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Small perturbations of Robin problems driven by the \(p\)-Laplacian plus a positive potential. (English) Zbl 1479.35473

Let \(\Omega \subseteq \mathbb{R}^N\) (\(N>2\)) be a bounded domain with a \(C^2\)-boundary \(\partial \Omega\). The authors study a parametric Robin problem of the form \[ \begin{cases} -\Delta_p u+V(x) u=a(x) |u|^{q-1}u+ \lambda g(x,u)+f(x) \mbox{ in } \Omega,\\ |\nabla u|^{p-2}\nabla u \cdot \nu + \beta(x)|u|^{p-2}u=0 \text{ on }\partial \Omega, \quad\lambda \in \mathbb{R}, \quad p \geq 2, \quad 0<q<p-1, \end{cases}\tag{\(P_\lambda\)}\] where \(\nu(x)\) denotes the outer unit normal of \(\Omega\) at \(x \in \partial \Omega\), and \(\Delta_p u= \operatorname{div}(|\nabla u|^{p-2}\nabla u)\) for all \(u \in W^{1,p}(\Omega)\) is the classical \(p\)-Laplace operator. Additionally, \(a,V,f \in L^\infty( \Omega)\), \(\beta \in L^\infty(\partial \Omega)\) and \(g:\overline{\Omega} \times \mathbb{R} \to \mathbb{R}\) satisfy precise assumptions. Thus, the authors develop a variational approach to study the solutions of \((P_\lambda)\). They prove the existence of a positive value \(\lambda_0\) such that \((P_\lambda)\) admits at least one nontrivial weak solution if the parameter \(\lambda\) satisfies the bound \(|\lambda|<\lambda_0\) (this means equivalently that a suitable perturbation of the second reaction term is sufficiently small).

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J25 Boundary value problems for second-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35J20 Variational methods for second-order elliptic equations
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