Existence results for a class of nonlinear degenerate \((p,q)\)-biharmonic operators

1. Introduction

In this paper, we prove the existence of (weak) solutions in the weighted Sobolev space \(X= W^{2,p}(\Omega , \omega)\,{\cap}\,W _0^{1,p}(\Omega , \omega)\) (see Definition 3 and Definition 4) for the Navier problem

where \(L\) is the partial differential operator \[Lu(x) = {\Delta}{\big[}{\omega}(x)\,{\big(}{\vert {\Delta}u\vert}^{p-2}{\Delta}u + {\vert {\Delta}u\vert}^{q-2}{\Delta}u{\big)}{\big]} -\sum_{j=1}^nD_j{\bigl[}{\omega}(x) {\mathcal{A}}_j(x, u(x), {\nabla}u(x)){\bigr]},\] where \(D_j = {\partial}/{\partial}x_j\), \(\Omega\) is a bounded open set in \(\mathbb{R}^n\), \(\omega\) is a weight function, \(\Delta\) is the usual Laplacian operator, \(2{\leq}\,q< p< {\infty}\) and the functions \({\mathcal{A}}_j: {\Omega}{\times}\mathbb{R}{\times}\mathbb{R}^n{\rightarrow}\mathbb{R}\) (\(j=1,…,n\)) satisfying the following conditions:

• (H1) \(x{\mapsto} {\mathcal{A}}_j(x, \eta , \xi)\) is measurable on \(\Omega\) for all \((\eta , \xi)\,{\in}\,\mathbb{R}{\times}\mathbb{R}^n\), \((\eta , \xi){\mapsto}{\mathcal{A}}_j(x,\eta, \xi)\) is continuous on \(\mathbb{R}{\times}\mathbb{R}^n\) for almost all \(x{\in}{\Omega}\).
• (H2) there exist a constant \({\theta}_1>0\) such that \[[{\mathcal{A}}(x,\eta,\xi) – {\mathcal{A}}(x, {\eta}’,{\xi}’) ].(\xi – {\xi}’)\,{\geq}\, {\theta}_1\, {\vert \xi – {\xi}’\vert}^p,\] whenever \({\xi},{\xi}'{\in}\mathbb{R}^n\), \(\xi{\not =}{\xi}’\), where \(\displaystyle {\mathcal{A}}(x, \eta, \xi) = ( {\mathcal{ A}}_1(x,\eta,\xi), …, {\mathcal{A}}_n(x,\eta,\xi))\) (where a dot denote here the Euclidian scalar product in \(\mathbb{R}^n\)).
• (H3) \({\mathcal{A}}(x,\eta,\xi).{\xi}\,{\geq}\, {\lambda}_1{\vert\xi\vert}^p\), where \({\lambda}_1\) is a positive constant.
• (H4) \({\vert} {\mathcal{ A}}(x,\eta,\xi){\vert}\,{\leq}\, K_1(x) + h_1(x){\vert\eta\vert}^{p/p\,’} + h_2(x){\vert\xi\vert}^{p/p\,’}\), where \(K_1, h_1\) and \(h_2\) are positive functions, with \(h_1\), \(h_2{\in}L^{\infty}(\Omega)\), and \(K_1{\in}L^{p\,’}(\Omega , \omega)\) (with \(1/p+1/p\,’=1\)).

By a weight, we shall mean a locally integrable function \(\omega\) on \(\mathbb{R}^n\) such that \(0< {\omega}(x)< {\infty}\) for a.e. \(x\,{\in}\,\mathbb{R}^n\). Every weight \(\omega\) gives rise to a measure on the measurable subsets on \(\mathbb{R}^n\) through integration. This measure will be denoted by \(\mu\). Thus, \({\mu}(E) = \int_E{\omega}(x)\,dx\) for measurable sets \(E\,{\subset}\,\mathbb{R}^n\).

In general, the Sobolev spaces \(W^{k,p}(\Omega)\) without weights occur as spaces of solutions for elliptic and parabolic partial differential equations. In the particular case where \(p=q=2\) and \({\omega}{\equiv}\, 1\), we have the equation

\[\displaystyle {\Delta}^2u – \sum_{j=1}^nD_j{\mathcal{A}}_j(x,u,{\nabla}u) = f,\] where \({\Delta}^2u\) is the biharmonic operator. If \(p=q\), \({\omega}{\equiv}\,1\) and \({\mathcal{A}}(x, \eta, \xi) = {\vert \xi \vert}^{p-2}\,{\xi}\), we have the equation \[{\Delta}({\vert{\Delta}\vert}^{p-2}\, {\Delta}u) – {\mathrm{div}}({\vert{\nabla}u\vert}^{p-2}{\nabla}u) = f.\] Biharmonic equations appear in the study of mathematical model in several real-life processes as, among others, radar imaging (see [1]) or incompressible flows (see [2]).

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 [3,4,5,6]). In various applications, we can meet boundary value problems for elliptic equations whose ellipticity is disturbed in the sense that some degeneration or singularity appears. There are several very concrete problems from practice which lead to such differential equations, e.g. from glaceology, non-Newtonian fluid mechanics, flows through porous media, differential geometry, celestial mechanics, climatology, petroleum extraction and reaction-diffusion problems (see some examples of applications of degenerate elliptic equations in [7,8]).

A class of weights, which is particularly well understood, is the class of \(A_p\)-weights (or Muckenhoupt class) that was introduced by B. Muckenhoupt (see [9]). These classes have found many useful applications in harmonic analysis (see [10]). Another reason for studying \(A_p\)-weights is the fact that powers of distance to submanifolds of \(\mathbb{R}^n\) often belong to \(A_p\) (see [11]). There are, in fact, many interesting examples of weights (see [12] 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

\begin{equation*} \begin{cases}\, – \, {\Delta}u = f(x), &\ {\mathrm{in}} \ {\Omega} \\ u(x) = 0, &\ {\mathrm{on}} \ {\partial\Omega} \end{cases} \end{equation*} is uniquely solvable in \(W^{2,p}(\Omega)\,{\cap}\, W_0^{1,p}(\Omega)\) (see [13]), and the nonlinear Dirichlet problem \begin{equation*} \begin{cases} – \, {\Delta}_p u = f(x), & {\mathrm{in}} \ {\Omega} \\ u(x) = 0, & {\mathrm{on}} \ {\partial\Omega} \end{cases} \end{equation*} is uniquely solvable in \(W_0^{1,p}(\Omega)\) (see [14]), where \({\Delta}_p u = {\div}({\vert {\nabla}u\vert}^{p-2}{\nabla}u)\) is the p-Laplacian operator. In the degenerate case, the weighted p-Biharmonic operator has been studied by many authors (see [15] and the references therein), and the degenerated p-Laplacian was studied in [6].

The following theorem will be proved in Section 3.

Theorem 1. Let \(2\,{\leq}\,q < p < {\infty}\) and assume (H1)-(H4). If \({\omega}\,{\in}\,A_p\), \({\dfrac{f_j}{\omega}}\,{\in}\,L^{p\,'}(\Omega , \omega)\) (\(j=0,1,…,n\)) then the problem (P) has a unique solution \(u{\in}\,X = W^{2,p}(\Omega , \omega) {\cap}\,W_0^{1,p}(\Omega , \omega)\). Moreover, we have \[{\Vert u \Vert}_X {\leq}{\dfrac{1}{{\gamma}^{p\,'/p}}} {\bigg(} C_{\Omega}{\Vert f_0/{\omega}\Vert}_{L^{p\,'}(\Omega , \omega)} + \sum_{j=1}^n {\Vert f_j/{\omega}\Vert}_{L^{p\,'}(\Omega , \omega)}{\bigg)}^{p\,'/p},\] where \({\gamma} = \min\,\{{\lambda}_1, 1\}\) and \(C_{\Omega}\) is the constant in Theorem 3.

2. Definitions and basic results

Let \(\omega\) be a locally integrable nonnegative function in \(\mathbb{R}^n\) and assume that \(0< {\omega}< {\infty}\) almost everywhere. We say that \({\omega}\) belongs to the Muckenhoupt class \(A_p\), \(1 < p < {\infty}\), or that \(\omega\) is an \(A_p\)-weight, if there is a constant \(C=C_{p, \omega}\) such that \[{\biggr(}\dfrac{1}{ {\vert B \vert} } \int_B {\omega}(x)dx{\biggr)}{\biggl(} \dfrac{1}{ {\vert B \vert} }\int_B{\omega}^{1/(1-p)}(x)dx{\biggr)}^{p-1}{\leq}C,\] for all balls \(B\,{\subset}\,\mathbb{R}^n\), 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 [10,12,16] for more information about \(A_p\)-weights). The weight \(\omega\) satisfies the doubling condition if there exists a positive constant \(C\) such that \({\mu}(B(x;2r))\, {\leq}\,C\, {\mu}(B(x;r))\), for every ball \(B=B(x;r)\, {\subset}\, \mathbb{R}^n\), where \({\mu}(B) = \int_B {\omega}(x)\,dx\). If \({\omega}{\in}A_p\), then \(\mu\) is doubling (see Corollary 15.7 in [12]).

As an example of \(A_p\)-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 Corollary 4.4, Chapter IX in [10]).

If \({\omega}{\in}A_p\), then \[{\biggl(} \dfrac{{\vert E \vert}}{{\vert B \vert}} {\biggr)}^p \,{\leq}\,C \dfrac{{\mu}(E)}{{\mu}(B)},\] whenever \(B\) is a ball in \(\mathbb{R}^n\) and \(E\) is a measurable subset of \(B\) (see 15.5 strong doubling property in [12]). Therefore, if \({\mu}(E) =0\) then \({\vert E \vert}=0\). The measure \(\mu\) and the Lebesgue measure \(\vert . \vert\) are mutually absolutely continuous, i.e., they have the same zero sets (\({\mu}(E)=0\) 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 1. Let \(\omega\) be a weight, and let \(\Omega\,{\subset}\,\mathbb{R}^n\) be open. For \(0< p< {\infty}\) we define \(L^p(\Omega , \omega)\) as the set of measurable functions \(f\) on \(\Omega\) such that \[{\Vert f \Vert}_{L^p(\Omega ,\omega)} = {\bigg(}\int_{\Omega} {\vert f(x) \vert}^p{\omega}(x)dx{\bigg)}^{1/p}< {\infty}.\] If \({\omega}\,{\in}\,A_p\), \(1< p< {\infty}\), then \({\omega}^{-1/(p-1)}\) is locally integrable and we have \(\displaystyle{L^p(\Omega , \omega)\,{\subset}\,L^1_{loc}(\Omega)}\) for every open set \(\Omega\) (see Remark 1.2.4 in [17]). It thus makes sense to talk about weak derivatives of functions in \(L^p(\Omega , \omega)\).

Definition 2. 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

If \({\omega}\, {\in}\, A_p\), then \(W^{k,p}(\Omega , \omega)\) is the closure of \(C^{\infty}(\Omega)\) with respect to the norm (1) (see Corollary 2.1.6 in [17]). 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.

The space \(W_0^{1,p}(\Omega , \omega)\) is the closure of \(C_0^{\infty}(\Omega)\) with respect to the norm (1). Equipped with this norm, \(W_0^{1,p}(\Omega , \omega)\) is a reflexive Banach space (see [18] for more information about the spaces \(W^{1,p}(\Omega , \omega)\)). The dual of space \(W_0^{1,p}(\Omega , \omega)\) is the space

\begin{eqnarray*} [W_0^{1,p}(\Omega , \omega)]^* = \{ T = f_0 – {\mathrm{div}}(F), \ F=(f_1,…,f_n): \ {\dfrac{f_j}{\omega}}\, {\in}\, L^{p\,’}(\Omega , \omega), j=0,1,…,n\}. \end{eqnarray*} It is evident that a weight function \(\omega\) which satisfies \(0< c_1\,{\leq}\,{\omega}(x)\,{\leq}\,c_2\) for \(x\,{\in}\,{\Omega}\) (where \(c_1\) and \(c_2\) are constants), give nothing new (the space \({W}_0^{1,p}(\Omega ,\omega)\) is then identical with the classical Sobolev space \({W}_0^{1,p}(\Omega)\)). Consequently, we shall be interested above all in such weight functions \(\omega\) which either vanish somewhere in \({\bar{\Omega}}\) or increase to infinity (or both).

In this paper we use the following results.

Theorem 2. Let \({\omega}\,{\in}\,A_p\), \(1< p< {\infty}\), and let \(\Omega\) be a bounded open set in \(\mathbb{R}^n\). If \(u_m{\rightarrow}\,u\) in \(L^p(\Omega , \omega)\) then there exist a subsequence \(\{ u_{m_k} \}\) and a function \({\Phi}\,{\in}\,L^p(\Omega , \omega)\) such that

• (i) \(u_{m_k}(x){\rightarrow}\,u(x)\), \(m_k\,{\to}\,{\infty}\) a.e. on \(\Omega\);
• (ii) \({\vert u_{m_k}(x) \vert}\,{\leq}\,{\Phi}(x)\) a.e. on \(\Omega\).

Proof. The proof of this theorem follows the lines of Theorem 2.8.1 in [19].

Theorem 3. (The weighted Sobolev inequality) Let \(\Omega\) be an open bounded set in \(\mathbb{R}^n\) and \({\omega}{\in}A_p\) (\(1< p< {\infty}\)). There exist constants \(C_{\Omega}\) and \(\delta\) positive such that for all \(u\,{\in}\,W_0^{1,p}(\Omega, \omega)\) and all \(k\) satisfying \(1\,{\leq}\,k\,{\leq}\,n/(n-1) + {\delta}\),

Proof. Its suffices to prove the inequality for functions \(u\, {\in}\, C_0^{\infty}(\Omega)\) (see Theorem 1.3 in [20]). 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 \(L^{kp}(\Omega , \omega)\). Consequently the limit function \(u\) will lie in the desired spaces and satisfy (2).

Lemma 1. Let \(1< p< {\infty}\).

• (a) There exists a constant \({\alpha}_p>0\) such that \[{\bigg\vert \,{\vert x\vert}^{p-2}x – {\vert y \vert}^{p-2}y\bigg\vert}\, {\leq}\, {\alpha}_p\, {\vert x-y\vert}({\vert x \vert}+{\vert y \vert})^{p-2},\] for all \(x,y\, {\in}\, \mathbb{R}^n\);
• (b) There exist two positive constants \({\beta}_{p}\), \({\gamma}_p\) such that for every \(x,y\, {\in}\, \mathbb{R}^n\) \[{\beta}_p\,({\vert x \vert} + {\vert y \vert})^{p-2}{\vert x – y\vert}^2\, {\leq}\, ({\vert x \vert}^{p-2}x – {\vert y \vert}^{p-2}y).(x-y)\, {\leq}\, {\gamma}_p\, ({\vert x \vert}+{\vert y \vert})^{p-2}{\vert x – y\vert}^2.\]

Proof. See [14], Proposition 17.2 and Proposition 17.3.

Definition 3. We denote by \(X = W^{2,p}(\Omega, \omega)\, {\cap}\, W_0^{1,p}(\Omega , \omega)\) with the norm \[{\Vert u \Vert}_X = {\bigg(}\int_{\Omega} {\vert{\nabla}u\vert}^p\, {\omega}\, dx + \int_{\Omega}{\vert{\Delta}u\vert}^p\, {\omega}\, dx{\bigg)}^{1/p}.\]

Definition 4. We say that an element \(u\,{\in}\,X = W^{2,p}(\Omega , \omega)\, {\cap}\,W_0^{1,p}(\Omega , ,\omega)\) is a (weak) solution of problem (P) if \begin{eqnarray*} & & \int_{\Omega}{\vert {\Delta}u\vert}^{p-2}\, {\Delta}u\, {\Delta}{\varphi}\, {\omega}\, dx + \int_{\Omega}{\vert {\Delta}u\vert}^{q-2}\, {\Delta}u\, {\Delta}{\varphi}\, {\omega}\, dx + \ \sum_{j=1}^n\int_{\Omega} {\mathcal{A}}_j(x, u(x), {\nabla}u(x))D_j{\varphi}(x)\,{\omega}(x)\,dx\\ & & = \int_{\Omega}f_0(x){\varphi}(x)dx + \sum_{j=1}^n\int_{\Omega}f_j(x)D_j{\varphi}(x)dx, \end{eqnarray*} for all \({\varphi}\,{\in}\,X\).

Remark 1. If \(0< {\eta}< p< {\infty}\) then, by Hölder's inequality, \[{\Vert u \Vert}_{L^{\eta}(\Omega , \omega)}\, {\leq}\,C_{p,{\eta}}{\Vert u \Vert}_{L^p(\Omega , \omega)},\] where \(\displaystyle C_{p,{\eta}} = {\bigg(}\int_{\Omega} {\omega}\, dx{\bigg)}^{(p-\eta)/p\,{\eta}}= {\Vert \omega \Vert}_{L^{p/(p-\eta)}(\Omega)}^{1/\eta}\). In fact, \begin{eqnarray*} {\Vert u \Vert}_{L^{\eta}(\Omega , {\omega})}^{\eta} & = & \int_{\Omega}{\vert u \vert}^{\eta}\,{\omega}\, dx\\ & {\leq}& {\bigg(}\int_{\Omega} {\vert u \vert}^{{\eta}\, p/{\eta}}\, {\omega}\, dx{\bigg)}^{{\eta}/p}{\bigg(}\int_{\Omega} {\omega}^{p/(p-\eta)}\, dx{\bigg)}^{(p-\eta)/p}\\ & = & {\Vert u \Vert}_{L^p(\Omega , {\omega})}^{\eta} \, {\Vert \omega\Vert}_{L^{p/(p-\eta)}(\Omega)}. \end{eqnarray*}

3. Proof of Theorem 1

The basic idea is to reduce the Problem (P) to an operator equation \(Au=T\) and apply the theorem below.

Theorem 4. Let \(A:X{\rightarrow} X^*\) be a monotone, coercive and hemicontinuous operator on the real, separable, reflexive Banach space \(X\). Then the following assertions hold:

• (a) For each \(T\,{\in}\,X^*\) the equation \(Au=T\) has a solution \(u{\in}X\);
• (b) If the operator \(A\) is strictly monotone, then equation \(A\,u=T\) is uniquely solvable in \(X\).

Proof. See Theorem 26.A in [21].

To prove Theorem 1, we define \(\displaystyle B, B_1, B_2, B_3:X\,{\times}\,X\, {\rightarrow}\, \mathbb{R}\) and \(T:X\, {\rightarrow}\,\mathbb{R}\) by

\begin{eqnarray*} B(u,\varphi) &=& B_1(u,\varphi) + B_2(u,\varphi)+ B_3(u, \varphi),\\ B_1(u, \varphi) & = & \sum_{j=1}^n\int_{\Omega} {\mathcal{A}}_j(x, u, {\nabla}u)D_j{\varphi}\,{\omega}\,dx = \int_{\Omega}{\mathcal{A}}(x,u,{\nabla}u). {\nabla}{\varphi}\,{\omega}\,dx \\ B_2(u, \varphi) & = & \int_{\Omega}{\vert {\Delta}u\vert}^{p-2}{\Delta}u\, {\Delta}{\varphi}\, {\omega}\, dx\\ B_3(u, \varphi) & = & \int_{\Omega}{\vert {\Delta}u\vert}^{q-2}{\Delta}u\, {\Delta}{\varphi}\, {\omega}\, dx\\ T(\varphi) & = & \int_{\Omega}f_0(x)\,{\varphi}(x)\,dx + \sum_{j=1}^n\int_{\Omega}f_j(x)\,D_j{\varphi}(x)\,dx. \end{eqnarray*} Then \(u\,{\in}\,X\) is a (weak) solution to problem (P) if, for all \({\varphi}\, {\in}\, X\), \[B(u, \varphi) = B_1(u, \varphi) + B_2(u, \varphi) + B_3(u , \varphi) = T(\varphi).\]

Step 1. For \(j=1,…,n\) we define the operator \( F_j :X\,{\rightarrow}L^{p\,’}(\Omega , \omega)\) as \[(F_ju)(x) = {\mathcal{A}}_j(x, u(x), {\nabla}u(x)).\] We now show that the operator \(F_j\) is bounded and continuous.

• (i) Using (H4), we obtain
  \begin{eqnarray} \label{eq3} {\Vert F_j u\Vert}^{p\,’}_{L^{p\,’}(\Omega , \omega)} & = & \int_{\Omega} {\vert F_ju(x) \vert}^{p\,’} {\omega}\,dx \nonumber \\ & = & \int_{\Omega}{\vert {\mathcal{ A}}_j(x, u , {\nabla}u )\vert}^{p\,’}{\omega}\,dx \nonumber \\ &{\leq}&\int_{\Omega} {\biggl(}K_1 + h_1{\vert u \vert}^{p/p\,’} + h_2{\vert\nabla u \vert}^{p/p\,’}{\biggr)}^{p\,’} {\omega}\,dx \nonumber \\ & {\leq}& C_p \int_{\Omega}{\biggl[}(K_1^{p\,’} + h_1^{p\,’}{\vert u \vert}^p + h_2^{p\,’}{\vert\nabla u \vert}^p){\omega} {\biggr]}dx \nonumber \\ & = & C_p{\biggl[} \int_{\Omega}K_1^{p\,’}\,{\omega}\,dx + \int_{\Omega}h_1^{p\,’}{\vert u \vert}^p\,{\omega}\,dx + \int_{\Omega}h_2^{p\,’}{\vert \nabla u \vert}^p{\omega}\,dx {\biggr]}, \end{eqnarray}

  (3)
  where the constant \(C_p\) depends only on \(p\). We have, by Theorem 3 (with \(k=1\)), \begin{eqnarray*} \int_{\Omega}h_1^{p\,’}{\vert u \vert}^p\,{\omega}\,dx & {\leq} & {\Vert h_1 \Vert}_{L^{\infty}(\Omega)}^{p\,’} \int_{\Omega}{\vert u \vert}^p\, {\omega}\,dx\\ &{\leq}& C_{\Omega}^p\, {\Vert h_1\Vert}_{L^{\infty}(\Omega)}^{p\,’}\int_{\Omega}{\vert{\nabla}u\vert}^p\, {\omega}\, dx\\ &{\leq}& C_{\Omega}^p\,{\Vert h_1 \Vert}_{L^{\infty}(\Omega)}^{p\,’} \, {\Vert u \Vert}_X^p, \end{eqnarray*} and \begin{eqnarray*} \int_{\Omega}h_2^{p\,’}{\vert {\nabla}u \vert}^p{\omega}\,dx & {\leq}& {\Vert h_2\Vert}_{L^{\infty}(\Omega)}^{p\,’} \int_{\Omega}{\vert {\nabla}u \vert}^p\,{\omega}\,dx\\ &{\leq}& {\Vert h_2 \Vert}_{L^{\infty}(\Omega)}^{p\,’} {\Vert u \Vert}_X^p. \end{eqnarray*} Therefore, in (3) we obtain \begin{eqnarray*} {\Vert F_ju \Vert}_{L^{p\,’}(\Omega , \omega)} &{\leq}&C_p^{1/p\,’}\,{\bigg(} {\Vert K \Vert}_{L^{p\,’}(\Omega , \omega)} + (C_{\Omega}^{p/p\,’}{\Vert h_1\Vert}_{L^{\infty}(\Omega)} + {\Vert h_2\Vert}_{L^{\infty}(\Omega)})\, {\Vert u \Vert}_X^{p/p\,’}{\bigg)}. \end{eqnarray*}
• (ii) Let \(u_m{\rightarrow}\,u\) in \(X\) as \(m\to\infty\). We need to show that \(F_ju_m{\rightarrow}F_ju\) in \(L^{p\,’}(\Omega , \omega)\). We will apply the Lebesgue Dominated Theorem. If \(u_m{\rightarrow}\,u\) in \(X\), then \({\vert}{\nabla}u_m{\vert}{\rightarrow}\,{\vert{\nabla}u\vert}\) in \(L^p(\Omega , \omega)\). Using Theorem 2, there exist a subsequence \(\{ u_{m_k} \}\) and a function \({\Phi}_1\) in \(L^p(\Omega , \omega)\) such that \begin{eqnarray*} & &D_ju_{m_k}(x)\,{\rightarrow}\,D_ju(x), \ {\mathrm{a.e.}} \ {\mathrm{in}} \ {\Omega},\\ & &{\vert{\nabla}u_{m_k}(x) \vert}{\leq}{\Phi}_1(x), \ {\mathrm{ a.e.}} \ {\mathrm{in}} \ {\Omega}. \end{eqnarray*} By Theorem 3 (with \(k=1\)), \[{\Vert u_{m_k}\Vert}_{L^p(\Omega , \omega)}\, {\leq}\, C_{\Omega}{\Vert \,\vert{\nabla}u_{m_k}\vert\,\Vert}_{L^p(\Omega , \omega)}\, {\leq}\, C_{\Omega}\, {\Vert {\Phi}_1\Vert}_{L^p(\Omega , \omega)}.\] Next, applying (H4) we obtain \begin{eqnarray*} {\Vert F_ju_{m_k} – F_ju \Vert}_{L^{p\,’}(\Omega , \omega)}^{p\,’}&=& \int_{\Omega}{\vert F_ju_{m_k}(x) – F_ju(x) \vert}^{p\,’}{\omega}\,dx \\ &=& \int_{\Omega} {\vert {\mathcal{A}}_j(x, u_{m_k}, {\nabla}u_{m_k}) – {\mathcal{A}}_j(x, u, {\nabla}u) \vert}^{p\,’}\,{\omega}\,dx\\ &\leq& C_p\,\int_{\Omega}{\biggl(} {\vert {\mathcal{A}}_j(x, u_{m_k}, {\nabla}u_{m_k}) \vert}^{p\,’} + {\vert{\mathcal{A}}_j(x, u, {\nabla}u) \vert}^{p\,’}{\biggr)}{\omega}\,dx\\ &\leq& C_p\,{\biggl[} \int_{\Omega} {\biggl(} K_1 + h_1{\vert u_{m_k} \vert}^{p/p\,’} + h_2{\vert {\nabla}u_{m_k} \vert}^{p/p\,’}{\biggr)}^{p\,’}\,{\omega}\,dx \\ & & + \int_{\Omega}{\biggl(} K_1 + h_1{\vert u \vert}^{p/p\,’} + h_2{\vert{\nabla} u \vert}^{p/p\,’}{\biggr)}^{p\,’}\,{\omega}\,dx {\biggr]}\\ &\leq& C_p{\bigg[}\int_{\Omega}K_1^{p\,’}{\omega}\, dx + {\Vert h_1\Vert}_{L^{\infty}(\Omega)}^{p\,’}\int_{\Omega}{\vert u_{m_k}\vert}^p\, {\omega}\, dx + {\Vert h_2\Vert}_{L^{\infty}(\Omega)}^{p\,’}\int_{\Omega}{\vert{\nabla}u_{m_k}\vert}^p\, {\omega}\, dx\\ & & + \int_{\Omega} K_1^{p\,’}\,{\omega}\, dx + {\Vert h_1\Vert}_{L^{\infty}(\Omega)}^{p\,’}\int_{\Omega}{\vert u \vert}^p\, {\omega}\, dx + {\Vert h_2 \Vert}_{L^{\infty}(\Omega)}^{p\,’}\int_{\Omega}{\vert{\nabla}u\vert}^p\,{\omega}\, dx{\bigg]}\\ &\leq& 2\,C_p{\bigg[}\int_{\Omega}K_1^{p\,’}\, {\omega}\, dx + {\Vert h_1\Vert}_{L^{\infty}(\Omega)}^{p\,’}\,C_{\Omega}^p\,\int_{\Omega}{\Phi}_1^p\, {\omega}\,dx + {\Vert h_2\Vert}_{L^{\infty}(\Omega)}^{p\,’}\int_{\Omega}{\Phi}_1^p\, {\omega}\, dx{\bigg]}\\ &=& 2\,C_p{\bigg[} {\Vert K_1 \Vert}_{L^{p\,’}(\Omega , \omega)}^{p\,’} + {\bigg(}C_{\Omega}^p{\Vert h_1\Vert}_{L^{\infty}(\Omega)}^{p\,’} + {\Vert h_2\Vert}_{L^{\infty}(\Omega)}^{p\,’}{\bigg)}{\Vert {\Phi}_1\Vert}_{L^p(\Omega , \omega)}^p{\bigg]}. \end{eqnarray*} By condition (H1), we have \[F_ju_{m_k}(x) = {\mathcal{A}}_j(x, u_{m_k}(x), {\nabla}u_{m_k}(x)){\rightarrow} \,{\mathcal{A}}_j(x, u(x), {\nabla}u(x)) = F_ju(x),\] as \(m_k\to +\infty\). Therefore, by the Lebesgue Dominated Convergence Theorem, we obtain \[{\Vert F_ju_{m_k} – F_ju \Vert}_{L^{p\,’}(\Omega , \omega)}{\rightarrow}\,0,\] that is, \(F_ju_{m_k} \,{\rightarrow}\, F_ju \ \ {\mathrm{in}} \ \ L^{p\,’}(\Omega , \omega)\). We conclude from the Convergence Principle in Banach spaces (see Proposition 10.13 in [22]) that
  \begin{equation} \label{eq4} F_ju_m{\rightarrow}\,F_ju \ \ {\mathrm{in}} \ \ L^{p\,’}(\Omega , \omega). \end{equation}

  (4)

Step 2. We define the operator \(G_1:X\, {\rightarrow}\, L^{p\,’}(\Omega, \omega)\) by \[(G_1 u)(x) = {\vert {\Delta}u(x)\vert}^{p-2}\, {\Delta}u(x).\] This operator is continuous and bounded. In fact,

• (i) We have \begin{eqnarray*} {\Vert G_1u\Vert}_{L^{p\,’}(\Omega , \omega)}^{p\,’} & = & \int_{\Omega}{\big\vert}\, {\vert{\Delta}u\vert}^{p-2}\, {\Delta}u {\big\vert}^{p\,’}\, {\omega}\, dx\\ & = & \int_{\Omega}{\vert{\Delta}u\vert}^{(p-1)\, p\,’}\, {\omega}\, dx \\ & = & \int_{\Omega}{\vert {\Delta}u\vert}^p\, {\omega}\, dx\\ &{\leq}& {\Vert u \Vert}_X^p. \end{eqnarray*} Hence, \(\displaystyle {\Vert G_1u \Vert}_{L^{p\,’}(\Omega , \omega)}\, {\leq}\, {\Vert u \Vert}_X^{p/p\,’}\).
• (ii) If \(u_m\,{\rightarrow}\,u\) in \(X\) then \({\Delta}u_m {\rightarrow}\, {\Delta}u\) in \(L^p(\Omega , \omega)\).By Theorem 2, there exist a subsequence \(\{u_{m_k}\}\) and a function \({\Phi}_2\, {\in}\, L^p(\Omega , \omega)\) such that
  \begin{eqnarray} & & {\Delta}u_{m_k}(x)\, {\rightarrow}\, {\Delta}u(x), \ a.e. \ {\mathrm{in}} \ \Omega,\label{eq5}\\ \end{eqnarray}

  (5)
  \begin{eqnarray} & & {\vert {\Delta}u_{m_k}(x)\vert}\, {\leq}\, {\Phi}_2(x), \ a.e.\label{eq6} \ {\mathrm{in}} \ \Omega. \end{eqnarray}

  (6)
  Hence, using Lemma 1(a), we obtain (since \(2\, {\leq}\, q < p< {\infty}\)) with \({\theta}=p/p\,' = p-1\) and \({\theta}\,' = (p-1)/(p-2)\), \begin{eqnarray*} & & {\Vert G_1u_{m_k} – G_1u\Vert}_{L^{p\,'}(\Omega , \omega)}^{p\,'} = \int_{\Omega}{\vert \, G_1u_{m_k} – G_1u\vert}^{p\,'}\,{\omega}\, dx\\ & & = \int_{\Omega}{\bigg\vert \, {\vert {\Delta}u_{m_k}\vert}^{p-2}\,{\Delta}u_{m_k} – {\vert\, {\Delta}u\vert}^{p-2}\, {\Delta}u\bigg\vert}^{p\,'}\, {\omega}\, dx \\ & & {\leq}\, \int_{\Omega} {\bigg[ {\alpha}_p\,{\vert {\Delta}u_{m_k} – {\Delta}u\vert}\,(\, {\vert{\Delta}u_{m_k}\vert} + {\vert{\Delta}u\vert})^{(p-2)}\bigg]}^{p\,'}\, {\omega}\, dx\\ & & {\leq}\, {\alpha}_p^{p\,'} \int_{\Omega}{\vert\, {\Delta}u_{m_k}-{\Delta}u\vert}^{p\,'}\, (2\, {\Phi}_2)^{(p-2)\,p\,'}\, {\omega}\, dx\\ & & {\leq}\, 2^{(p-2)\,p\,'}{\alpha}_p^{p\,'}{\bigg(}\int_{\Omega}{\vert{\Delta}u_{m_k} – {\Delta}u\vert}^{p\,'{\theta}}{\omega}\, dx{\bigg)}^{1/{\theta}}{\bigg(}\int_{\Omega}{\Phi}_2^{(p-2)\,p\,' {\theta}\,'}{\omega}\,dx{\bigg)}^{1/{\theta}\,'}\\ & & {\leq}\, {\alpha}_p^{p\,'}\, 2^{(p-2)p\,'}{\bigg(}\int_{\Omega}{\vert {\Delta}u_{m_k}-{\Delta}u\vert}^p\,{\omega}\, dx{\bigg)}^{p\,'/p}{\bigg(}\int_{\Omega}{\Phi}_2^p\, {\omega}\, dx{\bigg)}^{(p-2)/(p-1)}\\ & & {\leq}\, {\alpha}_p^{p\,'}\, 2^{(p-2)\,p\,'}\, {\Vert u_{m_k} – u \Vert}_X^{p\,'}\, {\Vert {\Phi}_2\Vert}_{L^p(\Omega , \omega)}^{(p-2)\,p\,'}, \end{eqnarray*} since \(\displaystyle (p-2)\,p\,'{\theta}\,' = (p-2){\dfrac{p}{(p-1)}}{\dfrac{(p-1)}{(p-2)}} = p\) if \(p\,{\neq}\,2\). Hence \[{\Vert G_1u_{m_k} – G_1u\Vert}_{L^{p\,'}(\Omega , \omega)}\, {\leq}\, 2^{p-2}{\alpha}_p\, {\Vert{\Phi}_2\Vert}_{L^p(\Omega , \omega)}^{p-2}{\Vert u_{m_k} – u \Vert}_X.\] Therefore, by the Lebesgue Dominated Convergence Theorem, we obtain (when \(m_k \to \infty\)) \[\displaystyle {\Vert G_1u_{m_k} – G_1u\Vert}_X\, {\rightarrow}\, 0,\] that is, \(\displaystyle G_1u_{m_k}{\rightarrow}\, G_1u\) in \(L^{p\,'}(\Omega , \omega)\). By the Convergence Principle in Banach spaces (see Proposition 10.13 in [22]), we have
  \begin{equation} \label{eq7} G_1u_m\,{\rightarrow}\, G_1u \ {\mathrm{in}} \ L^{p\,’}(\Omega, \omega). \end{equation}

  (7)

Step 3. We define the operator \(G_2:X{\rightarrow}\, L^{p\,’}(\Omega , \omega)\) by \[(G_2u)(x) = {\vert{\Delta}u(x)\vert}^{q-2}{\Delta}u(x).\] We also have that the operator \(G_2\) is continuous and bounded. In fact,

• (i) We have, with \(r=(p-1)/(q-1)>1\) and \(r’ =(p-1)/(p-q)\), that \begin{eqnarray*} {\Vert G_2 u \Vert}_{L^{p\,’}(\Omega , \omega)}^{p\,’} & = & \int_{\Omega}{\big\vert} {\vert{\Delta}u\vert}^{q-2}{\Delta}u{\big\vert}^{p\,’}\, {\omega}\, dx\\ & = & \int_{\Omega}{\vert{\Delta}u\vert}^{(q-1)\, p\,’}\, {\omega}\, dx\\ & {\leq} & {\bigg(}\int_{\Omega}{\vert{\Delta}u\vert}^{(q-1)\,p\,’\,r}\, {\omega}\, dx{\bigg)}^{1/r} {\bigg(}\int_{\Omega}{\omega}\, dx{\bigg)}^{1/r\,’}\\ & = & {\bigg(}\int_{\Omega}{\vert{\Delta}u\vert}^p\, {\omega}\, dx{\bigg)}^{(q-1)/(p-1)}{\bigg(}\int_{\Omega}{\omega}\, dx{\bigg)}^{(p-q)/(p-1)}\\ & = & C_{p,q}\, {\Vert {\Delta}u\Vert}_{L^p(\Omega , \omega)}^{(q-1)p\,’}\\ & {\leq} & C_{p,q}\, {\Vert u \Vert}_X^{(q-1)\, p\,’}, \end{eqnarray*} where \(\displaystyle C_{p,q} = {\bigg(}\int_{\Omega}{\omega}\, dx{\bigg)}^{(p-q)/(p-1)}\). Hence \(\displaystyle {\Vert G_2u \Vert}_{L^{p\,’}(\Omega , \omega)}\, {\leq}\, C_{p,q}^{1/p\,’}\, {\Vert u \Vert}_X^{q-1}\).
• (ii) If \(u_m{\rightarrow}\,u\) in \(X\) then \({\Delta}u_m{\rightarrow}\, {\Delta}u\) in \(L^p(\Omega , \omega)\). If \(2< q< p< {\infty}\), by (5), (6) and Lemma 1(a), we have
  \begin{eqnarray} {\Vert G_2u_{m_k} – G_2u\Vert}_{L^{p\,’}(\Omega , \omega)}^{p\,’} & = & \int_{\Omega}{\bigg\vert}{\vert{\Delta}u_{m_k}\vert}^{q-2}\,{\Delta}u_{m_k} – {\vert{\Delta}u\vert}^{q-2}{\Delta}u{\bigg\vert}^{p\,’}{\omega}\, dx\nonumber\\ & {\leq} & \int_{\Omega}{\bigg[}{\alpha}_q{\vert {\Delta}u_{m_k} – {\Delta}u\vert}\, {\bigg(}{\vert{\Delta}u_{m_k}\vert} + {\vert{\Delta}u\vert}{\bigg)}^{q-2}{\bigg]}^{p\,’}\, {\omega}\, dx\nonumber\\ & = & {\alpha}_q^{p\,’}\int_{\Omega}{\vert{\Delta}u_{m_k} – {\Delta}u\vert}^{p\,’}{\bigg(}{\vert{\Delta}u_{m_k}\vert} + {\vert{\Delta}u\vert}{\bigg)}^{(q-2)p\,’}\, {\omega}\,dx\nonumber\\ & {\leq} & \,2^{(q-2)p\,’}{\alpha}_q^{p\,’}\int_{\Omega} {\vert {\Delta}u_{m_k} – {\Delta}u\vert}^{p\,’}\, {\Phi}_2^{(q-2)p\,’}\,{\omega}\, dx.\label{eq8} \end{eqnarray}

  (8)
  For \(s = p/p\,’ = p-1>1\) and \(s\,’ = (p-1)/(p-2)\), we have in (8) \begin{eqnarray*} & & {\Vert G_2u_{m_k} – G_2u\Vert}_{L^{p\,’}(\Omega , \omega)}^{p\,’}\\ & & {\leq} \, 2^{(q-2)p\,’}{\alpha}_q^{p\,’}\int_{\Omega} {\vert {\Delta}u_{m_k} – {\Delta}u \vert}^{p\,’}\, {\Phi}_2^{(q-2)p\,’}\,{\omega}\, dx\\ & & {\leq}\, 2^{(q-2)p\,’}{\alpha}_q^{p\,’}{\bigg(} \int_{\Omega}{\vert{\Delta}u_{m_k} -{\Delta}u\vert}^{p\,’ s}{\omega}\, dx{\bigg)}^{1/s}{\bigg(}\int_{\Omega}{\Phi}_2^{(q-2)p\,’\, s\,’}\, {\omega}\, dx{\bigg)}^{1/s\,’}\\ & & = \ 2^{(q-2)p\,’}{\alpha}_q^{p\,’}{\bigg(} \int_{\Omega}{\vert{\Delta}u_{m_k} -{\Delta}u\vert}^p\,{\omega}\,dx{\bigg)}^{p\,’/p}{\bigg(}\int_{\Omega}{\Phi}_2^{(q-2)p/(p-2)}\, {\omega}\, dx{\bigg)}^{(p-2)/(p-1)}. \end{eqnarray*} Now, since \(\displaystyle 0< {\eta}={\dfrac{(q-2)p}{(p-2)}} < p\), then by Remark 1, we have \(\displaystyle {\Vert{\Phi}_2\Vert}_{L^{\eta}(\Omega , \omega)}\, {\leq}\, C_{p,\eta}{\Vert {\Phi}_2\Vert}_{L^p(\Omega , \omega)}.\) Therefore, we obtain \[{\Vert G_2u_{m_k} – G_2u\Vert}_{L^{p\,'}(\Omega , \omega)}^{p\,'}\, {\leq}\, 2^{(q-2)\,p\,'}{\alpha}_q^{p\,'}C_{p,\eta}^{(q-2)\,p\,'}{\Vert u_{m_k} – u\Vert}_X^{p\,'}{\Vert{\Phi}_2\Vert}_{L^p(\Omega , \omega)}^{(q-2)\,p\,'}.\] Hence \(\displaystyle {\Vert G_2u_{m_k} – G_2u\Vert}_{L^{p\,'}(\Omega , \omega)}\, {\leq}\, 2^{q-2}{\alpha}_q\, C_{p,\eta}^{q-2}{\Vert {\Phi}_2\Vert}_{L^p(\Omega , \omega)}^{q-2}{\Vert u_{m_k} – u \Vert}_X\).

  In the case \(2=q< p< {\infty}\), we have \((G_2u)(x) = {\Delta}u(x)\) and

  \begin{eqnarray*} {\Vert G_2u_{m_k} – G_2u\Vert}_{L^{p\,’}(\Omega , \omega)}^{p\,’} & = & \int_{\Omega}{\vert {\Delta}u_{m_k} – {\Delta}u\vert}^{p\,’}\, {\omega}\, dx\\ & {\leq}& {\bigg(}\int_{\Omega}{\vert{\Delta}u_{m_k} – {\Delta}u \vert}^p\, {\omega}\, dx{\bigg)}^{p\,’/p}{\bigg(}\int_{\Omega}{\omega}\, dx{\bigg)}^{(p-2)/(p-1)}\\ & {\leq} & {\Vert u_{m_k} – u \Vert}_X^{p\,’} {\bigg(}\int_{\Omega}{\omega}\, dx{\bigg)}^{(p-2)/(p-1)}. \end{eqnarray*} Therefore, for \(2\, {\leq}\, q < p< {\infty}\), by the Dominated Convergence Theorem we obtain (when \(m_k \to \infty\)) \[{\Vert G_2u_{m_k} – G_2u\Vert}_{L^{p\,'}(\Omega , \omega)}{\rightarrow}\, 0,\] that is, \(G_2u_{m_k} {\rightarrow}\, G_2u\) in \(L^{p\,'}(\Omega , \omega)\). By the Convergence Principle in Banach spaces, we have
  \begin{equation} \label{eq9} G_2u_m {\rightarrow}\, G_2u \ \ {in} \ \ L^{p\,’}(\Omega , \omega). \end{equation}

  (9)

Step 4. Since \(\displaystyle {\dfrac{f_j}{\omega}}\, {\in}\, L^{p\,’}(\Omega , \omega)\) (\(j=0,1,…,n\)), then \(T\,{\in}\, [W_0^{1.p}(\Omega , \omega)]^*\, {\subset}\, X^*\). Moreover, we have by Theorem 3 (with \(k=1\)), \begin{eqnarray*} {\vert T(\varphi) \vert} & {\leq}& \int_{\Omega} {\vert f_0 \vert}{\vert\varphi\vert}\,dx + \sum_{j=1}^n\int_{\Omega}{\vert f_j\vert}{\vert D_j{\varphi}\vert}\,dx \\ & = & \int_{\Omega} {\dfrac{\vert f_0 \vert} {\omega}}{\vert\varphi\vert}{\omega}\,dx + \sum_{j=1}^n \int_{\Omega}{\dfrac{\vert f_j \vert}{\omega}} {\vert D_j{\varphi}\vert}\,{\omega}\,dx \\ & {\leq}&{\Vert f_0/{\omega} \Vert}_{L^{p\,’}(\Omega , \omega)}{\Vert {\varphi} \Vert}_{L^p(\Omega , \omega)} + \sum_{j=1}^n {\Vert f_j/{\omega} \Vert}_{L^{p\,’}(\Omega , \omega)}{\Vert D_j\varphi \Vert}_{L^p(\Omega , \omega)} \\ & {\leq}& C_{\Omega}\,{\Vert f_0 /{\omega} \Vert}_{L^{p\,’}(\Omega , \omega)}{\Vert{\nabla}{\varphi}\Vert}_{L^p(\Omega , \omega)} + {\bigg(}\sum_{j=1}^n{\Vert f_j/{\omega} \Vert}_{L^{p\,’}(\Omega , \omega)}{\bigg)} {\Vert \nabla\varphi \Vert}_{L^p(\Omega , \omega)}\\ & {\leq}&{\biggl(} C_{\Omega}\,{\Vert f_0 /{\omega} \Vert}_{L^{p\,’}(\Omega , \omega)} + \sum_{j=1}^n{\Vert f_j/{\omega} \Vert}_{L^{p\,’}(\Omega , \omega)} {\biggr)} {\Vert \varphi \Vert}_X. \end{eqnarray*} Moreover, we also have

In (10) we have, using (H4), we have \begin{eqnarray*} & & \int_{\Omega}{\vert {\mathcal{A}}(x,u,{\nabla}u)\vert}\,{\vert {\nabla}{\varphi}\vert}\, {\omega}\, dx \,{\leq}\, \int_{\Omega}{\bigg(}K_1+ h_1{\vert u \vert}^{p/p\,’} + h_2{\vert{\nabla}u\vert}^{p/p\,’}{\bigg)}{\vert{\nabla}{\varphi}\vert}\, {\omega}\, dx\\ & & {\leq} \, {\Vert K_1\Vert}_{L^{p\,’}(\Omega , \omega)}{\Vert\,\vert {\nabla}\varphi\vert\,\Vert}_{L^p(\Omega , \omega)} + {\Vert h_1\Vert}_{L^{\infty}(\Omega)}{\Vert u \Vert}_{L^p(\Omega , \omega)}^{p/p\,’}{\Vert\,\vert {\nabla}{\varphi}\vert\,\Vert}_{L^p(\Omega , \omega)}\\ & & + \,{\Vert h_2\Vert}_{L^{\infty}(\Omega)}{\Vert\,\vert {\nabla}u\vert\,\Vert}_{L^p(\Omega , \omega)}^{p/p\,’}{\Vert\,\vert {\nabla}{\varphi}\vert\,\Vert}_{L^p(\Omega , \omega)}\\ & & {\leq} \, {\bigg(}{\Vert K_1\Vert}_{L^{p\,’}(\Omega , \omega)} + (C_{\Omega}^{p/p\,’}\,{\Vert h_1\Vert}_{L^{\infty}(\Omega)} + {\Vert h_2\Vert}_{L^{\infty}(\Omega)}){\Vert u \Vert}_X^{p/p\,’}{\bigg)}{\Vert \varphi \Vert}_X, \end{eqnarray*} and \begin{eqnarray*} \int_{\Omega}{\vert\, {\Delta}u\vert}^{p-2}\, {\vert\, {\Delta}u\vert}\, {\vert\, {\Delta}\,{\varphi}\vert}\, {\omega}\, dx & = & \int_{\Omega}{\vert\, {\Delta}u\vert}^{p-1}\, {\vert\, {\Delta}\,{\varphi}\vert}\, {\omega}\,dx\\ &{\leq}& {\bigg(}\int_{\Omega}{\vert\, {\Delta}u\vert}^p\, {\omega}\, dx{\bigg)}^{1/p\,’}{\bigg(}\int_{\Omega}{\vert\,{\Delta}{\varphi}\vert}^p\, {\omega}\,dx{\bigg)}^{1/p}\\ &{\leq}& {\Vert u \Vert}_X^{p/p\,’}{\Vert \varphi\Vert}_X, \end{eqnarray*} and \begin{eqnarray*} \int_{\Omega}{\vert\, {\Delta}u\vert}^{q-2}\, {\vert\, {\Delta}u\vert}\, {\vert\, {\Delta}\,{\varphi}\vert}\, {\omega}\, dx & = & \int_{\Omega}{\vert\, {\Delta}u\vert}^{q-1}\, {\vert\, {\Delta}\,{\varphi}\vert}\, {\omega}\,dx\\ & {\leq} & {\bigg(}\int_{\Omega}{\vert{\Delta}u\vert}^{(q-1)p\,’}\,{\omega}\, dx{\bigg)}^{1/p\,’}{\bigg(}\int_{\Omega}{\vert{\Delta}{\varphi}\vert}^p\, {\omega}\, dx{\bigg)}^{1/p}\\ & = & {\Vert{\Delta}u\Vert}_{L^{p\,'(q-1)}(\Omega , \omega)}^{q-1}{\Vert{\Delta}{\varphi}\Vert}_{L^p(\Omega , \omega)}. \end{eqnarray*} Since \(\displaystyle 0< {\eta} = p\,'(q-1) = {\dfrac{p(q-1)}{p-1}}< p\), then by Remark 1 we have \[{\Vert{\Delta}u\Vert}_{L^{\eta}(\Omega , \omega)}\, {\leq}\, C_{p,{\eta}}{\Vert {\Delta}u \Vert}_{L^p(\Omega , \omega)}.\] Hence \begin{eqnarray*} \int_{\Omega}{\vert{\Delta}u\vert}^{q-2}{\vert{\Delta}u\vert}{\vert{\Delta}{\varphi}\vert}\, {\omega}\,dx &{\leq}& {\Vert{\Delta}u\Vert}_{L^{p\,'(q-1)}(\Omega , \omega)}^{q-1}{\Vert{\Delta}{\varphi}\Vert}_{L^p(\Omega , \omega)}\\ & {\leq}& C_{p,{\eta}}^{q-1}{\Vert{\Delta}u\Vert}_{L^p(\Omega , \omega)}^{q-1}{\Vert{\Delta}{\varphi}\Vert}_{L^p(\Omega , \omega)}\\ & {\leq} & C_{p,{\eta}}^{q-1}{\Vert u \Vert}_X^{q-1}{\Vert \varphi \Vert}_X. \end{eqnarray*} Hence, in (10) we obtain, for all \(u,{\varphi}\, {\in}\, X\) \begin{eqnarray*} {\vert B(u,\varphi) \vert} {\leq} \, {\bigg[}{\Vert K_1\Vert}_{L^{p\,'}(\Omega , \omega)} + C_{\Omega}^{p/p\,'}\,{\Vert h_1\Vert}_{L^{\infty}(\Omega)}{\Vert u \Vert}_X^{p/p\,'} + {\Vert h_2\Vert}_{L^{\infty}(\Omega , \omega)}{\Vert u \Vert}_X^{p/p\,'} + {\Vert u \Vert}_X^{p/p\,'} + \, C_{p,{\eta}}^{q-1}{\Vert u \Vert}_X^{q-1} {\bigg]} {\Vert \varphi \Vert}_X. \end{eqnarray*} Since \(B(u, .)\) is linear, for each \(u\,{\in}\,X\), there exists a linear and continuous functional on \(X\) denoted by \(Au\) such that \({\langle}Au, \varphi{\rangle} = B(u , \varphi)\), for all \(u, \ {\varphi}\,{\in}\, X\) (here \({\langle}f,x{\rangle}\) denotes the value of the linear functional \(f\) at the point \(x\)). Moreover \begin{eqnarray*} {\Vert Au\Vert}_* {\leq} {\Vert K_1\Vert}_{L^{p\,'}(\Omega , \omega)} + C_{\Omega}^{p/p\,'}\,{\Vert h_1\Vert}_{L^{\infty}(\Omega)}{\Vert u \Vert}_X^{p/p\,'} + {\Vert h_2\Vert}_{L^{\infty}(\Omega , \omega)}{\Vert u \Vert}_X^{p/p\,'} + {\Vert u \Vert}_X^{p/p\,'} + C_{p,{\eta}}^{q-1}{\Vert u \Vert}_X^{q-1}, \end{eqnarray*} where \(\displaystyle {\Vert Au\Vert}_* = \sup\{{\vert{\langle}Au, {\varphi}{\rangle}\vert} = {\vert B(u,\varphi)\vert}: \ {\varphi}\, {\in}\, X, {\Vert {\varphi}\Vert}_X=1\}\) is the norm of the operators \(Au\). Hence, we obtain the operator \(A:X \,{\rightarrow}\, X^*\), \(u\,{\mapsto}\, Au\). Consequently, Problem (P) is equivalent to the operator equation \[Au = T, \ u\,{\in}\,X.\]

Step 5. Using condition (H2) and Lemma 1(b), we have \begin{eqnarray*} & & {\langle}Au_1 – Au_2, u_1 – u_2{\rangle} = B(u_1,u_1 – u_2) – B(u_2, u_1 – u_2) \\ & & = \int_{\Omega} {\mathcal{A}}(x, u_1, {\nabla}u_1).{\nabla}(u_1-u_2)\,{\omega}\,dx + \int_{\Omega}{\vert\,{\Delta}u_1\vert}^{p-2} \,{\Delta}u_1\, {\Delta}(u_1-u_2)\, {\omega}\, dx \\ && + \int_{\Omega}{\vert\,{\Delta}u_1\vert}^{q-2} \,{\Delta}u_1\, {\Delta}(u_1-u_2)\, {\omega}\, dx – \int_{\Omega} {\mathcal{A}}(x, u_2, {\nabla}u_2). {\nabla}(u_1-u_2)\,{\omega}\, dx\\&& – \int_{\Omega} {\vert\, {\Delta}u_2\vert}^{p-2}\, {\Delta}u_2\, {\Delta}(u_1-u_2)\,{\omega}\, dx – \int_{\Omega}{\vert\,{\Delta}u_2\vert}^{q-2} \,{\Delta}u_2\, {\Delta}(u_1-u_2)\, {\omega}\, dx \\ && = \int_{\Omega} {\biggl(}{\mathcal{A}}(x, u_1, {\nabla}u_1) – {\mathcal{A}}(x, u_2, {\nabla}u_2){\biggr)}.{\nabla}(u_1-u_2)\,{\omega}\,dx \\ && + \int_{\Omega}({\vert\,{\Delta}u_1\vert}^{p-2}\, {\Delta}u_1 – {\vert\,{\Delta}u_2\vert}^{p-2}\,{\Delta}u_2)\, {\Delta}(u_1-u_2)\, {\omega}\, dx\\ && + \int_{\Omega}({\vert\,{\Delta}u_1\vert}^{q-2}\, {\Delta}u_1 – {\vert\,{\Delta}u_2\vert}^{q-2}\,{\Delta}u_2)\, {\Delta}(u_1-u_2)\, {\omega}\, dx\\ &&{\geq}{\theta}_1\int_{\Omega} {\vert{\nabla}(u_1-u_2)\vert}^p\,{\omega}\, dx + {\beta}_p \int_{\Omega}({\vert\,{\Delta}u_1\vert} + {\vert\,{\Delta}u_2\vert})^{p-2}\, {\vert {\Delta}u_1-{\Delta}u_2\vert}^2\, {\omega}\, dx\\ && + {\beta}_q \int_{\Omega}({\vert\,{\Delta}u_1\vert} + {\vert\,{\Delta}u_2\vert})^{q-2}\, {\vert {\Delta}u_1-{\Delta}u_2\vert}^2\, {\omega}\, dx\\ &&{\geq}{\theta}_1\int_{\Omega} {\vert{\nabla}(u_1-u_2)\vert}^p\,{\omega}\, dx + {\beta}_p \int_{\Omega}({\vert\,{\Delta}u_1 – {\Delta}u_2\vert})^{p-2}\, {\vert {\Delta}u_1-{\Delta}u_2\vert}^2\, {\omega}\, dx\\ && + {\beta}_q \int_{\Omega}({\vert\,{\Delta}u_1 – {\Delta}u_2\vert})^{q-2}\, {\vert {\Delta}u_1-{\Delta}u_2\vert}^2\, {\omega}\, dx\\ && = {\theta}_1\int_{\Omega} {\vert{\nabla}(u_1-u_2)\vert}^p\,{\omega}\, dx + {\beta}_p \int_{\Omega} {\vert {\Delta}u_1-{\Delta}u_2\vert}^p\, {\omega}\, dx + {\beta}_q \int_{\Omega} {\vert {\Delta}u_1-{\Delta}u_2\vert}^q\, {\omega}\, dx\\ && {\geq} {\theta}_1\int_{\Omega} {\vert{\nabla}(u_1-u_2)\vert}^p\,{\omega}\, dx + {\beta}_p \int_{\Omega} {\vert {\Delta}u_1-{\Delta}u_2\vert}^p\, {\omega}\, dx\\ && {\geq} {\theta}\, {\Vert u_1 – u_2\Vert}_X^p \end{eqnarray*} where \({\theta} = \min\,\{{\theta}_1, {\beta}_p\}\). Therefore, the operator \(A\) is strongly monotone, and this implies that \(A\) is strictly monotone. Moreover, from (H3), we obtain \begin{eqnarray*} {\langle}Au,u{\rangle} &=& B(u,u) = B_1(u,u) + B_2(u,u)+ B_3(u,u)\\ &=& \int_{\Omega} {\mathcal{ A}}(x,u,{\nabla}u).{\nabla}u\,{\omega}\,dx + \int_{\Omega}{\vert\, {\Delta}u\vert}^{p-2}\, {\Delta}u\, {\Delta}u\, {\omega}\, dx + \int_{\Omega}{\vert\, {\Delta}u \vert}^{q-2}\, {\Delta}u\, {\Delta}u\, {\omega}\, dx\\ &{\geq}& \int_{\Omega} {\lambda}_1{\vert {\nabla} u \vert}^p \, {\omega}\, dx + \int_{\Omega} {\vert\, {\Delta}u\vert}^p\, {\omega}\, dx + \int_{\Omega} {\vert\, {\Delta}u\vert}^q\, {\omega}\, dx\\ &{\geq}& \int_{\Omega} {\lambda}_1{\vert {\nabla} u \vert}^p \, {\omega}\, dx + \int_{\Omega} {\vert\, {\Delta}u\vert}^p\, {\omega}\, dx\\ & {\geq}& {\gamma} \, {\Vert u \Vert}_X^p, \end{eqnarray*} where \( {\gamma} = \min\, \{{\lambda}_1,1 \}\). Hence, since \(2\, {\leq}\, q< p < \infty\), we have \[{\dfrac{{\langle}Au,u{\rangle}}{{\Vert u \Vert}_X}}\, {\rightarrow}+{\infty}, \ {as} \ {\Vert u \Vert}_X\,{\rightarrow}+{\infty},\] that is, \(A\) is coercive.

Step 6. We need to show that the operator \(A\) is continuous. Let \(u_m{\rightarrow}\,u\) in \(X\) as \(m\to\infty\). We have \begin{eqnarray*} {\vert B_1(u_m , \varphi) – B_1(u , \varphi) \vert} & {\leq}& \sum_{j=1}^n\int_{\Omega} {\vert {\mathcal{A}}_j(x,u_m, {\nabla}u_m) – {\mathcal{A}}_j(x,u,{\nabla}u) \vert}{\vert D_j{\varphi} \vert}\,{\omega}\,dx \\ & = & \sum_{j=1}^n \int_{\Omega} {\vert F_ju_m – F_ju \vert}{\vert D_j{\varphi} \vert}\,{\omega}\,dx\\ &{\leq}& \sum_{j=1}^n {\Vert F_ju_m – F_ju \Vert}_{L^{p\,’}(\Omega , \omega)}{\Vert D_j{\varphi} \Vert}_{L^p(\Omega , \omega)}\\ &{\leq}& {\bigg(}\sum_{j=1}^n{\Vert F_ju_m – F_ju \Vert}_{L^{p\,’}(\Omega , \omega)}{\bigg)}{\Vert \varphi \Vert}_X, \end{eqnarray*} and \begin{eqnarray*} {\vert B_2(u_m, {\varphi}) – B_2(u, \varphi)\vert} & =& {\bigg\vert}\int_{\Omega}{\vert\,{\Delta}u_m\vert}^{p-2}{\Delta}u_m \, {\Delta}{\varphi}\, {\omega}\, dx – \int_{\Omega}{\vert\,{\Delta}u\vert}^{p-2}{\Delta}u \, {\Delta}{\varphi}\, {\omega}\, dx\,{\bigg\vert}\\ &{\leq}& \int_{\Omega}{\bigg\vert}\,{\vert\,{\Delta}u_m\vert}^{p-2}\, {\Delta}u_m – {\vert\,{\Delta}u\vert}^{p-2}{\Delta}u\, {\bigg\vert}\, {\vert\, {\Delta}{\varphi}\vert}\, {\omega}\, dx\\ &=& \int_{\Omega}{\vert G_1u_m – G_1u\vert}\, {\vert {\Delta}{\varphi}\vert}\, {\omega}\, dx\\ &{\leq}& {\Vert G_1u_m -G_1u\Vert}_{L^p\,'(\Omega , \omega)}{\Vert{\Delta}{\varphi}\Vert}_{L^p(\Omega , \omega)}\\ &{\leq}&\, {\Vert G_1u_m – G_1u\Vert}_{L^{p\,’}(\Omega , \omega)}\, {\Vert \varphi\Vert}_X. \end{eqnarray*} and \begin{eqnarray*} {\vert B_3(u_m, {\varphi}) – B_3(u, \varphi)\vert}& =& {\bigg\vert}\int_{\Omega}{\vert\,{\Delta}u_m\vert}^{q-2}{\Delta}u_m \, {\Delta}{\varphi}\, {\omega}\, dx – \int_{\Omega}{\vert\,{\Delta}u\vert}^{q-2}{\Delta}u \, {\Delta}{\varphi}\, {\omega}\, dx\,{\bigg\vert}\\ &{\leq}& \int_{\Omega}{\bigg\vert}\,{\vert\,{\Delta}u_m\vert}^{q-2}\, {\Delta}u_m – {\vert\,{\Delta}u\vert}^{q-2}{\Delta}u\, {\bigg\vert}\, {\vert\, {\Delta}{\varphi}\vert}\, {\omega}\, dx\\ &=& \int_{\Omega}{\vert G_2u_m – G_2u\vert}\, {\vert {\Delta}{\varphi}\vert}\, {\omega}\, dx\\ &{\leq}&\, {\Vert G_2u_m – G_2u\Vert}_{L^{p\,’}(\Omega , \omega)}\, {\Vert \varphi\Vert}_X\,, \end{eqnarray*} for all \({\varphi}\,{\in}\,X\). Hence \begin{eqnarray*} {\vert B(u_m , \varphi) – B(u, \varphi)\vert} & {\leq}&\, {\vert B_1(u_m,\varphi) – B_1(u, \varphi)\vert} + {\vert B_2(u_m , \varphi) – B_2(u, \varphi)\vert} + {\vert B_3(u_m , \varphi) – B_3(u, \varphi)\vert}\\ &{\leq}&\, {\bigg[}\sum_{j=1}^n{\Vert F_ju_m – F_ju \Vert}_{L^{p\,’}(\Omega , \omega)} + \, {\Vert G_1u_m-G_1u\Vert}_{L^{p\,’}(\Omega , \omega)}+ \ {\Vert G_2u_m-G_2u\Vert}_{L^{p\,’}(\Omega , \omega)} {\bigg]} {\Vert \varphi \Vert}_X. \end{eqnarray*} Then we obtain \begin{eqnarray*} {\Vert Au_m – Au \Vert}_*\, {\leq} \, \sum_{j=1}^n {\Vert F_ju_m – F_ju \Vert}_{L^{p\,’}(\Omega , \omega)} + {\Vert G_1u_m-G_1u\Vert}_{L^{p\,’}(\Omega , \omega)}+ \ {\Vert G_2u_m – G_2u\Vert}_{L^{p\,’}(\Omega , \omega)}. \end{eqnarray*} Therefore, using (4), (7) and (9) we have \({\Vert Au_m – Au \Vert}_*{\rightarrow}\,0\) as \(m\to +{\infty}\), that is, \(A\) is continuous and this implies that \(A\) is hemicontinuous.

Therefore, by Theorem 4, the operator equation \(Au = T\) has a unique solution \(u\,{\in}\,X\) and it is the unique solution for problem (P).

Step 7. In particular, by setting \({\varphi}=u\) in Definition 4, we have

Hence, using (H3) and \({\gamma}=\min\,\{{\lambda}_1, 1\}\), we obtain \begin{eqnarray*} & & B_1(u,u) + B_2(u,u) + B_3(u,u)\\ & & = \int_{\Omega} {\mathcal{A}}(x,u,{\nabla}u).{\nabla}u\,{\omega}\, dx + \int_{\Omega} {\vert\,{\Delta}u\vert}^{p-2}\, {\Delta}u\, {\Delta}u\, {\omega}\, dx + \int_{\Omega} {\vert\,{\Delta}u\vert}^{q-2}\, {\Delta}u\, {\Delta}u\, {\omega}\, dx\\ & & {\geq} \int_{\Omega} {\lambda}_1\,{\vert {\nabla}u\vert}^p + \int_{\Omega}{\vert\, {\Delta}u\vert}^p\, {\omega}\, dx + \int_{\Omega}{\vert\, {\Delta}u\vert}^q\, {\omega}\, dx\\ & & {\geq} \int_{\Omega} {\lambda}_1\,{\vert {\nabla}u\vert}^p + \int_{\Omega}{\vert\, {\Delta}u\vert}^p\, {\omega}\, dx\\ & & {\geq} \, {\gamma} {\Vert u \Vert}_X^p \end{eqnarray*} and \begin{eqnarray*} T(u) & = & \int_{\Omega}f_0\, u \, dx + \sum_{j=1}^n\int_{\Omega} f_j\, D_ju\, dx\\ & {\leq} & {\Vert f_0/{\omega}\Vert}_{L^{p\,’}(\Omega , \omega)}{\Vert u \Vert}_{L^p(\Omega , \omega)} + \sum_{j=1}^n{\Vert f_j/{\omega}\vert}_{L^{p\,’}(\Omega)}{\Vert D_ju\Vert}_{L^p(\Omega , \omega)}\\ & {\leq} & {\bigg(} C_{\Omega}\, {\Vert f_0/{\omega}\Vert}_{L^{p\,’}(\Omega , \omega)} + \sum_{j=1}^n{\Vert f_j/{\omega}\Vert}_{L^{p\,’}(\Omega)}{\bigg)}\, {\Vert u \Vert}_X. \end{eqnarray*} Therefore, in (11), we obtain \[{\gamma}\, {\Vert u \Vert}_X^p \,{\leq}\, {\bigg(}C_{\Omega}\, {\Vert f_0/{\omega}\Vert}_{L^{p\,’}(\Omega , \omega)} + \sum_{j=1}^n{\Vert f_j/{\omega}\Vert}_{L^{p\,’}(\Omega , \omega)}{\bigg)}\,{\Vert u \Vert}_X,\] and we obtain \[{\Vert u \Vert}_X\, {\leq}\, {\dfrac{1}{{\gamma}^{p\,’/p}}}\, {\bigg(}C_{\Omega}\, {\Vert f_0/{\omega}\Vert}_{L^{p\,’}(\Omega , \omega)} + \sum_{j=1}^n{\Vert f_j/{\omega}\Vert}_{L^{p\,’}(\Omega , \omega)} {\bigg)}^{p\,’/p}.\]

Example 1. Let \({\Omega} = \{ (x,y)\, {\in}\, \mathbb{R}^2: x^2+y^2< 1\}\), the weight function \({\omega}(x,y) = (x^2 + y^2)^{-1/2}\) (\({\omega}\,{\in}\, A_4\), \(p=4\) and \(q=3\)), and the function \begin{eqnarray*} & & {\mathcal{A}}: {\Omega}\, {\times}\,\mathbb{R}\, {\times}\, \mathbb{R}^2 {\rightarrow}\, \mathbb{R}^2\;\;\text{defined by}\\ & & {\mathcal{A}}((x,y), \eta,\xi) = h_2(x,y)\, {\vert \xi\vert}\, {\xi}, \end{eqnarray*} where \(h(x,y) = 2\, {e}^{(x^2+y^2)}\). Let us consider the partial differential operator \[Lu(x,y) = {\Delta}{\big[}(x^2+y^2)^{-1/2}{\big(}{\vert{\Delta}u \vert}^2\,{\Delta} u + {\vert{\Delta}u\vert}{\Delta}u{\big)}{\big]} – { div}\,((x^2+y^2)^{-1/2}\,{\mathcal{A}}((x,y), u , {\nabla}u)).\] Therefore, by Theorem 1, the problem \[ \begin{cases}Lu(x) = {\dfrac{\cos(xy)}{(x^2+y^2)}} – {\dfrac{\partial}{\partial x}}{\bigg(}{\dfrac{\sin(xy)}{(x^2+y^2)}}{\bigg)} – {\dfrac{\partial}{\partial y}}{\bigg(}{\dfrac{\sin(xy)}{(x^2+y^2)}}{\bigg)},& \ \ { in} \ \ {\Omega} \\ u(x) = 0, &\ \ { on} \ \ {\partial\Omega} \end{cases} \] has a unique solution \(u\, {\in}\, X = W^{2,4}(\Omega , \omega)\, {\cap}\, W_0^{1,4}(\Omega , \omega)\).

Conflict of Interests

The author declares no conflict of interest.