POSITIVE SOLUTIONS TO ELLIPTIC PROBLEMS HAVING VARIABLE GROWTH STRUCTURE WITH NONLINEAR BOUNDARY CONDITIONS

We investigate the existence of positive solutions to a class of nonlinear boundary value problems. Our model is governed by an elliptic problem having nonlinear terms in both equation and boundary conditions, exhibiting p(x)-growth structure. Under suitable assumptions on the nonlinearities, we establish the existence of a positive weak solution in an anisotropic Sobolev space with a variable exponent. Our approach revolves around developing a novel nonvariational method, incorporating the so-called \(\beta\) $\beta$ -norm.