Existence Of Solutions For The Fractional(p,q)-Laplacian Equations

This thesis is concerned with the existence of solutions for two kinds of fractional(p,q)-Laplacian equations by the variational method,which can be divided into two parts.In the first part,we investigate the following fractional(p,q)-Laplacian equation:where ?(?)R^N is a smooth and bounded domain,0<s₂<s₁<1,2?q<p<N/(s₁).The nonlocal operator is defined as follows:(?)B_?(x)={y?R^N:|x-y|<?}.The function f:?×R?R is a Carathéodory function and satisfies the subcritical growth condition.By computing the critical groups at zero and at infinity,we obtain the existence of nontrivial solutions via Morse theory.In the second part,we consider the following fractional(p,q)-Laplacian equation involving the critical Sobolev exponent:(?)where ?,?>0,0<s₂<s₁<1<q<p<N/(s₁).p_s1^*=Np/(N-s₁p) is the critical Sobolev exponent.When the parameters ? and ? satisfy certain conditions,we establish the existence of nontrivial solutions by the Mountain Pass Theorem.To verify the compactness condition below a suitable threshold level,we estimate the critical value by certain asymptotic estimates for minimizers.