Existence And Multiplicity Of Near-resonance Solutions For Kirchhoff-type Equations

This paper studies the existence and multiplicity of solutions for two Kirchhoff type equation near resonance via critical point theory.Firstly, we study Kirchhoff equation with Hardy singular term where ? (?) R3 is an open bounded domain of smooth enough boundary,0 ? ?, b, ?> 0 is a parameter,4< p<6,0< ??< 1/4. The Kirchhoff equations are studied by local minimum theorem and the Mountain-Pass lemma. Namely, there exists a small enough ?> 0, the above Kirchhoff equation has at least two positive solutions for ? ? (?1,?1+?). And, we considered fourth-order Kirchhoff equation as follows where ?2= A(A) is a biharmonic operator, ?(?)RN RN(N> 4) is an open bounded domain of smooth enough boundary,?1 is the principal eigenvalue of operator ?2-a?, a, b> 0 are all real parameter, ?>0 is small enough parameter. The nonlinear term f satisfying the following three conditions(A1)There f:?×R ?R is a continuous function and exist a ?> 1/2, such that ?uf(x,u)-F(x,u)?-? as |u|? ?, uniformly for x??(A2)for all x? ?, there exists L>0,|u|> L, such that uf(x,u)>0;(A3)h(x)? L2(?) andThe above Kirchhoff equation with condition(A1), (A2) and (A3) has at least three solutions via a variant of Mountain-Pass lemma.