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The Nehari manifold for a class of concave-convex elliptic systems involving the \(p\)-Laplacian and nonlinear boundary condition. (English) Zbl 1200.35103

Summary: The existence and multiplicity of weak solutions is established for a class of concave-convex elliptic systems of the form:
\[ \begin{cases} -\Delta_pu+m(x)|u|^{p-2}u=\lambda a(x)|u|^{\gamma-2}u, &x\in\Omega,\\ -\Delta_pv+m(x)|v|^{p-2}u=\mu b(x)|v|^{\gamma-2}v, &x\in\Omega,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial n}= \frac{\alpha}{\alpha+\beta} |u|^{\alpha-2} u|v|^\beta, &x\in\partial\Omega,\\ |\nabla v|^{p-2} \frac{\partial v}{\partial n}= \frac{\beta}{\alpha+\beta} |u|^\alpha |v|^{\beta-2}v, &x\in\partial\Omega.\end{cases} \]
Here \(\Delta_p\) denotes the \(p\)-Laplacian operator defined by \(\Delta_pz= \text{div}(|\nabla z|^{p-2}\nabla z)\), \(p>2\), \(\Omega\subset\mathbb R^N\) is a bounded domain with smooth boundary, \(\alpha>1\), \(\beta>1\), \(2<\alpha+\beta<p<\gamma<p^*\) (\(p^*= \frac{pN}{N-p}\) if \(N>p\), \(p^*=\infty\) if \(N\leq p\)), \(\frac{\partial}{\partial n}\) is the outer normal derivative, \((\lambda,\mu)\in\mathbb R^2\setminus\{(0,0)\}\), the weight \(m(x)\) is a positive bounded function and \(a(x),b(x)\in C(\Omega)\) are functions which change sign in \(\Omega\). Our technical approach is based on the Nehari manifold which is similar to the fibering method of P. Drabek and S. I. Pohozaev [Positive solutions for the \(p\)-Laplacian: application of the fibering method, Proc. R. Soc. Edinb. Sect. A 127, 721–747 (1997)] together with the recent idea from K. J. Brown and T.-F. Wu [J. Math. Anal. Appl. 337, No. 2, 1326–1336 (2008; Zbl 1132.35361)].

MSC:

35J57 Boundary value problems for second-order elliptic systems
35J50 Variational methods for elliptic systems
35J66 Nonlinear boundary value problems for nonlinear elliptic equations
35D30 Weak solutions to PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 1132.35361
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References:

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