Solvability Of An Elliptic Problem With Nonlocal Operators

In recent years,the study of elliptic nonlocal operators has been widely con-cerned,especially the fractional Laplacian operators.In fact,nonlocal operators appear in many fields,such as conservation laws,super relative limits of quantum mechanics,finance,materials science,flame propagation,optimization and so on.In this paper,we mainly use the variational method,the mountain pass theo-rem,the fountain theorem and the fixed point index theory to obtain the nontrivial solutions,the existence of positive solutions and infinite solutions of elliptic equa-tions with nonlocal operators.The paper is divided into four chapters.In the first chapter,we mainly introduce the research background and current situation of nonlocal operators,and give some common symbols and preparatory lemmas.In the second chapter,we study the following elliptic problems with nonlocal operators Where ? RN(N>ps)is an open bounded set with Lipschitz boundary,s?(0,1),p>1,the nonlocal operator LK is defined by f(x,u)is asymptotically linear with respect to up 1 at infinity.By using the varia-tional methods and mountain pass lemma,we obtain that there exists at least one nontrivial solution for the above elliptic problem with nonlocal operators.In the third chapter,we consider the following elliptic problems with fractional p-Laplacian operators where ? RN(N>ps)is an open bounded set with Lipschitz boundary,s?(0,1),p>1,the fractional p-Laplacian operator(-?)ps is defined by By using the fixed point index theory,we obtain that there exists positive solutions for the above elliptic problem with fractional p-Laplacian operators.In the fourth chapter,we mainly study the above elliptic problem with fraction-al p-Laplacian operators,and obtain the existence of infinite solutions by using mountain pass theorem and fountain theorem.