Existence and regularity for elliptic equations under \(p,q\)-growth
The nonlinear elliptic PDE \[ \sum_{i=1}^n\frac{\partial}{\partial x_i}a^i\left(x,Du\right)=b(x) \] is considered in an open bounded subset \(\Omega\) of \(\mathbb{R}^n\) with Dirichlet boundary condition \(u=u_0\) on \(\partial \Omega\). Here \(b\in L^\infty_{\text{loc}}\left(\Omega\right)\), the functions \(a^i(x,\xi)\), \(i=1,2,\ldots,n\) are locally Lipshitz-continuous in \(\Omega\times\mathbb{R}^n\) and satisfy the following \(p,q\)-growth conditions, where \(1<p\leq q \): there exist positive constants \(m\) and \(M\) such that for every \(\xi, \lambda\in \mathbb{R}^n\), for a.e. \(x\in\Omega\) the inequalities \[ \begin{aligned} m\left(1+\left|\xi\right|^2\right)^{(p-2)/2}\left|\lambda\right|^2\leq \sum_{i,j=1}^na_{\xi_j}^i(x,\xi)\lambda_i\lambda_j, \left|a_{\xi_j}^i(x,\xi)\right|\leq M\left(1+\left|\xi\right|^2\right)^{(q-2)/2}, \\ \left|a_{\xi_j}^i(x,\xi)-a_{\xi_i}^j(x,\xi)\right|\leq M\left(1+\left|\xi\right|^2\right)^{(q+p-4)/4}\end{aligned} \] and \[ \left|a_{x_j}^i(x,\xi)\right|\leq M\left(1+\left|\xi\right|^2\right)^{(q+p-2)/4} \] hold for every \(i,j=1,2,\ldots,n\). The existence and regularity of a weak solution are studied under the following assumptions: let \(1<p\leq q\leq p+1\) and \(q<p(n-1)/(n-p)\) when \(p<n\), \(u_0\in W^{1,r}\) where \(r=p(q-1)/(p-1)\) and \(b\in L^{p/(p-1)}\left(\Omega\right) \bigcap L^\infty_{\text{loc}}\left(\Omega\right)\). Then there exists a weak solution \(u\in W_{\text{loc}}^{1,q}\) such that \(u-u_0\in W^{1,p}_{0}\left(\Omega\right)\bigcap W_{\text{loc}}^{1,q}\). Moreover, \(u\in W^{1,\infty}_{\text{loc}}\left(\Omega\right)\bigcap W^{2,2}_{\text{loc}}\left(\Omega\right)\).
scientific article; zbMATH DE number 6305506