Existence of solutions for a class of elliptic equations in \(\mathbb R^N\) with vanishing potentials (Q414824)

In this paper, the authors study the following semilinear elliptic problem: NEWLINE\[NEWLINE-\Delta u + V(x)u =f(s) \text{\;in \;} \mathbb{R}^N, \tag{P}NEWLINE\]NEWLINE where \(V(x) \geq 0\) for all \(x\in \mathbb{R}^N\), \(V(x)\) is bounded above by a positive constant on \(B_1(0)\) and there are \(\Lambda >0\), \(R>1\) such that NEWLINE\[NEWLINE\frac{1}{R^4} \lim\inf_{|x|\geq R} |x|^4 V(x) \geq \Lambda.NEWLINE\]NEWLINE Under certain conditons on \(f(s)\), such as, \(f(s)\) is superlinear and with subcritial growth on \(s\), and by applying a version of the mountain pass theorem, the authors prove that there exists \(\Lambda^*>0\) such that problem (P) has a positive solution for all \(\Lambda \geq \Lambda^*\).