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Existence of solutions of anisotropic elliptic equations with nonpolynomial nonlinearities in unbounded domains. (English. Russian original) Zbl 1332.35107

Sb. Math. 206, No. 8, 1123-1149 (2015); translation from Mat. Sb. 206, No. 8, 99-126 (2015).
Let \(\Omega\subseteq\mathbb R^n\) \((n\geq 2)\) be an unbounded domain and let \(a_0,a_1,\dots,a_n:\Omega \times\mathbb R\times\mathbb R^n\rightarrow\mathbb R\) be \(n+1\)-functions which are measurable in the first variable and continuous in the last two variables. The authors consider the following Dirichlet problem for an anisotropic quasilinear elliptic equation \[ \begin{cases}\sum_{\alpha=1}^n(a_\alpha(x,u(x),\nabla u(x)))_{x_\alpha}-a_0(x,u(x),\nabla u(x))=0\quad & \text{in }\Omega,\\ u=0 \quad &\text{on }\partial \Omega,\end{cases} \] and establish an existence result of weak solutions under the following assumptions: \[ \sum_{\alpha=0}^n(a_\alpha(x,s_0,s)-a_\alpha(x,t_0,t))(s_\alpha-t_\alpha)>0, \]
\[ \sum_{\alpha=0}^na_\alpha(x,s_0,s)s_\alpha\geq \varphi(x)\sum_{\alpha=0}^nB_\alpha(s_\alpha)-\psi(x) \]
\[ \sum_{\alpha=0}^n\overline{B}_\alpha(a_\alpha(x,s_0,s))\leq \varphi_1(x)\sum_{\alpha=0}^nB_\alpha(s_\alpha)+\psi_1(x) \] for almost all \(x\in\Omega\) and for all \((s_0,s_1,\dots,s_n),(t_0,t_1,\dots,t_n)\in\mathbb R^{n+1}\), with \((s_0,s_1,\dots,s_n)\neq(t_0,t_1,\dots,t_n)\).
Here, \(\psi,\psi_1,\varphi,\varphi_1:\Omega \rightarrow \mathbb R\) are nonnegative measurable functions, and, for each \(\alpha=0,1,\dots,n\), \[ B_\alpha(z)=\int_0^{|z|}b_\alpha(\theta)d\theta, \overline{B}_\alpha(z)=\sup_{y\geq 0}(y|z|-M(y)), \] for all \(z\in \mathbb R\), where \(b_\alpha:\mathbb R_+\rightarrow \mathbb R_+\) is a nondecreasing right-continuous function which is \(0\) at \(\theta=0\), positive at each \(\theta>0\), and satisfies \[ \lim_{\theta\rightarrow \infty}b_\alpha(\theta)=\lim_{\lambda\rightarrow +\infty}\inf_{\theta>0}\frac{b_\alpha(\lambda\theta)}{b_\alpha(\theta)}=+\infty. \] Moreover, the function \(B_\alpha\) is assumed to satisfy the further conditions \(B_\alpha(2z)\leq cB_\alpha(z)\) and \(B_\alpha(z^{1+\epsilon})\leq B_\alpha(lz)\), for some constants \(c,l>0\), \(\epsilon\in (0,1)\), and for all \(z\) greater than a given number \(z_0\geq 0\).
Applying an abstract surjectivity result due to J. L. Lions [Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod; Paris: Gauthier-Villars (1969; Zbl 0189.40603), Theorem 2.1], the authors first prove the existence of a unique solution \(u_Q\) to the above problem in a bounded domain \(Q\subset \Omega\). Then, the authors prove an a priori estimate for \(u_Q\) which allows to obtain, by an approximation argument, a solution to the problem in the domain \(\Omega\).

MSC:

35J47 Second-order elliptic systems
35D30 Weak solutions to PDEs
35J62 Quasilinear elliptic equations

Citations:

Zbl 0189.40603
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