On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents
This paper is concerned with the existence of at least two positive solutions for the following problem: \[ - \Delta u= c_1 u^{p- 1}+ f(x, u)+ h,\quad u> 0\quad \text{in } \Omega; \quad u= 0\quad \text{in } \partial\Omega, \] where \(c_1> 0\), \(\Omega\subset \mathbb{R}^N\) \((N\geq 3)\) is a bounded smooth domain, \(h\in H^{- 1}(\Omega)\), \(p= 2N/(N- 2)\) is the critical exponent of the embeddings from \(H^1_0(\Omega)\) to \(L^q(\Omega)\) and \(f\) satisfies some conditions. By using the super- and subsolution method, the existence of the minimum positive solution is proved. To obtain the second positive solution, the authors first apply the Mountain Pass Lemma ``around'' the minimum positive solution and get a sequence of approximated solutions, and then find a subsequence converging weakly to a limit which is another positive solution of this problem.
scientific article; zbMATH DE number 859675