Multiple solutions Bounded domain 2 Clarkson inequality 2 Critical Sobolev exponent 2 Critical growth 2 Dirichlet boundary condition 2 Laplace equation 2 Laplace transforms 2 Linear equations 2 Mathematical operators 2 Mountain pass Theorem 2 Nonlinear equations 2 Nonlinear systems 2 Nontrivial solution 2 P-Laplace equations 2 P-Laplace operator 2 P-Laplacian 2 Problem solving 2 Quasilinear elliptic equations 2 Set theory 2 Theorem proving 2 Uniform convexity 2 Variational methods 2 p-Laplace equations 2 p-Laplacian 2