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The aim of this paper is the study a class of double phase problems with variable exponent. Using the Clark's theorem and the symmetric mountain pass lemma, we prove the existence of infinitely many small solutions and infinitely many large solutions, respectively. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the existence of nontrivial solutions for double-phase problem without any growth and Ambrosetti–Rabinowitz conditions. Employing the cutoff function approach, we show that the auxiliary problem has at least one nontrivial solution. Furthermore, we obtain nontrivial solutions for original problems using the Moser iteration method. The results presented here extend some recent contributions obtained for problems driven by the p-Laplacian or even to more general differential operators. [ABSTRACT FROM AUTHOR]

The main goal of this work is to investigate the initial boundary value problem for a class of pseudo-parabolic p-Laplacian equations with singular potential and logarithmic nonlinearity. First of all, we prove the local existence of weak solutions. Second, we show the existence of the global solution and the weak solution converging to the stationary solution when the time tends to infinity, and we show blow-up phenomena of solutions with the initial energy less than the mountain pass level d by using the potential well method. Finally, we parallelly stretch all the conclusions for the subcritical case to the critical case. [ABSTRACT FROM AUTHOR]