Critical \(p\)-Laplacian problems in \(\mathbf R^ N\)
The authors consider the quasilinear elliptic problem \[ - \Delta_p u= \lambda a(x) u^{p- 1}+ f(x) u^{p^*- 1}+ g(x) u^q,\quad x\in \mathbb{R}^N, \] where \(1< p< N\), \(p^*= {pN\over N- p}\) (critical Sobolev exponent), \(p- 1< q< p^*- 1\), and \(\Delta_p u= \text{div}(|\nabla u|^{p- 2} \nabla u)\) is the \(p\)-Laplacian. Assume that \(a\in L^{N/p}(\mathbb{R}^N)\), \(f\in C^0_{\text{loc}}(\mathbb{R}^N)\), and \(g\in L^Q(\mathbb{R}^N)\) \((Q= {pN\over pN- (q+ 1)(N- p)})\), are nontrivial nonnegative bounded functions satisfying \(f(0)= \sup_{x\in \mathbb{R}^N} f(x)\) and \(f(x)= f(0)+ o(|x|^\delta)\), \(g(x)\geq g_0> 0\) in a neighborhood of \(x= 0\), where \(\delta= N/Q\) if \(N\geq {p(q+ 1)\over q- p+ 2}\), \(\delta= {N(Q- 1)\over Q(p- 1)}\) if \(N< {p(q+ 1)\over q- p+ 2}\). Then the authors prove the existence of a constant \(\lambda_0> 0\) such that for all \(0\leq \lambda< \lambda_0\) the quasilinear equation has a positive solution \(u_\lambda\in W^{1, p}_0(\mathbb{R}^N)\cap C^{1, \alpha}_{\text{loc}}(\mathbb{R}^N)\) with \(\lim_{|x|\to \infty} u_\lambda(x)= 0\). \(\lambda_0\) is defined to be the lowest eigenvalue of an associated quasilinear eigenvalue problem. A nontrivial nonnegative (weak) solution of the above-mentioned equation is then obtained via the mountain pass lemma as a critical point of the functional \[ J(u)= \int_{\mathbb{R}^N} \Biggl({1\over p} |\nabla u|^p- \lambda {a\over p} (u_+)^p- {f\over p^*} (u_+)^{p^*}- {g\over q+ 1} (u_+)^{q+ 1}\Biggr) dx,\quad u\in W^{1, p}_0(\mathbb{R}^N), \] where \(u_+(x)= \max\{u(x), 0\}\). The crucial step here is to derive a Palais-Smale compactness condition for the functional \(J\), if \(0\leq \lambda< \lambda_0\). To prove this, a concentration-compactness method of \textit{P. L. Lions} [Rev. Mat. Iberoam. 1, No. 1, 145-201 (1985; Zbl 0704.49005), No. 2, 45-121 (1985; Zbl 0704.49006)] is used.
scientific article; zbMATH DE number 878431
scientific article