## 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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### References:

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