p\,’}}{p\,’}}\)), we have that \begin{eqnarray*} J(\varphi) & {\geq}& {\dfrac{1}{p}}\int_{\Omega}{\vert {\Delta}{\varphi} \vert}^p\,{\omega}_1\, dx +{\dfrac{1}{q}}\int_{\Omega}{\vert {\Delta}{\varphi}\vert}^q\,{\nu}_1\,dx + {\dfrac{1}{p}}\,\int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^p\,{\omega}_2\, dx +{\dfrac{1}{s}}\,\int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^s\,{\nu}_2\, dx\\ && -{\bigg\Vert}{\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p’}(\Omega ,{\omega}_2)}{\Vert \varphi \Vert}_{L^p(\Omega , {\omega}_2)} -{\bigg\Vert}{\dfrac{\vert G\vert}{{\nu}_2}}{\bigg\Vert}_{L^{s’}(\Omega ,{\nu}_2)}{\Vert\,\vert {\nabla}{\varphi}\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}\\ & {\geq} & {\dfrac{1}{p}}\,\int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^p\,{\omega}_2\, dx+{\dfrac{1}{s}}\,\int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^s\,{\nu}_2\, dx – C_{\Omega}{\bigg\Vert}{\dfrac{ f}{{\omega}_2}}{\bigg\Vert}_{L^{p’}(\Omega , {\omega}_2)}{\Vert\,\vert{\nabla}{\varphi}\vert\,\Vert}_{L^p(\Omega , {\omega}_2)}\\ && – {\bigg\Vert}{\dfrac{\vert G\vert}{{\nu}_2}}{\bigg\Vert}_{L^{s’}(\Omega , {\nu}_2)} {\Vert\,\vert{\nabla}{\varphi}\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}\\ & {\geq} & {\dfrac{1}{p}}\, \int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^p\,{\omega}_2\, dx +{\dfrac{1}{s}}\, \int_{\Omega}{\vert}{\nabla}{\varphi}{\vert}^s\,{\nu}_2\, dx – {\dfrac{C_{\Omega}^{p’}}{p’}}{\bigg\Vert} {\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p’}(\Omega , {\omega}_2)}^{p’} \\&&- {\dfrac{1}{p}}\,{\Vert\,\vert {\nabla}{\varphi}\vert\,\Vert}_{L^p(\Omega , {\omega}_2)}^p – {\dfrac{1}{s’}}\,{\bigg\Vert}{\dfrac{\vert G\vert}{{\nu}_2}}{\bigg\Vert}_{L^{s’}(\Omega , {\nu}_2)}^{s’} – {\dfrac{1}{s}}\,{\Vert\,\vert{\nabla}{\varphi}\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}^s\\ & = & – {\dfrac{C_{\Omega}^{p’}}{p\,’}}{\bigg\Vert} {\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p’}(\Omega , {\omega}_2)}^{p’} – {\dfrac{1}{s\,’}}{\bigg\Vert} {\dfrac{\vert G \vert}{{\nu}_2}}{\bigg\Vert}_{L^{s’}(\Omega , {\nu}_2)}^{s’} \end{eqnarray*} that is, \(J\) is bounded from below. Let \(\{u_n\}\) be a minimizing sequence, that is, a sequence such that $$J(u_n)\,\,\, {\rightarrow}\, \inf_{{\varphi}\, {\in}\,X}J(\varphi)\,.$$ Then for \(n\) large enough, we obtain \begin{eqnarray*}0\, {\geq}\, J(u_n) & = & {\dfrac{1}{p}}\int_{\Omega}{\vert{\Delta}u_n\vert}^p\,{\omega}_1\, dx + {\dfrac{1}{q}}\int_{\Omega}{\vert{\Delta}u_n\vert}^q\,{\nu}_1\, dx + {\dfrac{1}{p}}\int_{\Omega}{\vert {\nabla}u_n\vert}^p\,{\omega}_2\, dx + {\dfrac{1}{s}}\, \int_{\Omega}{\vert}{\nabla}u_n{\vert}^s\, {\nu}_2\, dx\\ & – & \int_{\Omega}f\, u_n\, dx – \int_{\Omega}{\langle}G , {\nabla}u_n{\rangle}\, dx, \end{eqnarray*} and we have \begin{eqnarray*} & & {\dfrac{1}{p}}\int_{\Omega}{\vert{\Delta}u_n\vert}^p\,{\omega}_1\, dx + {\dfrac{1}{p}}\int_{\Omega}{\vert{\nabla}u_n\vert}^{p}\, {\omega}_2\, dx\end{eqnarray*} Hence, by Theorem 4 (with \({\eta}=1\)), Remark 3(a) and (6), we obtain \begin{eqnarray*} {\Vert} u_n{\Vert}_X^p &=& \int_{\Omega}{\vert {\Delta}u_n\vert}^p\,{\omega}_1\, dx + \int_{\Omega}{\vert {\nabla}u_n\vert}^p\,{\omega}_2\, dx\\ & & {\leq}\, p {\bigg(}\int_{\Omega}f\, u_n\, dx + \int_{\Omega}{\langle}G , {\nabla}u_n{\rangle}\, dx{\bigg)}\\ & & {\leq} \, p\,{\bigg(}\, {\bigg\Vert} {\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p’}(\Omega , {\omega}_2)}\, {\Vert u_n \Vert}_{L^p(\Omega , {\omega}_2)} + {\bigg\Vert {\dfrac{\vert G\vert}{{\nu}_2}} \bigg\Vert}_{L^{s’}(\Omega , {\nu}_2)} {\Vert\,\vert {\nabla}u_n\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}\,{\bigg)}\\ & & {\leq}\, p\, {\bigg(}C_{\Omega}\, {\bigg\Vert}{\dfrac{f}{{\omega}_2}} {\bigg\Vert}_{L^{p’}(\Omega , {\omega}_2)}{\Vert\,\vert {\nabla}u_n\vert\,\Vert}_{L^p(\Omega , {\omega}_2)} + M_2\,{\bigg\Vert {\dfrac{\vert G\vert}{{\nu}_2}} \bigg\Vert}_{L^{s’}(\Omega , {\nu}_2)}{\Vert\,\vert{\nabla}u_n\vert\,\Vert}_{L^p(\Omega , {\omega}_2)}{\bigg)}\\ & & {\leq}\, p\, {\bigg(}C_{\Omega}\, {\bigg\Vert} {\dfrac{f}{{\omega}_2}} {\bigg\Vert}_{L^{p’}(\Omega , {\omega}_2)} + M_2\,{\bigg\Vert {\dfrac{\vert G\vert}{{\nu}_2}} \bigg\Vert}_{L^{q’}(\Omega , {\nu}_2)}{\bigg)}{\Vert}u_n{\Vert}_X. \end{eqnarray*} Hence, $$\displaystyle {\Vert}u_n{\Vert}_X\, {\leq}\, {\bigg[}p\, {\bigg(}\,C_{\Omega}\, {\bigg\Vert} {\dfrac{f}{{\omega}_2}} {\bigg\Vert}_{L^{p\,’}(\Omega , {\omega}_2)} + M_2\, {\bigg\Vert} {\dfrac{\vert G\vert}{{\nu}_2}}{\bigg\Vert}_{L^{s’}(\Omega , {\nu}_2)}{\bigg)}{\bigg]}^{1/(p-1)}.$$ Therefore \(\{u_n\}\) is bounded in \(X\). Since \(X\) is reflexive, there exists a subsequence, still denoted by \(\{u_n\}\), and a function \(u\,{\in}\,X\) such that \(u_n{\rightharpoonup}\, u\) in \(X\). Since, $$ X\, {\ni}\,{\varphi}\, \mapsto \, \int_{\Omega}\, f\, {\varphi}\, dx + \int_{\Omega}{\langle}G,{\nabla}{\varphi}{\rangle}\, dx,$$ and $$ X\,{\ni}\,{\varphi}\, \mapsto \,{\Vert{\Delta}{\varphi}\Vert}_{L^p(\Omega , {\omega}_1)}^p + {\Vert{\Delta}{\varphi}\Vert}_{L^q(\Omega , {\nu}_1)}^q + {\Vert\,\vert{\nabla}{\varphi}\vert\,\Vert}_{L^p(\Omega , {\omega}_2)}^p + {\Vert\,\vert {\nabla}{\varphi}\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}^s,$$ are continuous then \(J\) is continuous. Moreover since \(1< q,s< p< {\infty}\) we have that \(J\) is convex and thus lower semi-continuous for the weak convergence. It follows that $$J(u)\, {\leq}\, \liminf_{n} J(u_n)\, = \, \inf_{{\varphi}\, {\in}\,X}J(\varphi),$$ and thus \(u\) is a minimizer of \(J\) on \( X \) (see Theorem 25.C and Corollary 25.15 in [14]). For any \({\varphi}\, {\in}\,X\) the function \begin{eqnarray*} {\lambda}\, \mapsto \, & & {\dfrac{1}{p}}\int_{\Omega}{\vert}{\Delta}(u+{\lambda}{\varphi}){\vert}^p\, {\omega}_1\, dx + {\dfrac{1}{q}}\int_{\Omega}{\vert}{\Delta}(u+{\lambda}{\varphi}){\vert}^q\, {\nu}_1\, dx + {\dfrac{1}{p}}\int_{\Omega}{\vert{\nabla}(u+{\lambda}\, {\varphi})\vert}^p\,{\omega}_2\, dx \\ & &+ {\dfrac{1}{s}} \, \int_{\Omega}{\vert}{\nabla}(u+{\lambda}{\varphi}){\vert}^s\, {\nu}_2\, dx – \int_{\Omega}(u+{\lambda}\, {\varphi})\, f\, dx – \int_{\Omega}{\langle}G , {\nabla}(u+{\lambda}\, {\varphi}){\rangle}\,dx \end{eqnarray*} has a minimum at \({\lambda}=0\). Hence, $${\dfrac{d}{d{\lambda}}}{\bigg(}J(u+{\lambda}\,{\varphi}){\bigg)} {\bigg\vert}_{{\lambda}=0} = 0, \ {\forall}\, {\varphi}\, {\in}\, X.$$ We have $${\dfrac{d}{d\,{\lambda}}} \, {\bigg(}{\vert}\,{\nabla}(u+{\lambda}\, {\varphi}){\vert}^p\, {\omega}_2{\bigg)} = p\, \{{\vert}{\nabla}(u+{\lambda}\, {\varphi}){\vert}^{p-2} ({\langle}{\nabla}u, {\nabla}{\varphi}{\rangle} + {\lambda}\, {\vert}{\nabla}{\varphi}{\vert}^2)\}\,{\omega}_2,$$ and $${\dfrac{d}{d\,{\lambda}}} \, {\bigg(}{\vert}\,{\Delta}(u+{\lambda}\, {\varphi}){\vert}^p\, {\omega}_1{\bigg)} = p\, {\vert{\Delta}u + {\lambda}{\Delta}{\varphi}\vert}^{p-2}(\,{\Delta}u + {\lambda}{\Delta}{\varphi})\,{\Delta}{\varphi}\,{\omega}_1,$$ and we obtain \begin{eqnarray*} 0 & = & {\dfrac{d}{d{\lambda}}}{\bigg(}J(u+{\lambda}\,{\varphi}){\bigg)} {\bigg\vert}_{{\lambda}=0} = {\bigg[}{\dfrac{1}{p}}{\bigg(}p\, \int_{\Omega}{\vert}{\nabla}(u+{\lambda}\,{\varphi}){\vert}^{p-2}({\langle}{\nabla}u, {\nabla}{\varphi}{\rangle} + {\lambda}\,{\vert}{\nabla}{\varphi}{\vert}^2)\,{\omega}_2\, dx\end{eqnarray*} \begin{eqnarray*} && + p\,\int_{\Omega}{\vert {\Delta}u + {\lambda}{\Delta}{\varphi}\vert}^{p-2}(\,{\Delta}u + {\lambda}{\Delta}{\varphi})\,{\Delta}{\varphi}\, {\omega}_1\, dx{\bigg)} + {\dfrac{1}{s}}{\bigg(}s\, \int_{\Omega}{\vert}{\nabla}(u+{\lambda}\,{\varphi}){\vert}^{s-2}({\langle}{\nabla}u, {\nabla}{\varphi}{\rangle} + {\lambda}\,{\vert}{\nabla}{\varphi}{\vert}^2)\,{\nu}_2\, dx{\bigg)}\nonumber\\ && + {\dfrac{1}{q}}{\bigg(} q\,\int_{\Omega}{\vert {\Delta}u + {\lambda}{\Delta}{\varphi}\vert}^{q-2}(\,{\Delta}u + {\lambda}{\Delta}{\varphi})\,{\Delta}{\varphi}\, {\nu}_1\, dx{\bigg)} – \int_{\Omega}{\varphi}\, f \, dx- \int_{\Omega}{\langle}G , {\nabla}{\varphi}{\rangle}\, dx {\bigg]} {\bigg\vert}_{{\lambda}=0}\\ & = & \int_{\Omega} {\vert{\Delta}u\vert}^{p-2}{\Delta}u\,{\Delta}{\varphi}\, {\omega}_1\, dx + \int_{\Omega}{\vert}{\nabla}u{\vert}^{p-2}\,{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\, {\omega}_2\, dx + \int_{\Omega} {\vert{\Delta}u\vert}^{q-2}{\Delta}u\,{\Delta}{\varphi}\, {\nu}_1\, dx \nonumber\\ &&+ \int_{\Omega}{\vert}{\nabla}u{\vert}^{s-2}\,{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\, {\nu}_2\, dx – \int_{\Omega} f\, {\varphi}\, dx – \int_{\Omega}{\langle}G , {\nabla}{\varphi}{\rangle}\, dx. \end{eqnarray*} Therefore \begin{eqnarray*} & & \int_{\Omega}{\vert{\Delta}u\vert}^{p-2}{\Delta}u\,{\Delta}{\varphi}\, {\omega}_1\,dx + \int_{\Omega} {\vert}{\nabla}u{\vert}^{p-2}{\langle}{\nabla}u\,{\nabla}{\varphi}{\rangle}\, {\omega}_2\, dx + \int_{\Omega}{\vert{\Delta}u\vert}^{q-2}{\Delta}u\,{\Delta}{\varphi}\, {\nu}_1\,dx + \int_{\Omega}{\vert}{\nabla}u{\vert}^{s-2}\,{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\, {\nu}_2\, dx\\&& = \int_{\Omega} f\, {\varphi}\, dx + \int_{\Omega}{\langle}G , {\nabla}{\varphi}{\rangle}\, dx, \end{eqnarray*} for all \({\varphi}\, {\in}\, X\), that is, \(u\, {\in}\, X\) is a solution of problem \((P)\).
(II) Uniqueness. If \(u_1, u_2\, {\in}\, X\) are two weak solutions of problem \((P)\), we have \begin{eqnarray*} & & \int_{\Omega}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1\,{\Delta}{\varphi}\,{\omega}_1\, dx + \int_{\Omega}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1\,{\Delta}{\varphi}\, {\nu}_1\, dx + \int_{\Omega}{\vert}{\nabla}u_1{\vert}^{p-2}\,{\langle}{\nabla}u_1, {\nabla}{\varphi}{\rangle}\, {\omega}_2\, dx\\&& + \int_{\Omega}{\vert}{\nabla}u_1{\vert}^{s-2}\,{\langle}{\nabla}u_1, {\nabla}{\varphi}{\rangle}\, {\nu}_2\, dx = \int_{\Omega}f\, {\varphi}\, dx + \int_{\Omega}{\langle} G , {\nabla}{\varphi}{\rangle}\, dx, \end{eqnarray*} and \begin{eqnarray*} & & \int_{\Omega}{\vert{\Delta}u_2\vert}^{p-2}{\Delta}u_2\,{\Delta}{\varphi}\,{\omega}_1\, dx + \int_{\Omega}{\vert{\Delta}u_2\vert}^{q-2}{\Delta}u_2\,{\Delta}{\varphi}\, {\nu}_1\, dx + \int_{\Omega}{\vert}{\nabla}u_2{\vert}^{p-2}\,{\langle}{\nabla}u_2, {\nabla}{\varphi}{\rangle}\, {\omega}_2\, dx \\&&+ \int_{\Omega}{\vert}{\nabla}u_2{\vert}^{s-2}\,{\langle}{\nabla}u_2, {\nabla}{\varphi}{\rangle}\, {\nu}_2\, dx = \int_{\Omega}f\, {\varphi}\, dx + \int_{\Omega}{\langle} G , {\nabla}{\varphi}{\rangle}\, dx, \end{eqnarray*} for all \({\varphi}\in X\). Hence \begin{eqnarray*} & &\int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1 – {\vert{\Delta}u_2\vert}^{p-2}{\Delta}u_2{\bigg)}{\Delta}{\varphi}\, {\omega}_1\, dx + \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1 – {\vert{\Delta}u_2\vert}^{q-2}{\Delta}u_2{\bigg)}{\Delta}{\varphi}\, {\nu}_1\, dx\\ & & + \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert}^{p-2}{\langle}{\nabla}u_1, {\nabla}{\varphi}{\rangle} – {\vert}{\nabla}u_2{\vert}^{p-2}{\langle}{\nabla}u_2, {\nabla}{\varphi}{\rangle}{\bigg)}\,{\omega}_2\, dx + \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert}^{s-2}{\langle}{\nabla}u_1, {\nabla}{\varphi}{\rangle} \\&&- {\vert}{\nabla}u_2{\vert}^{s-2}{\langle}{\nabla}u_2, {\nabla}{\varphi}{\rangle}{\bigg)}{\nu}_2\, dx = 0. \end{eqnarray*} Taking \({\varphi}= u_1-u_2\), and using Lemma 5 (b) there exist positive constants \({\alpha}_p, {\tilde{\alpha}_p}, {\alpha}_q, {\alpha}_s\) such that \begin{eqnarray*} 0 & = & \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1-{\vert{\Delta}u_2\vert}^{p-2}{\Delta}u_2{\bigg)} (\,{\Delta}u_1 – {\Delta}u_2)\, {\omega}_1\, dx\\ && + \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1-{\vert{\Delta}u_2\vert}^{q-2} {\Delta}u_2{\bigg)}(\,{\Delta}u_1 – {\Delta}u_2)\, {\nu}_1\, dx\\ && + \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert}^{p-2}{\langle}{\nabla}u_1, {\nabla}u_1 – {\nabla}u_2{\rangle} – {\vert}{\nabla}u_2{\vert}^{p-2}{\langle}{\nabla}u_2, {\nabla}u_1-{\nabla}u_2{\rangle}{\bigg)}\,{\omega}_2\, dx\\ && + \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert}^{s-2}{\langle}{\nabla}u_1, {\nabla}u_1 – {\nabla}u_2{\rangle} – {\vert}{\nabla}u_2{\vert}^{s-2}{\langle}{\nabla}u_2, {\nabla}u_1-{\nabla}u_2{\rangle}{\bigg)}\,{\nu}_2\, dx\\ & = & \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1-{\vert{\Delta}u_2\vert}^{p-2}{\Delta}u_2{\bigg)} (\,{\Delta}u_1 – {\Delta}u_2)\, {\omega}_1\, dx\\ && + \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1-{\vert{\Delta}u_2\vert}^{q-2} {\Delta}u_2{\bigg)}(\,{\Delta}u_1 – {\Delta}u_2)\, {\nu}_1\, dx\\ && + \int_{\Omega}{\langle}\, {\vert}{\nabla}u_1{\vert}^{p-2}{\nabla}u_1 – {\vert}{\nabla}u_2{\vert}^{p-2}{\nabla}u_2, {\nabla}u_1 – {\nabla}u_2{\rangle}\, {\omega}_2\, dx\end{eqnarray*}\begin{eqnarray*} && + \int_{\Omega}{\langle}\, {\vert}{\nabla}u_1{\vert}^{s-2}{\nabla}u_1 – {\vert}{\nabla}u_2{\vert}^{s-2}{\nabla}u_2, {\nabla}u_1 – {\nabla}u_2{\rangle}\, {\nu}_2\, dx\\ & {\geq} & {\alpha}_p \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert} + {\vert{\Delta}u_2\vert}{\bigg)}^{p-2} {\vert {\Delta}u_1 – {\Delta}u_2\vert}^2\,{\omega}_1\, dx + \, {\tilde{\alpha}_p} \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert} + {\vert}{\nabla}u_2{\vert}{\bigg)}^{p-2}{\vert {\nabla}u_1 – {\nabla}u_2 \vert}^2\, {\omega}_2\, dx \\ && + {\alpha}_q\, \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert} + {\vert{\Delta}u_2\vert}{\bigg)}^{q-2} {\vert {\Delta}u_1 – {\Delta}u_2\vert}^2\,{\nu}_1\, dx + {\alpha}_s \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert} + {\vert}{\nabla}u_2{\vert}{\bigg)}^{s-2}{\vert {\nabla}u_1 – {\nabla}u_2 \vert}^2\, {\nu}_2\, dx\\ & {\geq} & {\alpha}_p \int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert} + {\vert{\Delta}u_2\vert}{\bigg)}^{p-2} {\vert {\Delta}u_1 – {\Delta}u_2\vert}^2\,{\omega}_1\, dx + {\tilde{\alpha}_p} \int_{\Omega}{\bigg(}{\vert}{\nabla}u_1{\vert} + {\vert}{\nabla}u_2{\vert}{\bigg)}^{p-2}{\vert {\nabla}u_1 – {\nabla}u_2 \vert}^2\, {\omega}_2\, dx. \end{eqnarray*} Therefore \({\Delta}u_1={\Delta}u_2\) and \({\nabla}u_1 = {\nabla}u_2\) a.e. and since \(u_1,u_2\, {\in}\,X\), then \(u_1 = u_2\) a.e. (by Remark 1).
(III) Estimate for \({\Vert u \Vert}_X\).
In particular, for \({\varphi}=u\, {\in}\, X\) in Definition 6 we have \begin{eqnarray*} \int_{\Omega}{\vert{\Delta}u\vert}^p\,{\omega}_1\, dx + \int_{\Omega}{\vert{\Delta}u\vert}^q\,{\nu}_1\, dx + \int_{\Omega}{\vert {\nabla}u\vert}^p\,{\omega}_2\, dx + \int_{\Omega}{\vert}{\nabla}u{\vert}^s\, {\nu}_2\, dx= \int_{\Omega}f\, u\, dx + \int_{\Omega}{\langle}G , {\nabla}u{\rangle}\, dx. \end{eqnarray*} Then, by Theorem 4 and Remark 3(a), we obtain \begin{eqnarray*} {\Vert u \Vert}_X^p & = & \int_{\Omega}{\vert{\Delta}u\vert}^p\,{\omega}_1\, dx + \int_{\Omega}{\vert{\nabla}u\vert}^p\,{\omega}_2\, dx\\ &{\leq} & \int_{\Omega}{\vert{\Delta}u\vert}^p\,{\omega}_1\, dx + \int_{\Omega}{\vert{\Delta}u\vert}^q\,{\nu}_1\, dx + \int_{\Omega}{\vert {\nabla}u\vert}^p\,{\omega}_2\, dx + \int_{\Omega}{\vert}{\nabla}u{\vert}^s\, {\nu}_2\, dx\\ & = & \int_{\Omega}f\,u\,dx + \int_{\Omega}{\langle}G, {\nabla}u{\rangle}\, dx\\ & {\leq} & {\bigg\Vert}{\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p’}(\Omega , {\omega}_2)}{\Vert u \Vert}_{L^p(\Omega , {\omega}_2)} + {\bigg\Vert}{\dfrac{\vert G \vert}{{\nu}_2}}{\bigg\Vert}_{L^{s’}(\Omega , {\nu}_2)}{\Vert\,\vert{\nabla}u\vert\,\Vert}_{L^s(\Omega , {\nu}_2)}\\ & {\leq} & C_{\Omega}{\bigg\Vert}{\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p\,’}(\Omega , {\omega}_2)}{\Vert\,\vert{\nabla}u\vert\,\Vert}_{L^p(\Omega , {\omega}_2)} + M_2 {\bigg\Vert}{\dfrac{\vert G \vert}{{\nu}_2}}{\bigg\Vert}_{L^{s’}(\Omega , {\nu}_2)} {\Vert\,\vert{\nabla}u\vert\,\Vert}_{L^p(\Omega , {\omega}_2)}\\ & {\leq} & {\bigg(}C_{\Omega}{\bigg\Vert}{\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p\,’}(\Omega , {\omega}_2)} + M_2\,{\bigg\Vert}{\dfrac{\vert G \vert}{{\nu}_2}}{\bigg\Vert}_{L^{s’}(\Omega , {\nu}_2)}{\bigg)}{\Vert u \Vert}_X. \end{eqnarray*} Therefore, $${\Vert u \Vert}_X {\leq}\,{\bigg(}C_{\Omega}{\bigg\Vert}{\dfrac{f}{{\omega}_2}}{\bigg\Vert}_{L^{p\,’}(\Omega , {\omega}_2)} + M_2\,{\bigg\Vert}{\dfrac{\vert G \vert}{{\nu}_2}}{\bigg\Vert}_{L^{s\,’}(\Omega , {\nu}_2)}{\bigg)}^{1/(p-1)}.$$

Corollary 8. Under the assumptions of Theorem 7 with \(2\, {\leq}\,q,s < p < {\infty}\). If \(u_1,u_2\, {\in}\, X\) are solutions of \[ (P_1)\left\{ \begin{array}{lll} & Lu_1(x) = f(x) – {\textrm{div}}(G(x)), \ \ {\textrm{in}} \ \ {\Omega}, \\ & u_1(x) = {\Delta}u_1(x) = 0, \ \ {\textrm{in}} \ \ {\partial\Omega}, \end{array} \right. \] and \[ (P_2)\left\{ \begin{array}{lll} & Lu_2(x) = {\tilde{f}}(x) – {\textrm{div}}({\tilde{G}}(x)), \ \ {\textrm{in}} \ \ {\Omega}, \\ & u_2(x) = {\Delta}u_2(x) = 0, \ \ {\textrm{in}} \ \ {\partial\Omega}, \end{array} \right. \] then $${\Vert u_1 – u_2\Vert}_X\, {\leq}\, {\dfrac{1}{{{\gamma}}^{1/(p-1)}}}\, {\bigg(}C_{\Omega}\, {\bigg\Vert} {\dfrac{f-{\tilde{f}}}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2{\bigg\Vert} {\dfrac{{\vert G – {\tilde{G}} \vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\bigg)}^{1/(p-1)},$$ where \({\gamma}\) is a positive constant, \(C_{\Omega}\) and \(M_2\) are the same constants of Theorem 7.

Proof. If \(u_1\) and \(u_2\) are solutions of \((P1)\) and \((P2)\) then for all \({\varphi}\, {\in}\, X\) we have \begin{eqnarray*} \int_{\Omega}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1\,{\Delta}{\varphi}\, {\omega}_1\, dx + \int_{\Omega}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1\, {\Delta}{\varphi}\, {\nu}_1\, dx + \int_{\Omega}{\vert{\nabla}u_1\vert}^{p-2}{\langle}{\nabla}u_1, {\nabla}{\varphi}{\rangle}\, {\omega}_2\, dx \end{eqnarray*}

In particular, for \({\varphi}= u_1 – u_2\), we obtain in (7).
(i) By Lemma 5(b) and since \(2\, {\leq}\, q,s< p< {\infty}\), there exist two positive constants \({\alpha}_p\) and \({\alpha}_q\) such that \begin{eqnarray*} &&\int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{p-2}{\Delta}u_1 – {\vert{\Delta}_2\vert}^{p-2}{\Delta}u_2{\bigg)}\, {\Delta}(u_1 – u_2)\, {\omega}_1\, dx \geq {\alpha}_p\int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert} +{\vert{\Delta}u_2\vert}{\bigg)}^{p-2}\, {\vert}{\Delta}u_1 – {\Delta}u_2{\vert}^2\, {\omega}_1\, dx\\ & & {\geq}\, {\alpha}_p\, \int_{\Omega}{\vert {\Delta}u_1 – {\Delta}u_2\vert}^{p-2}{\vert}{\Delta} u_1 – {\Delta}u_2{\vert}^2\, {\omega}_1\, dx = {\alpha}_p \int_{\Omega}{\vert{\Delta}(u_1 – u_2)\vert}^p\, {\omega}_1\, dx, \end{eqnarray*} and analogously $$\int_{\Omega}{\bigg(}{\vert{\Delta}u_1\vert}^{q-2}{\Delta}u_1 – {\vert{\Delta}u_2\vert}^{q-2}{\Delta}u_2{\bigg)}\, {\Delta}(u_1 – u_2)\, {\nu}_1\, dx \, {\geq}\, {\alpha}_q\int_{\Omega}{\vert{\Delta}(u_1 – u_2)\vert}^q\, {\nu}_1\, dx\, {\geq}\, 0.$$ (ii) Since \(2\, {\leq}\, q,s< p< {\infty}\) and by Lemma 5(b), there exit two positive constants \({\tilde{\alpha}}_p\) and \({\alpha}_s\) such that \begin{eqnarray*} & & \int_{\Omega} {\bigg(}{\vert{\nabla}u_1\vert}^{p-2}{\langle}{\nabla}u_1, {\nabla}(u_1 – u_2){\rangle} – {\vert{\nabla}u_2\vert}^{p-2}{\langle}{\nabla}u_2, {\nabla}(u_1 – u_2){\rangle}{\bigg)}\, {\omega}_2\, dx\\ & & = \int_{\Omega}{\langle} {\vert{\nabla}u_1\vert}^{p-2}{\nabla}u_1 – {\vert{\nabla}u_2\vert}^{p-2}{\nabla}u_2 , {\nabla}(u_1 – u_2){\rangle}\, {\omega}_2\, dx\\ & & {\geq}\, {\tilde{\alpha}}_p\,\int_{\Omega}({\vert{\nabla}u_1\vert}+{\vert{\nabla}u_2\vert})^{p-2}{\vert{\nabla}u_1 – {\nabla}u_2\vert}^2\, {\omega}_2\, dx\\ & & {\geq}\,{\tilde{\alpha}}_p\int_{\Omega}{\vert{\nabla}u_1 – {\nabla}u_2\vert}^{p-2}\, {\vert{\nabla}u_1 – {\nabla}u_2\vert}^2\, {\omega}_2\, dx = \ {\tilde{\alpha}}_p \int_{\Omega}{\vert {\nabla}(u_1 – u_2)\vert}^p\, {\omega}_2\, dx, \end{eqnarray*} and analogously, \begin{eqnarray*} \int_{\Omega} {\bigg(}{\vert{\nabla}u_1\vert}^{s-2}{\langle}{\nabla}u_1, {\nabla}(u_1 – u_2){\rangle} – {\vert{\nabla}u_2\vert}^{s-2}{\langle}{\nabla}u_2, {\nabla}(u_1 – u_2){\rangle}{\bigg)}\, {\nu}_2\, dx {\geq}\, {\alpha}_s \int_{\Omega}{\vert {\nabla}(u_1 – u_2)\vert}^s\, {\nu}_2\, dx\, {\geq}\, 0. \end{eqnarray*} (iii) By Remark 3(a) we have \begin{eqnarray*} & & {\bigg\vert}\int_{\Omega} (f – {\tilde{f}})\, (u_1 – u_2)\, dx + \int_{\Omega}{\langle} G – {\tilde{G}}, {\nabla}(u_1 – u_2){\rangle}\, dx{\bigg\vert}\\ & & {\leq} \ {\bigg(}C_{\Omega} {\bigg\Vert} {\dfrac{f – {\tilde{f}}}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2{\bigg\Vert}{\dfrac{{\vert G – {\tilde{G}}\vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\bigg)}\, {\Vert u_1 – u_2\Vert}_X. \end{eqnarray*} Hence, with \({\gamma} = \min\{ {\alpha}_p, {\tilde{\alpha}}_p\}\), we obtain \begin{eqnarray*} & & {\gamma}\, {\Vert u_1 – u_2 \Vert}_X^p\, {\leq}\, {\alpha}_p\int_{\Omega}{\vert{\Delta}(u_1 – u_2)\vert}^p\, {\omega}_1\, dx + {\tilde{\alpha}}_p\int_{\Omega}{\vert{\nabla}(u_1 – u_2)\vert}^p\, {\omega}_2\, dx\\ & & {\leq}\,{\bigg(}C_{\Omega}{\bigg\Vert}{\dfrac{f – {\tilde{f}}}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2 \, {\bigg\Vert}{\dfrac{{\vert G – {\tilde{G}}\vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\bigg)}\, {\Vert u_1 – u_2\Vert}_X. \end{eqnarray*} Therefore, $${\Vert u_1 – u_2\Vert}_X\, {\leq}\, {\dfrac{1}{{\gamma}^{1/(p-1)}}}\, {\bigg(}C_{\Omega}{\bigg\Vert}{\dfrac{f – {\tilde{f}}}{{\omega}_2}}{\bigg\Vert}_{L^{p\,'}(\Omega , {\omega}_2)} + M_2 \, {\bigg\Vert}{\dfrac{{\vert G – {\tilde{G}}\vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\bigg)}^{1/(p-1)}.$$

Corollary 9. Assume \(2\,{\leq}\, q, s < p< {\infty}\). Let the assumptions of Theorem 7 be fulfilled, and let \(\{f_m\}\) and \(\{G_m\}\) be sequences of functions satisfying \(\displaystyle {\dfrac{f_m}{{\omega}_2}}\,{\rightarrow}\, {\dfrac{f}{{\omega}_2}}\) in \(L^{p\,'}(\Omega , {\omega}_2) \) and \(\displaystyle {\Bigg\Vert {\dfrac{\vert G_m – G \vert}{{\nu}_2}} \Bigg\Vert}_{L^{s\,'}(\Omega , {\nu}_2)}{\rightarrow}\,0\) as \(m\to\infty\). If \(u_m\,{\in}\, X\) is a solution of the problem \[ (P_m)\left\{ \begin{array}{lll} & Lu_m(x) = f_m(x) – {\textrm{div}}(G_m(x)), \ \ {\textrm{in}} \ \ {\Omega}, \\ & u_m(x) = {\Delta}u_m(x) = 0, \ \ {\textrm{in}} \ \ {\partial\Omega}, \end{array} \right. \] then \(u_m {\rightarrow}\, u\) in \(X\) and \(u\) is a solution of problem \((P)\).

Proof. By Corollary 8 we have $${\Vert u_m – u_r\Vert}_X\, {\leq}\, {\dfrac{1}{{\gamma}^{1/(p-1)}}}\, {\bigg(}C_{\Omega}{\bigg\Vert}{\dfrac{f_m – {f}_r}{{\omega}_2}}{\bigg\Vert}_{L^{p\,’}(\Omega , {\omega}_2)} + M_2 \, {\bigg\Vert}{\dfrac{{\vert G_m – {G}_r\vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,’}(\Omega , {\nu}_2)}{\bigg)}^{1/(p-1)}.$$ Therefore \(\{u_m\}\) is a Cauchy sequence in \(X\). Hence, there is \(u\, {\in}\, X\) such that \(u_m\,{\rightarrow}\, u\) in \(X\). We have that \(u\) is a solution of problem \((P)\). In fact, since \(u_m\) is a solution of \((P_m)\), for all \({\varphi}\, {\in}\, X\) we have

where
\(I_1 =\int_{\Omega}{\bigg(}{\vert{\Delta} u\vert}^{p-2}{\Delta}u – {\vert{\Delta}u_m\vert}^{p-2}{\Delta}u_m{\bigg)}\, {\Delta}{\varphi}\,{\omega}_1\,dx,\)
\(I_2 = \int_{\Omega}{\bigg(}{\vert{\Delta} u\vert}^{q-2}{\Delta}u – {\vert{\Delta}u_m\vert}^{q-2}{\Delta}u_m{\bigg)}{\Delta}{\varphi}\, {\nu}_1\,dx,\)
\(I_3 = \int_{\Omega} {\bigg(}{\vert {\nabla}u\vert}^{p-2}{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle} -{\vert{\nabla}u_m\vert}^{p-2}{\langle}{\nabla}u_m, {\nabla}{\varphi}{\rangle}{\bigg)} \,{\omega}_2\,dx,\)
\( I_4 = \int_{\Omega} {\bigg(}{\vert {\nabla}u\vert}^{s-2}{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle} – {\vert{\nabla}u_m\vert}^{s-2}{\langle}{\nabla}u_m , {\nabla}{\varphi}{\rangle}{\bigg)}\,{\nu}_2\,dx.\)
We have that:
(1) By Lemma 5 (a) there exists \(C_p>0\) such that \begin{eqnarray*} {\vert I_1 \vert} & {\leq} & \int_{\Omega}{\big\vert} {\vert{\Delta}u\vert}^{p-2}{\Delta}u – {\vert{\Delta}u_m\vert}^{p-2}{\Delta}u_m{\big\vert}\, {\vert{\Delta}{\varphi}\vert}\, {\omega}_1\, dx\\ & {\leq} & \, C_p\, \int_{\Omega}{\vert{\Delta}u – {\Delta}u_m\vert}\,({\vert{\Delta}u\vert}+{\vert{\Delta}u_m\vert})^{p-2}\, {\vert{\Delta}{\varphi}\vert}\, {\omega}_1\, dx. \end{eqnarray*} Let \(r = p/(p-2)\). Since \(\displaystyle {\dfrac{1}{p}} + {\dfrac{1}{p}} + {\dfrac{1}{r}} = 1\), by the Generalized Hölder’s inequality we obtain \begin{eqnarray*} {\vert I_1\vert} & \leq& C_p\, {\bigg(}\int_{\Omega}{\vert{\Delta}u – {\Delta}u_m\vert}^p\, {\omega}_1\, dx{\bigg)}^{1/p}{\bigg(}\int_{\Omega}{\vert{\Delta}{\varphi}\vert}^p\, {\omega}_1\, dx{\bigg)}^{1/p} {\bigg(}\int_{\Omega}({\vert{\Delta}u\vert}+{\vert{\Delta}u_m\vert})^{(p-2)r}\, {\omega}_1\, dx{\bigg)}^{1/r}\end{eqnarray*}\begin{eqnarray*} {\leq} \, C_p {\Vert u – u_m\Vert}_X\, {\Vert \varphi\Vert}_X {\Vert {\vert{\Delta}u\vert} + {\vert{\Delta}u_m\vert}\Vert}_{L^p(\Omega , {\omega}_1)}^{(p-2)}. \end{eqnarray*} Now, since \(u_m{\rightarrow}\, u\) in \(X\), then exists a constant \(M>0\) such that \({\Vert u_m\Vert}_X\, {\leq}\, M\). Hence, Therefore, \begin{eqnarray*} {\vert I_1 \vert} & {\leq} & C_p\, (2M)^{p-2}\, {\Vert u – u_m \Vert}_X\, {\Vert \varphi \Vert}_X = C_1 \, {\Vert u – u_m\Vert}_X\, {\Vert{\varphi}\Vert}_X. \end{eqnarray*} Analogously, there exists a constant \(C_3\) such that $${\vert I_3 \vert}\, {\leq}\, C_3 {\Vert u – u_m\Vert}_X\, {\Vert \varphi \Vert}_X.$$ (2) By Lemma 5 (a) there exists a positive constant \(C_q\) such that \begin{eqnarray*} {\vert I_2 \vert} & {\leq} & \int_{\Omega}{\big\vert} {\vert{\Delta}u\vert}^{q-2}{\Delta}u – {\vert{\Delta}u_m\vert}^{q-2}{\Delta}u_m{\big\vert}\, {\vert{\Delta}{\varphi}\vert}\, {\nu}_1\, dx\\ & {\leq} & C_q\, \int_{\Omega}{\vert{\Delta}u – {\Delta}u_m\vert}\,({\vert{\Delta}u\vert}+{\vert{\Delta}u_m\vert})^{q-2}\, {\vert{\Delta}{\varphi}\vert}\, {\nu}_1\, dx. \end{eqnarray*} Let \({\alpha} = q/(q-2)\) (if \(2< q< p< {\infty}\)). Since \(\displaystyle {\dfrac{1}{q}} + {\dfrac{1}{q}} + {\dfrac{1}{\alpha}} = 1\), by the Generalized Hölder's inequality we obtain \begin{eqnarray*} {\vert I_2\vert} & \leq &C_q\, {\bigg(}\int_{\Omega}{\vert{\Delta}u – {\Delta}u_m\vert}^q\, {\nu}_1\, dx{\bigg)}^{1/q}{\bigg(}\int_{\Omega}{\vert{\Delta}{\varphi}\vert}^q\, {\nu}_1\, dx{\bigg)}^{1/q} {\bigg(}\int_{\Omega}({\vert{\Delta}u\vert}+{\vert{\Delta}u_m\vert})^{(q-2){\alpha}}\, {\nu}_1\, dx{\bigg)}^{1/{\alpha}}\\ & =& C_q\, {\Vert {\Delta}u – {\Delta}u_m\Vert}_{L^q(\Omega,{\nu}_1)}\, {\Vert {\Delta}\varphi\Vert}_{L^q(\Omega , {\nu}_1)} {\Vert {\vert{\Delta}u\vert} + {\vert{\Delta}u_m\vert}\Vert}_{L^q(\Omega , {\nu}_1)}^{q-2}. \end{eqnarray*} Now, by Remark 3(a) and (9) we have \begin{eqnarray*} {\vert I_2\vert} & {\leq}& C_q\ M_1 {\Vert {\Delta}u – {\Delta}u_m\Vert}_{L^p(\Omega,{\omega}_1)}\,M_1\, {\Vert {\Delta}\varphi\Vert}_{L^p(\Omega , {\omega}_1)} M_1^{q-2}\,{\Vert {\vert{\Delta}u\vert} + {\vert{\Delta}u_m\vert}\Vert}_{L^p(\Omega , {\omega}_1)}^{q-2}\\ & {\leq} & C_q \ M_1^q {\Vert u – u_m \Vert}_X {\Vert \varphi \Vert}_X \ (2M)^{q-2}\\ & = & C_2 \, {\Vert u – u_m\Vert}_X \ {\Vert \varphi \Vert}_X. \end{eqnarray*} Analogously, if \(2< s< p< {\infty}\), there exists a positive constant \(C_4\) such that $${\vert I_4 \vert}\, {\leq}\, C_4\, {\Vert u – u_m \Vert}_X \ {\Vert \varphi \Vert}_X.$$ In case \(q=2\) and \(s=2\), we have \({\vert I_2 \vert}, {\vert I_4\vert}\, {\leq}\,M_1^2\, {\Vert u -u_m\Vert}_X\, {\Vert \varphi \Vert}_X\).
Therefore, we have \(I_1,I_2, I_3, I_4{\rightarrow}\, 0\) when \(m{\rightarrow}\, {\infty}\).
(3) We also have \begin{eqnarray*} {\bigg\vert}\int_{\Omega}(f_m – f)\,{\varphi}\, dx + \int_{\Omega}{\langle} G_m – G, {\nabla}{\varphi}{\rangle}\, dx{\bigg\vert} {\bigg(} C_{\Omega}{\bigg\Vert}{\dfrac{f_m – f}{{\omega}_2}}{\bigg\Vert}_{L^{p\,’}(\Omega , {\omega}_2)} + M_2{\bigg\Vert}{\dfrac{{\vert G_m – G \vert}}{{\nu}_2}}{\bigg\Vert}_{L^{s\,’}(\Omega, {\nu}_2)}{\bigg)}{\Vert \varphi \Vert}_X {\rightarrow}\, 0, \end{eqnarray*} when \(m{\rightarrow}\, {\infty}\).
Therefore, in (8), we obtain when \(m{\rightarrow}\, {\infty}\) that \begin{eqnarray*} & & \int_{\Omega}{\vert{\Delta} u\vert}^{p-2}{\Delta}u\, {\Delta}{\varphi}\,{\omega}_1\,dx + \int_{\Omega}{\vert{\Delta} u\vert}^{q-2}{\Delta}u\,{\Delta}{\varphi}\, {\nu}_1\,dx + \int_{\Omega} {\vert {\nabla}u\vert}^{p-2}{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\,{\omega}_2\,dx\\ & & + \int_{\Omega} {\vert {\nabla}u\vert}^{s-2}{\langle}{\nabla}u , {\nabla}{\varphi}{\rangle}\,{\nu}_2\,dx = \int_{\Omega}f\, {\varphi}\, dx + \int_{\Omega}{\langle}G, {\nabla}{\varphi}{\rangle}\, dx, \end{eqnarray*} i.e., \(u\) is a solution of problem \((P)\).

Example 1. Let \({\Omega} = \{ (x,y)\,{\in}\,{\mathbb{R}}^2 \, : \, x^2+y^2 < 1 \}\), \({\omega}_1(x,y) = (x^2+y^2)^{-1/2}\), \({\omega}_2(x,y) = (x^2+y^2)^{-1/4}\) (\({\omega}_i\, {\in}\, A_4\), \(p=4\) and \(q=s=3\)), \({\nu}_1(x,y) = (x^2+y^2)^{-1/3}\), \({\nu}_2(x,y) = (x^2+y^2)^{1/8}\), \(\displaystyle f(x,y) = {\dfrac{\cos(xy)}{(x^2+y^2)^{1/6}}}\) and \(\displaystyle G(x,y) ={\bigg(} {\dfrac{\sin(x+y)}{(x^2+y^2)^{1/6}}},{\dfrac{\sin(xy)}{(x^2+y^2)^{1/6}}}{\bigg)}\). By Theorem 7 , the problem \[ \left\{ \begin{array}{llll} & {\Delta}{\bigg[}(x^2+y^2)^{-1/2}\,{\vert{\Delta}u\vert}^2{\Delta}u + (x^2+y^2)^{-1/3} {\vert{\Delta}u\vert}{\Delta}u{\bigg]}\\ & -\,{\textrm{div}}{\bigg[}(x^2+y^2)^{-1/4}{\vert{\nabla}u\vert}^2{\nabla}u + (x^2+y^2)^{-1/8}{\vert{\nabla}u\vert}{\nabla}u{\bigg]}\\ & = f(x) – {\textrm{div}}(G(x)),\ \ {\textrm{in}} \ \ {\Omega} \\ & u(x) = {\Delta}u = 0, \ \ {\textrm{in}} \ \ {\partial\Omega} \end{array} \right. \] has a unique solution \(u\, {\in}\, W^{2,4}(\Omega , {\omega}_1)\, {\cap}\, W_0^{1,4}(\Omega , {\omega}_2)\).

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.