On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents (Q1909920)

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.