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Keywords:
$\Delta _\gamma $-Laplace problem; Cerami condition; variational method; weak solution; Mountain Pass Theorem
Summary:
In this article, we study the existence of nontrivial weak solutions for the following boundary value problem: $$ -\Delta _\gamma u=f(x,u) \ \text {in} \ \Omega , \quad u=0 \ \text {on} \ \partial \Omega , $$ where $\Omega $ is a bounded domain with smooth boundary in $\mathbb {R}^N$, $\Omega \cap \{x_j=0\}\ne \emptyset $ for some $j$, $\Delta _{\gamma }$ is a subelliptic linear operator of the type $$ \Delta _\gamma : =\sum _{j=1}^{N}\partial _{x_j} (\gamma _j^2 \partial _{x_j} ), \quad \partial _{x_j}:=\frac {\partial }{\partial x_{j}}, \quad N\ge 2, $$ where $\gamma (x) = (\gamma _1(x), \gamma _2(x),\dots ,\gamma _N(x))$ satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity $f(x,\xi )$ is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.
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