Asymptotic behaviour of small solutions of quasilinear elliptic equations with critical and supercritical growth (Q1863483)

The author considers a perturbation of the equation \[ - \Delta_p u = u^k \quad {\text{in }} B,\qquad u = 0 \quad{\text{in }}\partial B, \] (\(B\) is the unit ball in \(\mathbb{R}^N\), \(N>2\), and \(\Delta_p\) the \(p\)-Laplacian) which has no positive solutions. The singularly perturbed equation is \[ - \Delta_p u = u^k - \varepsilon u^q \quad {\text{in }} B, \qquad u = 0\quad {\text{in }} \partial B, \tag \(*\) \] which has at least two positive radial solutions, one a small solution, the other a large solution. A small solution to \((*)\) is a function \(u_{\epsilon}\) such that \[ \limsup_{\varepsilon \to 0}{\varepsilon}^{1/(q-k)} \| u_{\varepsilon} \| _{\infty} = c, \quad c \in [0,1). \] The author studies the behaviour of the small solutions \(u_{\varepsilon}\) to \((*)\) as \(\varepsilon \to 0\) giving first an upper bound on \(u_{\varepsilon}\) (which implies that \(u_{\varepsilon}(x) \to 0\) for every \(x \in B \setminus \{0\}\)) and then giving the behaviour of \(\| u_{\varepsilon} \| _{\infty}\).