## Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth.(English)Zbl 1479.35429

Summary: We are concerned with Dirichlet problems of the form $\mathrm{div} (|Du|^{p-2} Du)+f(u)=0 \quad \text{in }\Omega,\qquad u=0\quad \text{on }\partial\Omega,$ where $$\Omega$$ is a bounded domain of $$\mathbb{R}^n, n\geq 2, 1< p<n$$ and $$f$$ is a continuous function with supercritical growth from the viewpoint of the Sobolev embedding. In particular, if $$n=2$$ and $$\gamma\colon [a,b]\to\mathbb{R}^2$$ is a smooth curve such that $$\gamma (t_1)\neq\gamma (t_2)$$ for $$t_1 \neq t_2$$, we prove that, for $$\varepsilon >0$$ small enough, there exists a unique solution of the Dirichlet problem in the domain $$\Omega =\Omega^{\Gamma}_{\varepsilon} =\{ (x_1, x_2)\in\mathbb{R}^2 :\mathrm{dist} \big( (x_1,x_2),\Gamma\big) < \varepsilon\}$$, where $$\Gamma =\{\gamma (t):t\in [a,b]\}$$. Moreover, we extend this uniqueness result to the case where $$n>2$$ and $$\Omega$$ is, for example, a domain of the type $\Omega =\widetilde{\Omega}^{\Gamma}_{\varepsilon,s} =\{ (x_1,x_2,y) : (x_1,x_2)\in\Omega^{\Gamma}_{\varepsilon}, y\in\mathbb{R}^{n-2}, |y|< s\}.$

### MSC:

 35J62 Quasilinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35J20 Variational methods for second-order elliptic equations
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### References:

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