MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION∗

doi:10.1007/s10473-023-0421-z

Abstract

Abstract: The existence and multiplicity of positive solutions for equation (1.1) with the new critical exponent $4<p<2\cdot 2^*$ shall be investigated in a high dimension. The conclusions extend the relative results recently attained in \cite{1} for the one-dimensional case. More precisely, as the coefficient $a(x)$ in the nonlinearity is sign-changing, the modified term $2(\Delta (|u|^2))u$ is still helpful for obtaining multiple positive solutions in a high dimension, even if a sign condition like $\int_{\mathbb{R}^N}a(x)e_1^p{\rm d}x<0$ (also named “a necessary condition” see [2,3]) does not hold.

Key words: variational methods, nonlinear Schrö, dinger equation, sign-changing

Yansheng ZHONG, Yongqing LI. MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION^∗[J].Acta mathematica scientia,Series B, 2023, 43(4): 1819-1840.

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MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION^∗

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