Semilinear elliptic variational inequalities with dependence on the gradient via Mountain Pass techniques

Stability for semilinear elliptic variational inequalities depending on the gradient

“…Semilinear variational inequalities are widely studied in the literature (see, for instance, [1][2][3][4][5][6][7][8][9][10][11][12][13] and references therein) via different methods and techniques, like variational and topological methods, sub-solution and super-solution methods, fixed point theorem methods, penalization techniques, methods using a priori estimates on the solutions, and approximation and regularization argument methods.…”

“…This paper can be seen as the second part of the work Matzeu and Servadei (2010) [9], in the sense that here we give a stability result for the C 1,α -weak solutions of problem (P n ) found in Matzeu and Servadei (2010) [9] through variational techniques. To be precise, we show that the solutions of (P n ), found with the arguments of Matzeu and Servadei (2010) [9], converge to a solution of the limiting problem (P ), under suitable convergence assumptions on the data.…”

On the existence of a solution to a class of variational inequalities

“…Also, our assumptions on the nonlinear part of (1.1) and on the obstacle function are different than those in papers [5,8]. In particular, a nonlinearity in [8] is sub-quadratic and in paper [5] it satisfies the Ambrosetti-Rabinowitz condition. Moreover, the obstacle function in [5] belongs to the space L r (Ω) for some r > 2 * .…”

“…In this situation we establish the existence of at least two solutions for a perturbed problem (1.1). In our approach we use some ideas from papers [3,5,8], where nonlinear variational inequalities of elliptic type have been investigated on the Sobolev space H 1 • (Ω). Also, our assumptions on the nonlinear part of (1.1) and on the obstacle function are different than those in papers [5,8].…”

“…In our approach we use some ideas from papers [3,5,8], where nonlinear variational inequalities of elliptic type have been investigated on the Sobolev space H 1 • (Ω). Also, our assumptions on the nonlinear part of (1.1) and on the obstacle function are different than those in papers [5,8]. In particular, a nonlinearity in [8] is sub-quadratic and in paper [5] it satisfies the Ambrosetti-Rabinowitz condition.…”