Existence Of Multiple Positive Solutions For P-Q-Laplacian Problems With Singular Term And Critical Nonlinear

In this article,we consider the following p-q-Laplacian system with singular and critical nonlinearity,where ? is a bounded domain in Rn with smooth boundary (?)?.11,??(0,?*)is parameter and h1(x),h2(x)?L?,h1(x),h2(x)>0.Laplace equation is widely used in chemistry,physics,biology,probability statistics and other disciplines.Due to its extremely important theoretical significance and great scientific research value,more and more scholars began to pay attention to and study the relevant problems about the solution of p-q Laplace equations in recent years.In this paper,the existence and multiplicity of positive weak solutions of p-q Laplace equations with singular terms and critical nonlinear terms are proved The main method used is variation.First of all,we only need weak solutions.Under the condition of tightly supported sets,the singularity has been greatly weakened,and the integrable singular terms can be obtained by comparison principle and control convergence theorem.In order to overcome the difficulty of lack of compactness caused by the critical term,we adopt the property of cutoff function and establish the exact estimate of the upper bound of parameter ?.Finally,the estimation of the reach function is applied to the p-q-Laplacian equations with critical nonlinear terms,and the solution can be obtained.By limiting the range of p and q and locating the solutions by using reductive proof and the special structure of manifold,it can be proved that there are at least two different solutions of the equations.