Singularly perturbed nonlinear Dirichlet problems involving critical growth
The authors consider the following singularly perturbed elliptic problem (\(P_{\epsilon}\)): \[ -\epsilon^2\Delta v+v=f(v),\, v>0 \text{ in } \Omega,\quad v=0 \text{ on } \partial \Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\), \(N\geq 3\), with a \(C^2\) boundary \(\partial \Omega\). The nonlinearity \(f:\mathbb R\to\mathbb R,f\in C(\mathbb R)\) satisfies the following conditions: \[ f(t)=0 \text{ for } t<0,\quad \underset{t\to 0} {\lim} \frac{f(t)}{t}=0,\,\underset{t\to\infty} {\lim}\frac{f(t)}{t^{\frac{N+2}{N-2}}}=\kappa>0. \] There exist \(C>0\) and \(p<\frac{2N}{N-2}\) such that \[ f(t) \geq \kappa t^{\frac{N+2}{N-2}}+Ct^{p-1} ,\text{for all }t\geq 0. \] The authors assume \(p>2\) for \(N\geq 4\) and \(p>4\) for \(N=3\), and prove that, for \(\epsilon\) small enough, \(P_{\epsilon}\) admits a solution \(v_{\epsilon}\), with a maximum point \(x_{\epsilon} \in\Omega\) such that \(\underset{\epsilon\to 0}{\lim}~ dist(x_{\epsilon},\partial\Omega) =\underset{x\in\Omega} \max ~ dist(x,\partial \Omega)\). Moreover, \(w_{\epsilon}(x)=v_{\epsilon} (\epsilon x+x_{\epsilon})\) converges, (up to a subsequence), uniformly to a least energy solution to problem (P): \[ -\Delta u+u=f(u),\quad u>0, \quad u\in H^1(\mathbb R^N). \] The main ingredients in the proof are a study of problem (P) by \textit{J. Zhang} and \textit{W. Zou} [Commun. Contemp. Math. 14, No. 5, 1250033, 14 p. (2012; Zbl 1254.35068)] and a subtle adaptation of the ideas of \textit{J. Byeon} [Trans. Am. Math. Soc. 362, No. 4, 1981--2001 (2010; Zbl 1188.35082)].
scientific article; zbMATH DE number 6161263
scientific article