Solutions, concentrating on spheres, to symmetric singularly perturbed problems (Q699224)

The following singularly perturbed problem is considered \[ -\varepsilon^2\nabla u+V(| x| ) u = u^p,\quad u\in H^1_r,\quad u>0, \] where \(| x| \) denotes the Euclidean norm of \(x\in \mathbb R^n\) and \(H^1_r(\mathbb R^n)\) denotes the subspace of the radial functions in \(H^1(\mathbb R^n)\). If \(1<p<(n+2)/(n-2)\) \((n\geq 3\)) the embedding of \(H^1_r\) into \(L^p\) is compact. Therefore in case of \(V\) differentiable and bounded from below by a positive constant a straight application of the Mountain-Pass theorem yields the existence of a radial solutions concentrating on \(x=0\). The paper establishes existence of radial solutions concentrating on a sphere, without any limitation on the exponent \(p\). It is assumed that \(V\in C^1(\mathbb R^+, \mathbb R)\), \(\lambda_0^2:=\inf\{V(| x| )\mid x\in R^n\}>0\) and \(p>1\). Under these assumptions it is shown that the equation admits a radial solution concentrated near the sphere \(\{| x| =\bar r\}\), where \(\bar r\) is the strict local maximum or the minimum of the auxiliary potential \(M(r)=r^{n-1} V^\theta(r)\), \(\theta=(p+1)/(p-1)-1/2\). Moreover, if in addition to the above assumptions there exists \(r^*>0\) such that \[ (n-1)V(r^*)+\theta r^* V^\prime(r^*)<0\,, \] it is shown that there exists a pair if solutions concentrating on spheres.