The paper deals with the equation -Δu+a(x)u=|u|p-1u\documentclass[12pt]{minimal}
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\begin{document}$$-\Delta u+a(x) u =|u|^{p-1}u $$\end{document}, u∈H1(RN)\documentclass[12pt]{minimal}
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\begin{document}$$u \in H^1({\mathbb {R}}^N)$$\end{document}, with N≥2\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 2$$\end{document}, p>1,p<N+2N-2\documentclass[12pt]{minimal}
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\begin{document}$$p> 1,\ p< {N+2\over N-2}$$\end{document} if N≥3\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 3$$\end{document}, a∈LlocN/2(RN)\documentclass[12pt]{minimal}
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\begin{document}$$a\in L^{N/2}_{loc}({\mathbb {R}}^N)$$\end{document}, infa>0\documentclass[12pt]{minimal}
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\begin{document}$$\inf a> 0$$\end{document}, lim|x|→∞a(x)=a∞\documentclass[12pt]{minimal}
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\begin{document}$$\lim _{|x| \rightarrow \infty } a(x)= a_\infty $$\end{document}. Assuming that the potential a(x) satisfies lim|x|→∞[a(x)-a∞]eη|x|=∞∀η>0\documentclass[12pt]{minimal}
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\begin{document}$$\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta > 0$$\end{document}, limρ→∞supa(ρθ1)-a(ρθ2):θ1,θ2∈RN,|θ1|=|θ2|=1eη~ρ=0forsomeη~>0\documentclass[12pt]{minimal}
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\begin{document}$$ \lim _{\rho \rightarrow \infty } \sup \left\{ a(\rho \theta _1) - a(\rho \theta _2) \ :\ \theta _1, \theta _2 \in {\mathbb {R}}^N,\ |\theta _1|= |\theta _2|=1 \right\} e^{\tilde{\eta }\rho } = 0 \quad \text{ for } \text{ some } \ \tilde{\eta }> 0$$\end{document} and other technical conditions, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers because we do not require that a(x) has radial symmetry, or that N=2\documentclass[12pt]{minimal}
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\begin{document}$$N=2$$\end{document}, or that |a(x)-a∞|\documentclass[12pt]{minimal}
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\begin{document}$$|a(x)-a_\infty |$$\end{document} is uniformly small in RN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^N$$\end{document}, etc. ....
