Jinmyung Seok

Abstract Consider nonlinear Choquard equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="f

In this paper we study the existence, regularity, radial symmetry and decay property of a mountain pass solution for nonlinear scalar field equations involving the fractional Laplacian under an almost optimal class of continuous nonlinearities.

In this article, we are interested in singularly perturbed nonlinear elliptic problems involving a fractional Laplacian. Under a class of nonlinearity which is believed to be almost optimal, we construct a positive solution which exhibits multiple spikes near any given local minimum components of an exterior potential of the problem.

We prove the existence of infinitely many solutions of the nonlinear Chern-Simons-Schrödinger equations under a wide class of nonlinearities. This class includes the standard power-type nonlinearity with exponent<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>4</mml:mn></mml:math>. This extends the previous result which covers the exponent<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>p</mml:mi><mml:mo>&gt;