Laplace transforms 6 Nonlinear equations 6 P-Laplace operator Optimization 4 Optimization problems 4 Shape derivative 4 p-Laplacian 4 Bounded domain 2 Critical Sobolev exponent 2 Critical growth 2 Dirichlet boundary condition 2 Laplace equation 2 Mathematical operators 2 Multiple solutions 2 Nontrivial solution 2 P-Laplace equations 2 P-Laplacian 2 Quasilinear elliptic equations 2 Variational methods 2 p-Laplace equations 2