Existence of Sign-Changing Solutions for a Class of p(x)-Biharmonic Kirchhoff-Type Equations - Rui Deng; Qing Miao (Axioms)
This paper mainly studies the existence of sign-changing solutions for the following <semantics>p<mfenced open="(" close=")">x</mfenced></semantics>-biharmonic Kirchhoff-type equations: <semantics>−<mfenced separators="" open="(" close=")">a+b∫RN<mstyle scriptlevel="0" displaystyle="true">1p(x)</mstyle>|Δu|p(x)dx</mfenced><msubsup>Δp(x)2</msubsup>u+V(x)|u|p(x)−2u</semantics> = <semantics>K<mfenced open="(" close=")">x</mfenced>f(u),<mspace width="1.em"></mspace>x∈RN</semantics>, where <semantics><msubsup>Δp(x)2</msubsup>u=Δ<mfenced separators="" open="(" close=")">|Δu|p(x)−2Δu</mfenced></semantics> is the <semantics>p(x)</semantics> biharmonic operator, <semantics>a,b>0</semantics> are constants, <semantics>N≥2</semantics>, <semantics>V(x),K(x)</semantics> are positive continuous functions which vanish at infinity, and the nonlinearity f has subcritical growth. Using the Nehari manifold method, deformation lemma, and other techniques of analysis, it is demonstrated that there are precisely two nodal domains in the problem’s least energy sign-changing solution <semantics>ub</semantics>. In addition, the convergence property of <semantics>ub</semantics> as <semantics>b→0</semantics> is also established.