Multiplicity of solutions for a class of elliptic problem in with Neumann conditions
The paper is concerned with the semilinear elliptic problem \[ -\Delta u + u = Q(x)f(u)~~\text{in } \mathbb{R}^2\setminus\Omega,~~~\frac{\partial u}{\partial \eta} = 0~~\text{on } \partial\Omega, \] where \(\Omega \subset \mathbb{R}^2\) is a bounded smooth domain, \(Q\) is a positive continuous function defined on \({\mathbb R}^2 \setminus \Omega\), with \(Q(x)\to \overline{Q}>0\) as \(|x|\to \infty\), and finally the nonlinear function \(f: \mathbb{R} \to \mathbb{R}\) is continuously differentiable and such that: \((f_0)\) there exists \(\alpha_0>0\) such that \[ \lim_{|s|\to\infty} \frac{|f(s)|}{\exp(\alpha s^2)} = 0\quad\forall~\alpha>\alpha_0,\qquad\lim_{|s|\to\infty} \frac{|f(s)|}{\exp(\alpha s^2)} = +\infty\quad\forall~\alpha<\alpha_0, \] \((f_1)\) \(|f(s)| \leq C \exp(4\pi s^2)\) for all \(s \in \mathbb{R}\), \((f_2)\) there exists \(\theta>2\) such that \(0 < \theta F(s) \leq sf(s)\) for all \(s \in \mathbb{R}\), \((f_3)\) there exists \(q>1\) such that \( \limsup_{s \to 0} \frac{|f(s)|}{|s|^q} < \infty, \) \((f_4)\) the function \(s \mapsto f(s)/s\) is increasing in \((0,\infty)\), \((f_5)\) there exists \(p>2\) and \(C_p>0\) such that \[ f(s) \geq C_ps^{p-1}~~~\forall~s \in [0,\infty), \] where \(C_p\) is a suitable number which depends on \(\Omega,p,\theta\) and \(\bar Q\). The author, motivated by a paper of \textit{D. M. Cao} [Commun. Partial Differ. Equations 18, No. 3--4, 687--700 (1993; Zbl 0803.35049)] proves the existence of a positive and a nodal solution. The positive solution is a ground state solution and it is obtained under the additional condition (\(Q_2\)) there exists \( R_0>0\) and \(m>2\), such that \[ Q(x) \geq \bar Q - C\text{e}^{-m|x|}~~~|x|>R_0. \] In the second result the author obtained a nodal solution under the additional conditions (\(f_6\)) there exists \(\sigma \geq 2\) such that \(f'(s)s^2 - f(s)s \geq C|s|^{\sigma}\) for all \(s\in\mathbb{R}\), (\(f_7\)) \(|f'(s)s| \leq C\exp(4\pi s^2)\) for all \(s \in\mathbb{R}\), (\(Q_3\)) there exists \(\mu < 1/(q+1)\), such that \[ Q(x) \geq \bar Q + C\text{e}^{-\mu|x|}\quad\forall~x \in \mathbb{R}^N, \] The positive solution is obtained by applying the Mountain Pass Theorem. For the nodal solution, following the above paper of D. M. Cao, the author uses the implicit function theorem to obtain a Palais-Smale sequence of functions which change sign. By using careful estimates the author shows that the minimax levels of the solutions belong in a range where the associated functional satisfies the Palais-Smale condition. The compactness condition is proved by a variant of the global compactness lemma due to \textit{V. Benci} and \textit{G. Cerami} [Arch. Ration. Mech. Anal. 99, 283--300 (1987; Zbl 0635.35036)]. In order to overcome the critical growth of the nonlinearity the author use some estimates proved in [\textit{C. O. Alves, J. M. do Ó} and \textit{O. H. Miyagaki}, Nonlinear Anal., Theory Methods Appl. 56, 781--791 (2004; Zbl 1134.35329)]. The proofs presented are closely related with a previous paper of \textit{C. O. Alves, P. C. Carrião} and \textit{E. S. Medeiros} [Abstr. Appl. Anal. 2004, No. 3, 251--268 (2004; Zbl 1133.35304)] where the subcritical case for dimension \(N \geq 3\) was considered.
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