Neumann problem of quasilinear elliptic equations with limit nonlinearity in boundary condition. (Q2764274)

The author studies the problem of existence of solutions to partial differential equations of the form NEWLINE\[NEWLINED_{j}(a^{ij}(x,y)D_{i}u)-{1\over 2}D_{s}a^{ij}(x,y)D_{i}uD_{j}u+\lambda u=0,\qquad x\in\Omega,NEWLINE\]NEWLINE verifying the Neumann type nonlinear boundary condition NEWLINE\[NEWLINEa^{ij}(x,u)D_{i}u\gamma_{j}=| u| ^{2^{+}-2}u,\qquad x\in\partial\Omega,NEWLINE\]NEWLINE where \(\Omega\in \mathbb R^{n}\) (\(n\geq 3\)) is a smooth bounded domain, \(\gamma\in \mathbb R\), \(2^{+}={2n-2\over n-2}\) and \(\gamma(x)\) is the unit toward normal to \(\partial \Omega\) at \(x\). Assuming on the set of coefficients \(a^{ij}:\Omega\times \mathbb R\to \mathbb R\) thatNEWLINENEWLINE(1) \(a^{ij}=a^{ji}\), \(a^{ij}\) is a (\(C^{1}\)) Carathéodory function and NEWLINE\[NEWLINE\| a^{ij}(\cdot,s)\| _{L^\infty(\Omega)}+ \| D_{s}a^{ij}(\cdot,s)\| _{L^\infty(\Omega)}\leq C,\quad \forall s\in \mathbb RNEWLINE\]NEWLINE;NEWLINENEWLINE(2) \(\exists \nu>0\) such that \(a^{ij}(x,s)\xi_{i}\xi_{j}\geq \nu\| \xi\| ^{2}\) and \(sD_{s}a^{ij}(x,s)\xi_{i}\xi_{j}\geq 0\) for a.e. \(x\in\Omega\), \(\forall s\in \mathbb R\), \(\xi\in \mathbb R^{n}\);NEWLINENEWLINE(3) \(\exists a_{\infty}>0\) such that \( \lim_{| s| \to\infty}a^{ij}(x,s)=a_{\infty}\delta^{ij}\), and \( \lim_{| s| \to\infty}sD_{s}a^{ij}(x,s)= 0\) unif. a.e. on \(\Omega\); NEWLINENEWLINE\noindent the Author proves that the above posed problem admits a nontrivial solution for any \(\lambda <0\). The result is even generalized to cases in which the term \(\lambda u\) in the equation is replaced by one of the form \(f(x,u)\), asymptotically linear in \(u\) as \(u\to 0\) and a lower perturbation of \(u^{n+2\over n-2}\) as \(u\to\infty\).NEWLINENEWLINETo prove the result the author makes use of the Ekeland Variational Principle to construct what he calls \textit{concrete} Palais-Smale sequences for the continuous but not differentiable action functional associated to the problem, \(J:H^{1}(\Omega)\to R\), NEWLINE\[NEWLINEJ(u)={1\over 2}\int_{\Omega}a^{ij}(x,u)D_{i}uD_{j}u\, dx- {\lambda\over 2}\int_{\Omega}u^{2}\, dx-{1\over 2^{+}}\int_{\partial\Omega} | u| ^{2^{+}}\, d\sigma.NEWLINE\]